Bounds for the second Hankel determinant of certain univalent functions

  • See Keong Lee1Email author,

    Affiliated with

    • V Ravichandran2 and

      Affiliated with

      • Shamani Supramaniam1

        Affiliated with

        Journal of Inequalities and Applications20132013:281

        DOI: 10.1186/1029-242X-2013-281

        Received: 11 December 2012

        Accepted: 11 March 2013

        Published: 5 June 2013

        Abstract

        The estimates for the second Hankel determinant a 2 a 4 a 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq1_HTML.gif of the analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq2_HTML.gif , for which either z f ( z ) / f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq3_HTML.gif or 1 + z f ( z ) / f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq4_HTML.gif is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.

        MSC:30C45, 30C80.

        Dedication

        Dedicated to Professor Hari M Srivastava

        1 Introduction

        Let A http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq5_HTML.gif denote the class of all analytic functions
        f ( z ) = z + a 2 z 2 + a 3 z 3 + http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ1_HTML.gif
        (1)
        defined on the open unit disk D : = { z C : | z | < 1 } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq6_HTML.gif. The Hankel determinants H q ( n ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq7_HTML.gif ( n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq8_HTML.gif , q = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq9_HTML.gif) of the function f are defined by
        H q ( n ) : = [ a n a n + 1 a n + q 1 a n + 1 a n + 2 a n + q a n + q 1 a n + q a n + 2 q 2 ] ( a 1 = 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equa_HTML.gif
        Hankel determinants are useful, for example, in showing that a function of bounded characteristic in D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif, i.e., a function which is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational [1]. For the use of Hankel determinants in the study of meromorphic functions, see [2], and various properties of these determinants can be found in [[3], Chapter 4]. In 1966, Pommerenke [4] investigated the Hankel determinant of areally mean p-valent functions, univalent functions as well as of starlike functions. In [5], he proved that the Hankel determinants of univalent functions satisfy
        | H q ( n ) | < K n ( 1 2 + β ) q + 3 2 ( n = 1 , 2 , , q = 2 , 3 , ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equb_HTML.gif

        where β > 1 / 4000 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq11_HTML.gif and K depends only on q. Later, Hayman [6] proved that | H 2 ( n ) | < A n 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq12_HTML.gif ( n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq8_HTML.gif ; A an absolute constant) for areally mean univalent functions. In [79], the estimates for the Hankel determinant of areally mean p-valent functions were investigated. ElHosh obtained bounds for Hankel determinants of univalent functions with a positive Hayman index α [10] and of k-fold symmetric and close-to-convex functions [11]. For bounds on the Hankel determinants of close-to-convex functions, see [1214]. Noor studied the Hankel determinant of Bazilevic functions in [15] and of functions with bounded boundary rotation in [1619]. In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [2027]. The Hankel determinant H 2 ( 1 ) = a 3 a 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq13_HTML.gif is the well-known Fekete-Szegö functional. For results related to this functional, see [28, 29]. The second Hankel determinant H 2 ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq14_HTML.gif is given by H 2 ( 2 ) = a 2 a 4 a 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq15_HTML.gif.

        An analytic function f is subordinate to an analytic function g, written f ( z ) g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq16_HTML.gif, if there is an analytic function w : D D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq17_HTML.gif with w ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq18_HTML.gif satisfying f ( z ) = g ( w ( z ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq19_HTML.gif. Ma and Minda [30] unified various subclasses of starlike ( S http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq20_HTML.gif) and convex functions ( C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq21_HTML.gif) by requiring that either of the quantity z f ( z ) / f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq22_HTML.gif or 1 + z f ( z ) / f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq23_HTML.gif is subordinate to a function φ with a positive real part in the unit disk D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif, φ ( 0 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq24_HTML.gif, φ ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq25_HTML.gif, φ maps D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif onto a region starlike with respect to 1 and symmetric with respect to the real axis. He obtained distortion, growth and covering estimates as well as bounds for the initial coefficients of the unified classes.

        The bounds for the second Hankel determinant H 2 ( 2 ) = a 2 a 4 a 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq26_HTML.gif are obtained for functions belonging to these subclasses of Ma-Minda starlike and convex functions in Section 2. In Section 3, the problem is investigated for two other related classes defined by subordination. In proving our results, we do not assume the univalence or starlikeness of φ as they were required only in obtaining the distortion, growth estimates and the convolution theorems. The classes introduced by subordination naturally include several well-known classes of univalent functions and the results for some of these special classes are indicated as corollaries.

        Let P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq27_HTML.gif be the class of functions with positive real part consisting of all analytic functions p : D C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq28_HTML.gif satisfying p ( 0 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq29_HTML.gif and Re p ( z ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq30_HTML.gif. We need the following results about the functions belonging to the class P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq27_HTML.gif.

        Lemma 1 [31]

        If the function p P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq31_HTML.gif is given by the series
        p ( z ) = 1 + c 1 z + c 2 z 2 + c 3 z 3 + , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ2_HTML.gif
        (2)
        then the following sharp estimate holds:
        | c n | 2 ( n = 1 , 2 , ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ3_HTML.gif
        (3)

        Lemma 2 [32]

        If the function p P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq31_HTML.gif is given by the series (2), then
        2 c 2 = c 1 2 + x ( 4 c 1 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ4_HTML.gif
        (4)
        4 c 3 = c 1 3 + 2 ( 4 c 1 2 ) c 1 x c 1 ( 4 c 1 2 ) x 2 + 2 ( 4 c 1 2 ) ( 1 | x | 2 ) z , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ5_HTML.gif
        (5)

        for some x, z with | x | 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq32_HTML.gif and | z | 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq33_HTML.gif.

