Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli

  • Mohsan Raza1Email author and

    Affiliated with

    • Sarfraz Nawaz Malik2

      Affiliated with

      Journal of Inequalities and Applications20132013:412

      DOI: 10.1186/1029-242X-2013-412

      Received: 15 February 2013

      Accepted: 8 August 2013

      Published: 28 August 2013

      Abstract

      In this paper, the upper bound of the Hankel determinant H 3 ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq1_HTML.gif for a subclass of analytic functions associated with right half of the lemniscate of Bernoulli ( x 2 + y 2 ) 2 2 ( x 2 y 2 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq2_HTML.gif is investigated.

      MSC:30C45, 30C50.

      Keywords

      starlike functions subordination lemniscate of Bernoulli Toeplitz determinants Hankel determinants

      1 Introduction and preliminaries

      Let A be the class of functions f of the form
      f ( z ) = z + n = 2 a n z n , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ1_HTML.gif
      (1.1)

      which are analytic in the open unit disk E = { z : | z | < 1 } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq3_HTML.gif. A function f is said to be subordinate to a function g, written as f g http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq4_HTML.gif, if there exists a Schwartz function w with w ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq5_HTML.gif and | w ( z ) | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq6_HTML.gif such that f ( z ) = g ( w ( z ) ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq7_HTML.gif. In particular, if g is univalent in E, then f ( 0 ) = g ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq8_HTML.gif and f ( E ) g ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq9_HTML.gif.

      Let P denote the class of analytic functions p normalized by
      p ( z ) = 1 + n = 1 p n z n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ2_HTML.gif
      (1.2)
      such that Re p ( z ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq10_HTML.gif. Let SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq11_HTML.gif be the class of functions defined by
      SL = { f A : | ( z f ( z ) f ( z ) ) 2 1 | < 1 } , z E . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equa_HTML.gif
      Thus a function f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif is such that z f ( z ) f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq13_HTML.gif lies in the region bounded by the right half of the lemniscate of Bernoulli given by the relation | w 2 1 | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq14_HTML.gif. It can easily be seen that f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif if it satisfies the condition
      z f ( z ) f ( z ) 1 + z , z E . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ3_HTML.gif
      (1.3)

      This class of functions was introduced by Sokół and Stankiewicz [1] and further investigated by some authors. For details, see [2, 3].

      Noonan and Thomas [4] have studied the q th Hankel determinant defined as
      H q ( n ) = | a n a n + 1 a n + q 1 a n + 1 a n + 2 a n + q 2 a n + q 1 a n + q 2 a n + 2 q 2 | , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ4_HTML.gif
      (1.4)

      where n 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq15_HTML.gif and q 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq16_HTML.gif. The Hankel determinant plays an important role in the study of singularities; for instance, see [[5], p.329] and Edrei [6]. This is also important in the study of power series with integral coefficients [[5], p.323] and Cantor [7]. For the use of the Hankel determinant in the study of meromorphic functions, see [8], and various properties of these determinants can be found in [[9], Chapter 4]. It is well known that the Fekete-Szegö functional | a 3 a 2 2 | = H 2 ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq17_HTML.gif. This functional is further generalized as | a 3 μ a 2 2 | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq18_HTML.gif for some μ (real as well as complex). Fekete and Szegö gave sharp estimates of | a 3 μ a 2 2 | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq18_HTML.gif for μ real and f S http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq19_HTML.gif, the class of univalent functions. It is a very great combination of the two coefficients which describes the area problems posted earlier by Gronwall in 1914-15. Moreover, we also know that the functional | a 2 a 4 a 3 2 | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq20_HTML.gif is equivalent to H 2 ( 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq21_HTML.gif. The q th Hankel determinant for some subclasses of analytic functions was recently studied by Arif et al. [10] and Arif et al. [11]. The functional | a 2 a 4 a 3 2 | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq20_HTML.gif has been studied by many authors, see [1214]. Babalola [15] studied the Hankel determinant H 3 ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq22_HTML.gif for some subclasses of analytic functions. In the present investigation, we determine the upper bounds of the Hankel determinant H 3 ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq22_HTML.gif for a subclass of analytic functions related with lemniscate of Bernoulli by using Toeplitz determinants.

      We need the following lemmas which will be used in our main results.