        2 Second Hankel determinant of Ma-Minda starlike/convex functions

        Subclasses of starlike functions are characterized by the quantity z f ( z ) / f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq3_HTML.gif lying in some domain in the right half-plane. For example, f is strongly starlike of order β if z f ( z ) / f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq3_HTML.gif lies in a sector | arg w | < β π / 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq34_HTML.gif, while it is starlike of order α if z f ( z ) / f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq3_HTML.gif lies in the half-plane Re w > α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq35_HTML.gif. The various subclasses of starlike functions were unified by subordination in [30]. The following definition of the class of Ma-Minda starlike functions is the same as the one in [30] except for the omission of starlikeness assumption of φ.

        Definition 1 Let φ : D C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq36_HTML.gif be analytic, and let the Maclaurin series of φ be given by
        φ ( z ) = 1 + B 1 z + B 2 z 2 + B 3 z 3 + ( B 1 , B 2 R , B 1 > 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ6_HTML.gif
        (6)
        The class S ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq37_HTML.gif of Ma-Minda starlike functions with respect to φ consists of functions f A http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq38_HTML.gif satisfying the subordination
        z f ( z ) f ( z ) φ ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equc_HTML.gif
        For the function φ given by φ α ( z ) : = ( 1 + ( 1 2 α ) z ) / ( 1 z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq39_HTML.gif , 0 < α 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq40_HTML.gif, the class S ( α ) : = S ( φ α ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq41_HTML.gif is the well-known class of starlike functions of order α. Let
        φ PAR ( z ) : = 1 + 2 π 2 ( log 1 + z 1 z ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equd_HTML.gif
        Then the class
        S P : = S ( φ PAR ) = { f A : Re ( z f ( z ) f ( z ) ) > | z f ( z ) f ( z ) 1 | } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Eque_HTML.gif
        is the parabolic starlike functions introduced by Rønning [33]. For a survey of parabolic starlike functions and the related class of uniformly convex functions, see [34]. For 0 < β 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq42_HTML.gif, the class
        S β : = S ( ( 1 + z 1 z ) β ) = { f A : | arg ( z f ( z ) f ( z ) ) | < β π 2 } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equf_HTML.gif
        is the familiar class of strongly starlike functions of order β. The class
        S L : = S ( 1 + z ) = { f A : | ( z f ( z ) f ( z ) ) 2 1 | < 1 } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equg_HTML.gif

        is the class of lemniscate starlike functions studied in [35].

        Theorem 1 Let the function f S ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq43_HTML.gif be given by (1).
        1. 1.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          | B 2 | B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | 3 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equh_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | B 1 2 4 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equi_HTML.gif
        1. 2.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          | B 2 | B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | B 1 | B 2 | 2 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equj_HTML.gif
           
        or the conditions
        | B 2 | B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | 3 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equk_HTML.gif
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | 1 12 | 4 B 1 B 3 B 1 4 3 B 2 2 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equl_HTML.gif
        1. 3.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          | B 2 | > B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | B 1 | B 2 | 2 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equm_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | B 1 2 12 ( 3 | 4 B 1 B 3 B 1 4 3 B 2 2 | 4 B 1 | B 2 | + 4 B 1 2 B 2 2 | 4 B 1 B 3 B 1 4 3 B 2 2 | 2 B 1 | B 2 | B 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equn_HTML.gif
        Proof Since f S ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq47_HTML.gif, there exists an analytic function w with w ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq48_HTML.gif and | w ( z ) | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq49_HTML.gif in  D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif such that
        z f ( z ) f ( z ) = φ ( w ( z ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ7_HTML.gif
        (7)
        Define the functions p 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq50_HTML.gif by
        p 1 ( z ) : = 1 + w ( z ) 1 w ( z ) = 1 + c 1 z + c 2 z 2 + , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equo_HTML.gif
        or, equivalently,
        w ( z ) = p 1 ( z ) 1 p 1 ( z ) + 1 = 1 2 ( c 1 z + ( c 2 c 1 2 2 ) z 2 + ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ8_HTML.gif
        (8)
        Then p 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq50_HTML.gif is analytic in D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif with p 1 ( 0 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq51_HTML.gif and has a positive real part in D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif. By using (8) together with (6), it is evident that
        φ ( p 1 ( z ) 1 p 1 ( z ) + 1 ) = 1 + 1 2 B 1 c 1 z + ( 1 2 B 1 ( c 2 c 1 2 2 ) + 1 4 B 2 c 1 2 ) z 2 + . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ9_HTML.gif
        (9)
        Since
        z f ( z ) f ( z ) = 1 + a 2 z + ( a 2 2 + 2 a 3 ) z 2 + ( 3 a 4 3 a 2 a 3 + a 2 3 ) z 3 + , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ10_HTML.gif
        (10)
        it follows by (7), (9) and (10) that
        a 2 = B 1 c 1 2 , a 3 = 1 8 [ ( B 1 2 B 1 + B 2 ) c 1 2 + 2 B 1 c 2 ] , a 4 = 1 48 [ ( 4 B 2 + 2 B 1 + B 1 3 3 B 1 2 + 3 B 1 B 2 + 2 B 3 ) c 1 3 a 4 = + 2 ( 3 B 1 2 4 B 1 + 4 B 2 ) c 1 c 2 + 8 B 1 c 3 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equp_HTML.gif
        Therefore
        a 2 a 4 a 3 2 = B 1 96 [ c 1 4 ( B 1 3 2 + B 1 2 B 2 + 2 B 3 3 B 2 2 2 B 1 ) + 2 c 2 c 1 2 ( B 2 B 1 ) + 8 B 1 c 1 c 3 6 B 1 c 2 2 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equq_HTML.gif
        Let
        d 1 = 8 B 1 , d 2 = 2 ( B 2 B 1 ) , d 3 = 6 B 1 , d 4 = B 1 3 2 + B 1 2 B 2 + 2 B 3 3 B 2 2 2 B 1 , T = B 1 96 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ11_HTML.gif
        (11)
        Then
        | a 2 a 4 a 3 2 | = T | d 1 c 1 c 3 + d 2 c 1 2 c 2 + d 3 c 2 2 + d 4 c 1 4 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ12_HTML.gif
        (12)
        Since the function p ( e i θ z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq52_HTML.gif ( θ R http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq53_HTML.gif) is in the class P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq27_HTML.gif for any p P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq31_HTML.gif, there is no loss of generality in assuming c 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq54_HTML.gif. Write c 1 = c http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq55_HTML.gif, c [ 0 , 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq56_HTML.gif. Substituting the values of c 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq57_HTML.gif and c 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq58_HTML.gif respectively from (4) and (5) in (12), we obtain
        | a 2 a 4 a 3 2 | = T 4 | c 4 ( d 1 + 2 d 2 + d 3 + 4 d 4 ) + 2 x c 2 ( 4 c 2 ) ( d 1 + d 2 + d 3 ) + ( 4 c 2 ) x 2 ( d 1 c 2 + d 3 ( 4 c 2 ) ) + 2 d 1 c ( 4 c 2 ) ( 1 | x | 2 ) z | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equr_HTML.gif
        Replacing | x | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq59_HTML.gif by μ and substituting the values of d 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq60_HTML.gif, d 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq61_HTML.gif, d 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq62_HTML.gif and d 4 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq63_HTML.gif from (11) yield
        | a 2 a 4 a 3 2 | T 4 [ c 4 | 2 B 1 3 + 8 B 3 6 B 2 2 B 1 | + 4 | B 2 | μ c 2 ( 4 c 2 ) + μ 2 ( 4 c 2 ) ( 2 B 1 c 2 + 24 B 1 ) + 16 B 1 c ( 4 c 2 ) ( 1 μ 2 ) ] = T [ c 4 4 | 2 B 1 3 + 8 B 3 6 B 2 2 B 1 | + 4 B 1 c ( 4 c 2 ) + | B 2 | ( 4 c 2 ) μ c 2 + B 1 2 μ 2 ( 4 c 2 ) ( c 6 ) ( c 2 ) ] F ( c , μ ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ13_HTML.gif
        (13)
        Note that for ( c , μ ) [ 0 , 2 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq64_HTML.gif, differentiating F ( c , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq65_HTML.gif in (13) partially with respect to μ yields
        F μ = T [ | B 2 | ( 4 c 2 ) + B 1 μ ( 4 c 2 ) ( c 2 ) ( c 6 ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ14_HTML.gif
        (14)
        Then, for 0 < μ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq66_HTML.gif and for any fixed c with 0 < c < 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq67_HTML.gif, it is clear from (14) that F μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq68_HTML.gif, that is, F ( c , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq69_HTML.gif is an increasing function of μ. Hence, for fixed c [ 0 , 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq70_HTML.gif, the maximum of F ( c , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq69_HTML.gif occurs at μ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq71_HTML.gif, and
        max F ( c , μ ) = F ( c , 1 ) G ( c ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equs_HTML.gif
        Also note that
        G ( c ) = B 1 96 [ c 4 4 ( | 2 B 1 3 + 8 B 3 6 B 2 2 B 1 | 4 | B 2 | 2 B 1 ) + 4 c 2 ( | B 2 | B 1 ) + 24 B 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equt_HTML.gif
        Let
        P = 1 4 ( | 2 B 1 3 + 8 B 3 6 B 2 2 B 1 | 4 | B 2 | 2 B 1 ) , Q = 4 ( | B 2 | B 1 ) , R = 24 B 1 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ15_HTML.gif
        (15)
        Since
        max 0 t 4 ( P t 2 + Q t + R ) = { R , Q 0 , P Q 4 ; 16 P + 4 Q + R , Q 0 , P Q 8  or  Q 0 , P Q 4 ; 4 P R Q 2 4 P , Q > 0 , P Q 8 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ16_HTML.gif
        (16)
        we have
        | a 2 a 4 a 3 2 | B 1 96 { R , Q 0 , P Q 4 ; 16 P + 4 Q + R , Q 0 , P Q 8  or  Q 0 , P Q 4 ; 4 P R Q 2 4 P , Q > 0 , P Q 8 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equu_HTML.gif

        where P, Q, R are given by (15). □

        Remark 1 When B 1 = B 2 = B 3 = 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq72_HTML.gif, Theorem 1 reduces to [[24], Theorem 3.1].

        Corollary 1
        1. 1.

          If f S ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq73_HTML.gif, then | a 2 a 4 a 3 2 | ( 1 α ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq74_HTML.gif.

           
        2. 2.

          If f S L http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq75_HTML.gif, then | a 2 a 4 a 3 2 | 1 / 16 = 0.0625 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq76_HTML.gif.

           
        3. 3.

          If f S P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq77_HTML.gif, then | a 2 a 4 a 3 2 | 16 / π 4 0.164255 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq78_HTML.gif.

           
        4. 4.

          If f S β http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq79_HTML.gif, then | a 2 a 4 a 3 2 | β 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq80_HTML.gif.

           
        Definition 2 Let φ : D C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq36_HTML.gif be analytic, and let φ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq81_HTML.gif be given as in (6). The class C ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq82_HTML.gif of Ma-Minda convex functions with respect to φ consists of functions f satisfying the subordination
        1 + z f ( z ) f ( z ) φ ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equv_HTML.gif
        Theorem 2 Let the function f C ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq83_HTML.gif be given by (1).
        1. 1.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          B 1 2 + 4 | B 2 | 2 B 1 0 , | 6 B 1 B 3 + B 1 2 B 2 B 1 4 4 B 2 2 | 4 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equw_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | B 1 2 36 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equx_HTML.gif
        1. 2.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          B 1 2 + 4 | B 2 | 2 B 1 0 , 2 | 6 B 1 B 3 + B 1 2 B 2 B 1 4 4 B 2 2 | B 1 3 4 B 1 | B 2 | 6 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equy_HTML.gif
           
        or the conditions
        B 1 2 + 4 | B 2 | 2 B 1 0 , | 6 B 1 B 3 + B 1 2 B 2 B 1 4 4 B 2 2 | 4 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equz_HTML.gif
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | 1 144 | 6 B 1 B 3 + B 1 2 B 2 B 1 4 4 B 2 2 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equaa_HTML.gif
        1. 3.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          B 1 2 + 4 | B 2 | 2 B 1 > 0 , 2 | 6 B 1 B 3 + B 1 2 B 2 B 1 4 4 B 2 2 | B 1 3 4 B 1 | B 2 | 6 B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equab_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | B 1 2 576 ( 16 | 6 B 1 B 3 + B 1 2 B 2 B 1 4 4 B 2 2 | 12 B 1 3 48 B 1 | B 2 | 36 B 1 2 B 1 4 8 B 1 2 | B 2 | 16 B 2 2 | 6 B 1 B 3 + B 1 2 B 2 B 1 4 4 B 2 2 | B 1 3 4 B 1 | B 2 | 2 B 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equac_HTML.gif
        Proof Since f C ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq84_HTML.gif, there exists an analytic function w with w ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq48_HTML.gif and | w ( z ) | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq49_HTML.gif in D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif such that
        1 + z f ( z ) f ( z ) = φ ( w ( z ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ17_HTML.gif
        (17)
        Since
        1 + z f ( z ) f ( z ) = 1 + 2 a 2 z + ( 4 a 2 2 + 6 a 3 ) z 2 + ( 8 a 2 3 18 a 2 a 3 + 12 a 4 ) z 3 + , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ18_HTML.gif
        (18)
        equations (9), (17) and (18) yield
        a 2 = B 1 c 1 4 , a 3 = 1 24 [ ( B 1 2 B 1 + B 2 ) c 1 2 + 2 B 1 c 2 ] , a 4 = 1 192 [ ( 4 B 2 + 2 B 1 + B 1 3 3 B 1 2 + 3 B 1 B 2 + 2 B 3 ) c 1 3 a 4 = + 2 ( 3 B 1 2 4 B 1 + 4 B 2 ) c 1 c 2 + 8 B 1 c 3 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equad_HTML.gif
        Therefore
        a 2 a 4 a 3 2 = B 1 768 [ c 1 4 ( 4 3 B 2 + 2 3 B 1 1 3 B 1 3 1 3 B 1 2 + 1 3 B 1 B 2 + 2 B 3 4 3 B 2 2 B 1 ) + 2 3 c 2 c 1 2 ( B 1 2 4 B 1 + 4 B 2 ) + 8 B 1 c 1 c 3 16 3 B 1 c 2 2 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equae_HTML.gif
        By writing
        d 1 = 8 B 1 , d 2 = 2 3 ( B 1 2 4 B 1 + 4 B 2 ) , d 3 = 16 3 B 1 , d 4 = 4 3 B 2 + 2 3 B 1 1 3 B 1 3 1 3 B 1 2 + 1 3 B 1 B 2 + 2 B 3 4 3 B 2 2 B 1 , T = B 1 768 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ19_HTML.gif
        (19)
        we have
        | a 2 a 4 a 3 2 | = T | d 1 c 1 c 3 + d 2 c 1 2 c 2 + d 3 c 2 2 + d 4 c 1 4 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ20_HTML.gif
        (20)
        Similar as in Theorems 1, it follows from (4) and (5) that
        | a 2 a 4 a 3 2 | = T 4 | c 4 ( d 1 + 2 d 2 + d 3 + 4 d 4 ) + 2 x c 2 ( 4 c 2 ) ( d 1 + d 2 + d 3 ) + ( 4 c 2 ) x 2 ( d 1 c 2 + d 3 ( 4 c 2 ) ) + 2 d 1 c ( 4 c 2 ) ( 1 | x | 2 ) z | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equaf_HTML.gif
        Replacing | x | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq59_HTML.gif by μ and then substituting the values of d 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq60_HTML.gif, d 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq61_HTML.gif, d 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq62_HTML.gif and d 4 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq63_HTML.gif from (19) yield
        | a 2 a 4 a 3 2 | T 4 [ c 4 | 4 3 B 1 3 + 4 3 B 1 B 2 + 8 B 3 16 3 B 2 2 B 1 | + 2 μ c 2 ( 4 c 2 ) ( 2 3 B 1 2 + 8 3 | B 2 | ) + μ 2 ( 4 c 2 ) ( 8 3 B 1 c 2 + 64 3 B 1 ) + 16 B 1 c ( 4 c 2 ) ( 1 μ 2 ) ] = T [ c 4 3 | B 1 3 + B 1 B 2 + 6 B 3 4 B 2 2 B 1 | + 4 B 1 c ( 4 c 2 ) + 1 3 μ c 2 ( 4 c 2 ) ( B 1 2 + 4 | B 2 | ) + 2 B 1 3 μ 2 ( 4 c 2 ) ( c 4 ) ( c 2 ) ] F ( c , μ ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ21_HTML.gif
        (21)
        Again, differentiating F ( c , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq69_HTML.gif in (21) partially with respect to μ yields
        F μ = T [ c 2 3 ( 4 c 2 ) ( B 1 2 + 4 | B 2 | ) + 4 B 1 3 μ ( 4 c 2 ) ( c 4 ) ( c 2 ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ22_HTML.gif
        (22)
        It is clear from (22) that F μ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq85_HTML.gif. Thus F ( c , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq69_HTML.gif is an increasing function of μ for 0 < μ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq66_HTML.gif and for any fixed c with 0 < c < 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq67_HTML.gif. So, the maximum of F ( c , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq69_HTML.gif occurs at μ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq71_HTML.gif and
        max F ( c , μ ) = F ( c , 1 ) G ( c ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equag_HTML.gif
        Note that
        G ( c ) = T [ c 4 3 ( | B 1 3 + B 1 B 2 + 6 B 3 4 B 2 2 B 1 | B 1 2 4 | B 2 | 2 B 1 ) + 4 3 c 2 ( B 1 2 + 4 | B 2 | 2 B 1 ) + 64 3 B 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equah_HTML.gif
        Let
        P = 1 3 ( | B 1 3 + B 1 B 2 + 6 B 3 4 B 2 2 B 1 | B 1 2 4 | B 2 | 2 B 1 ) , Q = 4 3 ( B 1 2 + 4 | B 2 | 2 B 1 ) , R = 64 3 B 1 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ23_HTML.gif
        (23)
        By using (16), we have
        | a 2 a 4 a 3 2 | B 1 768 { R , Q 0 , P Q 4 ; 16 P + 4 Q + R , Q 0 , P Q 8  or  Q 0 , P Q 4 ; 4 P R Q 2 4 P , Q > 0 , P Q 8 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equai_HTML.gif

        where P, Q, R are given in (23). □

        Remark 2 For the choice of φ ( z ) = ( 1 + z ) / ( 1 z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq86_HTML.gif, Theorem 2 reduces to [[24], Theorem 3.2].

        3 Further results on the second Hankel determinant

        Definition 3 Let φ : D C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq36_HTML.gif be analytic, and let φ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq81_HTML.gif be as given in (6). Let 0 γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq87_HTML.gif and τ C { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq88_HTML.gif. A function f A http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq89_HTML.gif is in the class R γ τ ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq90_HTML.gif if it satisfies the following subordination:
        1 + 1 τ ( f ( z ) + γ z f ( z ) 1 ) φ ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equaj_HTML.gif
        Theorem 3 Let 0 γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq87_HTML.gif, τ C { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq88_HTML.gif, and let the function f as in (1) be in the class R γ τ ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq91_HTML.gif. Also, let
        p = 8 9 ( 1 + γ ) ( 1 + 3 γ ) ( 1 + 2 γ ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equak_HTML.gif
        1. 1.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          2 | B 2 | ( 1 p ) + B 1 ( 1 2 p ) 0 , | B 1 B 3 p B 2 2 | p B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equal_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | | τ | 2 B 1 2 9 ( 1 + 2 γ ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equam_HTML.gif
        1. 2.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          2 | B 2 | ( 1 p ) + B 1 ( 1 2 p ) 0 , 2 | B 1 B 3 p B 2 2 | 2 ( 1 p ) B 1 | B 2 | B 1 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equan_HTML.gif
           
        or the conditions
        2 | B 2 | ( 1 p ) + B 1 ( 1 2 p ) 0 , | B 1 B 3 p B 2 2 | B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equao_HTML.gif
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | | τ | 2 8 ( 1 + γ ) ( 1 + 3 γ ) | B 3 B 1 p B 2 2 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equap_HTML.gif
        1. 3.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          2 | B 2 | ( 1 p ) + B 1 ( 1 2 p ) > 0 , 2 | B 1 B 3 p B 2 2 | 2 ( 1 p ) B 1 | B 2 | B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equaq_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | | τ | 2 B 1 2 32 ( 1 + γ ) ( 1 + 3 γ ) × ( 4 p | B 3 B 1 p B 2 2 | 4 ( 1 p ) B 1 [ | B 2 | ( 3 2 p ) + B 1 ] 4 B 2 2 ( 1 p ) 2 B 1 2 ( 1 2 p ) 2 | B 3 B 1 p B 2 2 | ( 1 p ) B 1 ( 2 | B 2 | + B 1 ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equar_HTML.gif
        Proof For f R γ τ ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq92_HTML.gif, there exists an analytic function w with w ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq48_HTML.gif and | w ( z ) | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq49_HTML.gif in D http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq10_HTML.gif such that
        1 + 1 τ ( f ( z ) + γ z f ( z ) 1 ) = φ ( w ( z ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ24_HTML.gif
        (24)
        Since f has the Maclaurin series given by (1), a computation shows that
        1 + 1 τ ( f ( z ) + γ z f ( z ) 1 ) = 1 + 2 a 2 ( 1 + γ ) τ z + 3 a 3 ( 1 + 2 γ ) τ z 2 + 4 a 4 ( 1 + 3 γ ) τ z 3 + . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ25_HTML.gif
        (25)
        It follows from (24), (9) and (25) that
        a 2 = τ B 1 c 1 4 ( 1 + γ ) , a 3 = τ B 1 12 ( 1 + 2 γ ) [ 2 c 2 + c 1 2 ( B 2 B 1 1 ) ] , a 4 = τ 32 ( 1 + 3 γ ) [ B 1 ( 4 c 3 4 c 1 c 2 + c 1 3 ) + 2 B 2 c 1 ( 2 c 2 c 1 2 ) + B 3 c 1 3 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equas_HTML.gif
        Therefore
        a 2 a 4 a 3 2 = τ 2 B 1 c 1 128 ( 1 + γ ) ( 1 + 3 γ ) [ B 1 ( 4 c 3 4 c 1 c 2 + c 1 3 ) + 2 B 2 c 1 ( 2 c 2 c 1 2 ) + B 3 c 1 3 ] τ 2 B 1 2 144 ( 1 + 2 γ ) 2 [ 4 c 2 2 + c 1 4 ( B 2 B 1 1 ) 2 + 4 c 2 c 1 2 ( B 2 B 1 1 ) ] = τ 2 B 1 2 128 ( 1 + γ ) ( 1 + 3 γ ) { [ ( 4 c 1 c 3 4 c 1 2 c 2 + c 1 4 ) + 2 B 2 c 1 2 B 1 ( 2 c 2 c 1 2 ) + B 3 B 1 c 1 4 ] 8 9 ( 1 + γ ) ( 1 + 3 γ ) ( 1 + 2 γ ) 2 [ 4 c 2 2 + c 1 4 ( B 2 B 1 1 ) 2 + 4 c 2 c 1 2 ( B 2 B 1 1 ) ] } , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equat_HTML.gif
        which yields
        | a 2 a 4 a 3 2 | = T | 4 c 1 c 3 + c 1 4 [ 1 2 B 2 B 1 p ( B 2 B 1 1 ) 2 + B 3 B 1 ] 4 p c 2 2 4 c 1 2 c 2 [ 1 B 2 B 1 + p ( B 2 B 1 1 ) ] | , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ26_HTML.gif
        (26)
        where
        T = | τ | 2 B 1 2 128 ( 1 + γ ) ( 1 + 3 γ ) and p = 8 9 ( 1 + γ ) ( 1 + 3 γ ) ( 1 + 2 γ ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equau_HTML.gif

        It can be easily verified that p [ 64 81 , 8 9 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq93_HTML.gif for 0 γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq87_HTML.gif.

        Let
        d 1 = 4 , d 2 = 4 [ 1 B 2 B 1 + p ( B 2 B 1 1 ) ] , d 3 = 4 p , d 4 = 1 2 B 2 B 1 p ( B 2 B 1 1 ) 2 + B 3 B 1 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ27_HTML.gif
        (27)
        Then (26) becomes
        | a 2 a 4 a 3 2 | = T | d 1 c 1 c 3 + d 2 c 1 2 c 2 + d 3 c 2 2 + d 4 c 1 4 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ28_HTML.gif
        (28)
        It follows that
        | a 2 a 4 a 3 2 | = T 4 | c 4 ( d 1 + 2 d 2 + d 3 + 4 d 4 ) + 2 x c 2 ( 4 c 2 ) ( d 1 + d 2 + d 3 ) + ( 4 c 2 ) x 2 ( d 1 c 2 + d 3 ( 4 c 2 ) ) + 2 d 1 c ( 4 c 2 ) ( 1 | x | 2 ) z | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equav_HTML.gif
        Application of the triangle inequality, replacement of | x | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq59_HTML.gif by μ and substituting the values of d 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq60_HTML.gif, d 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq61_HTML.gif, d 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq62_HTML.gif and d 4 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq63_HTML.gif from (27) yield
        | a 2 a 4 a 3 2 | T 4 [ 4 c 4 | B 3 B 1 p B 2 2 B 1 2 | + 8 | B 2 B 1 | μ c 2 ( 4 c 2 ) ( 1 p ) + ( 4 c 2 ) μ 2 ( 4 c 2 + 4 p ( 4 c 2 ) ) + 8 c ( 4 c 2 ) ( 1 μ 2 ) ] = T [ c 4 | B 3 B 1 p B 2 2 B 1 2 | + 2 c ( 4 c 2 ) + 2 μ | B 2 B 1 | c 2 ( 4 c 2 ) ( 1 p ) + μ 2 ( 4 c 2 ) ( 1 p ) ( c α ) ( c β ) ] F ( c , μ ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ29_HTML.gif
        (29)

        where α = 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq94_HTML.gif, β = 2 p / ( 1 p ) > 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq95_HTML.gif.

        Similarly as in the previous proofs, it can be shown that F ( c , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq69_HTML.gif is an increasing function of μ for 0 < μ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq96_HTML.gif. So, for fixed c [ 0 , 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq56_HTML.gif, let
        max F ( c , μ ) = F ( c , 1 ) G ( c ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equaw_HTML.gif
        which is
        G ( c ) = T { c 4 [ | B 3 B 1 p B 2 2 B 1 2 | ( 1 p ) ( 2 | B 2 B 1 | + 1 ) ] + 4 c 2 [ 2 | B 2 B 1 | ( 1 p ) + 1 2 p ] + 16 p } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equax_HTML.gif
        Let
        P = | B 3 B 1 p B 2 2 B 1 2 | ( 1 p ) ( 2 | B 2 B 1 | + 1 ) , Q = 4 [ 2 | B 2 B 1 | ( 1 p ) + 1 2 p ] , R = 16 p . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ30_HTML.gif
        (30)
        Using (16), we have
        | a 2 a 4 a 3 2 | T { R , Q 0 , P Q 4 ; 16 P + 4 Q + R , Q 0 , P Q 8  or  Q 0 , P Q 4 ; 4 P R Q 2 4 P , Q > 0 , P Q 8 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equay_HTML.gif

        where P, Q, R are given in (30). □

        Remark 3 For the choice φ ( z ) : = ( 1 + A z ) / ( 1 + B z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq97_HTML.gif with 1 B < A 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq98_HTML.gif, Theorem 3 reduces to [[36], Theorem 2.1].

        Definition 4 Let φ : D C http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq36_HTML.gif be analytic, and let φ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq81_HTML.gif be as given in (6). For a fixed real number α, the function f A http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq89_HTML.gif is in the class G α ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq99_HTML.gif if it satisfies the following subordination:
        ( 1 α ) f ( z ) + α ( 1 + z f ( z ) f ( z ) ) φ ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equaz_HTML.gif

        Al-Amiri and Reade [37] introduced the class G α : = G α ( ( 1 + z ) / ( 1 z ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq100_HTML.gif and they showed that G α S http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq101_HTML.gif for α < 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq102_HTML.gif. Univalence of the functions in the class G α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq103_HTML.gif was also investigated in [38, 39]. Singh et al. also obtained the bound for the second Hankel determinant of functions in G α http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq103_HTML.gif. The following theorem provides a bound for the second Hankel determinant of the functions in the class G α ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq99_HTML.gif.

        Theorem 4 Let the function f given by (1) be in the class G α ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq104_HTML.gif, 0 α 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq105_HTML.gif. Also, let
        p = 8 9 ( 1 + 2 α ) ( 1 + α ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equba_HTML.gif
        1. 1.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) + B 1 ( 1 + α 2 p ) 0 , | B 1 4 α ( 2 α 1 p α ) + α B 1 2 B 2 ( 3 2 p ) + ( α + 1 ) B 1 B 3 p B 2 2 | p B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbb_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | B 1 2 9 ( 1 + α ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbc_HTML.gif
        1. 2.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) + B 1 ( 1 + α 2 p ) 0 , 2 | B 1 4 α ( 2 α 1 p α ) + α B 1 2 B 2 ( 3 2 p ) + ( α + 1 ) B 1 B 3 p B 2 2 | B 1 3 α ( 3 2 p ) 2 ( 1 + α p ) B 1 | B 2 | ( α + 1 ) B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbd_HTML.gif
           
        or
        B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) + B 1 ( 1 + α 2 p ) 0 , | B 1 4 α ( 2 α 1 p α ) + α B 1 2 B 2 ( 3 2 p ) + ( α + 1 ) B 1 B 3 p B 2 2 | p B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Eqube_HTML.gif
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | | B 1 4 α ( 2 α 1 p α ) + α B 1 2 B 2 ( 3 2 p ) + ( α + 1 ) B 1 B 3 p B 2 2 | 8 ( 1 + α ) ( 1 + 2 α ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbf_HTML.gif
        1. 3.
          If B 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq44_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq45_HTML.gif and B 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq46_HTML.gif satisfy the conditions
          B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) + B 1 ( 1 + α 2 p ) > 0 , 2 | B 1 4 α ( 2 α 1 p α ) + α B 1 2 B 2 ( 3 2 p ) + ( α + 1 ) B 1 B 3 p B 2 2 | B 1 3 α ( 3 2 p ) 2 ( 1 + α p ) B 1 | B 2 | ( α + 1 ) B 1 2 0 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbg_HTML.gif
           
        then the second Hankel determinant satisfies
        | a 2 a 4 a 3 2 | B 1 2 32 ( 1 + α ) ( 1 + 2 α ) × [ 4 p [ B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) + B 1 ( 1 + α 2 p ) ] 2 | B 1 4 α ( 2 α 1 p α ) + α B 1 2 B 2 ( 3 2 p ) + ( α + 1 ) B 1 B 3 p B 2 2 | B 1 3 α ( 3 2 p ) ( 1 + α p ) B 1 ( 2 | B 2 | + B 1 ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbh_HTML.gif
        Proof For f G α ( φ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq106_HTML.gif, a calculation shows that
        | a 2 a 4 a 3 2 | = T | 4 ( 1 + α ) B 1 c 1 c 3 + c 1 4 [ 3 α B 1 2 + α ( 2 α 1 ) B 1 3 + B 1 ( 1 + α ) + 3 α B 1 B 2 + ( 1 + α ) ( B 3 2 B 2 ) p ( α B 1 2 B 1 + B 2 ) 2 B 1 ] 4 p B 1 c 2 2 + 2 c 1 2 c 2 [ 2 ( 1 + α ) B 1 + 3 α B 1 2 + 2 ( 1 + α ) B 2 2 p ( α B 1 2 B 1 + B 2 ) ] | , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ31_HTML.gif
        (31)
        where
        T = B 1 128 ( 1 + α ) ( 1 + 2 α ) and p = 8 9 ( 1 + 2 α ) ( 1 + α ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbi_HTML.gif
        It can be easily verified that for 0 α 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq105_HTML.gif, p [ 8 9 , 4 3 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq107_HTML.gif. Let
        d 1 = 4 ( 1 + α ) B 1 , d 2 = 2 [ 2 ( 1 + α ) B 1 + 3 α B 1 2 + 2 ( 1 + α ) B 2 2 p ( α B 1 2 B 1 + B 2 ) ] , d 3 = 4 p B 1 , d 4 = 3 α B 1 2 + α ( 2 α 1 ) B 1 3 + B 1 ( 1 + α ) + 3 α B 1 B 2 d 4 = + ( 1 + α ) ( B 3 2 B 2 ) p ( α B 1 2 B 1 + B 2 ) 2 B 1 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ32_HTML.gif
        (32)
        Then
        | a 2 a 4 a 3 2 | = T | d 1 c 1 c 3 + d 2 c 1 2 c 2 + d 3 c 2 2 + d 4 c 1 4 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ33_HTML.gif
        (33)
        Similarly as in earlier theorems, it follows that
        | a 2 a 4 a 3 2 | = T 4 | c 4 ( d 1 + 2 d 2 + d 3 + 4 d 4 ) + 2 x c 2 ( 4 c 2 ) ( d 1 + d 2 + d 3 ) + ( 4 c 2 ) x 2 ( d 1 c 2 + d 3 ( 4 c 2 ) ) + 2 d 1 c ( 4 c 2 ) ( 1 | x | 2 ) z | T [ c 4 | B 1 3 α ( 2 α 1 p α ) + α B 1 B 2 ( 3 2 p ) + ( α + 1 ) B 3 p B 2 2 B 1 | + μ c 2 ( 4 c 2 ) [ B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) ] + 2 c ( 4 c 2 ) B 1 ( 1 + α ) + μ 2 ( 4 c 2 ) B 1 ( 1 + α p ) ( c 2 ) ( c 2 p 1 + α p ) ] F ( c , μ ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ34_HTML.gif
        (34)
        and for fixed c [ 0 , 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq56_HTML.gif, max F ( c , μ ) = F ( c , 1 ) G ( c ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq108_HTML.gif with
        G ( c ) = T [ c 4 [ | B 1 3 α ( 2 α 1 p α ) + α B 1 B 2 ( 3 2 p ) + ( α + 1 ) B 3 p B 2 2 B 1 | B 1 2 α ( 3 2 p ) ( 1 + α p ) ( 2 | B 2 | + B 1 ) ] + 4 c 2 [ B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) + B 1 ( 1 + α 2 p ) ] + 16 p B 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbj_HTML.gif
        Let
        P = | B 1 3 α ( 2 α 1 p α ) + α B 1 B 2 ( 3 2 p ) + ( α + 1 ) B 3 p B 2 2 B 1 | P = B 1 2 α ( 3 2 p ) ( 1 + α p ) ( 2 | B 2 | + B 1 ) , Q = 4 [ B 1 2 α ( 3 2 p ) + 2 | B 2 | ( 1 + α p ) + B 1 ( 1 + α 2 p ) ] , R = 16 p B 1 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equ35_HTML.gif
        (35)
        By using (16), we have
        | a 2 a 4 a 3 2 | T { R , Q 0 , P Q 4 ; 16 P + 4 Q + R , Q 0 , P Q 8  or  Q 0 , P Q 4 ; 4 P R Q 2 4 P , Q > 0 , P Q 8 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_Equbk_HTML.gif

        where P, Q, R are given in (35). □

        Remark 4 For α = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq109_HTML.gif, Theorem 4 reduces to Theorem 2. For 0 α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq110_HTML.gif, let φ ( z ) : = ( 1 + ( 1 2 α ) z ) / ( 1 z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq111_HTML.gif. For this function φ, B 1 = B 2 = B 3 = 2 ( 1 α ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-281/MediaObjects/13660_2012_Article_729_IEq112_HTML.gif. In this case, Theorem 4 reduces to [[40], Theorem 3.1].

        Declarations

        Acknowledgements

        The work presented here was supported in part by research grants from Universiti Sains Malaysia (FRGS grants) and University of Delhi as well as MyBrain MyPhD programme of the Ministry of Higher Education, Malaysia.

        Authors’ Affiliations

        (1)
        School of Mathematical Sciences, Universiti Sains Malaysia
        (2)
        Department of Mathematics, University of Delhi

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