      Lemma 1.1 [16]

      Let p P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq23_HTML.gif and of the form (1.2). Then
      | p 2 v p 1 2 | { 4 v + 2 , v < 0 , 2 , 0 v 1 , 4 v 2 , v > 1 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equb_HTML.gif
      When v < 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq24_HTML.gif or v > 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq25_HTML.gif, the equality holds if and only if p ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq26_HTML.gif is 1 + z 1 z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq27_HTML.gif or one of its rotations. If 0 < v < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq28_HTML.gif, then the equality holds if and only if p ( z ) = 1 + z 2 1 z 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq29_HTML.gif or one of its rotations. If v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq30_HTML.gif, the equality holds if and only if p ( z ) = ( 1 2 + η 2 ) 1 + z 1 z + ( 1 2 η 2 ) 1 z 1 + z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq31_HTML.gif ( 0 η 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq32_HTML.gif) or one of its rotations. If v = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq33_HTML.gif, the equality holds if and only if p is the reciprocal of one of the functions such that the equality holds in the case of v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq30_HTML.gif. Although the above upper bound is sharp, when 0 < v < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq28_HTML.gif, it can improved as follows:
      | p 2 v p 1 2 | + v | p 1 | 2 2 ( 0 < v 1 / 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equc_HTML.gif
      and
      | p 2 v p 1 2 | + ( 1 v ) | p 1 | 2 2 ( 1 / 2 < v 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equd_HTML.gif

      Lemma 1.2 [16]

      If p ( z ) = 1 + p 1 z + p 2 z 2 + http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq34_HTML.gif is a function with positive real part in E, then for v a complex number
      | p 2 v p 1 2 | 2 max ( 1 , | 2 v 1 | ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Eque_HTML.gif
      This result is sharp for the functions
      p ( z ) = 1 + z 2 1 z 2 , p ( z ) = 1 + z 1 z . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equf_HTML.gif

      Lemma 1.3 [17]

      Let p P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq23_HTML.gif and of the form (1.2). Then
      2 p 2 = p 1 2 + x ( 4 p 1 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equg_HTML.gif
      for some x, | x | 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq35_HTML.gif, and
      4 p 3 = p 1 3 + 2 ( 4 p 1 2 ) p 1 x ( 4 p 1 2 ) p 1 x 2 + 2 ( 4 p 1 2 ) ( 1 | x | 2 ) z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equh_HTML.gif

      for some z, | z | 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq36_HTML.gif.

      2 Main results

      Although we have discussed the Hankel determinant problem in the paper, the first two problems are specifically related with the Fekete-Szegö functional, which is a special case of the Hankel determinant.

      Theorem 2.1 Let f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif and of the form (1.1). Then
      | a 3 μ a 2 2 | { 1 16 ( 1 4 μ ) , μ < 3 4 , 1 4 , 3 4 μ 5 4 , 1 16 ( 4 μ 1 ) , μ > 5 4 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equi_HTML.gif
      Furthermore, for 3 4 < μ 1 4 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq37_HTML.gif,
      | a 3 μ a 2 2 | + 1 4 ( 4 μ + 3 ) | a 2 | 2 1 4 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equj_HTML.gif
      and for 1 4 < μ 5 4 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq38_HTML.gif,
      | a 3 μ a 2 2 | + 1 4 ( 5 4 μ ) | a 2 | 2 1 4 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equk_HTML.gif

      These results are sharp.

      Proof If f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif, then it follows from (1.3) that
      z f ( z ) f ( z ) ϕ ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ5_HTML.gif
      (2.1)
      where ϕ ( z ) = 1 + z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq39_HTML.gif. Define a function
      p ( z ) = 1 + w ( z ) 1 w ( z ) = 1 + p 1 z + p 2 z 2 + . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equl_HTML.gif
      It is clear that p P http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq23_HTML.gif. This implies that
      w ( z ) = p ( z ) 1 p ( z ) + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equm_HTML.gif
      From (2.1), we have
      z f ( z ) f ( z ) = ϕ ( w ( z ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equn_HTML.gif
      with
      ϕ ( w ( z ) ) = ( 2 p ( z ) p ( z ) + 1 ) 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equo_HTML.gif
      Now
      ( 2 p ( z ) p ( z ) + 1 ) 1 2 = 1 + 1 4 p 1 z + [ 1 4 p 2 5 32 p 1 2 ] z 2 + [ 1 4 p 3 5 16 p 1 p 2 + 13 128 p 1 3 ] z 3 + . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equp_HTML.gif
      Similarly,
      z f ( z ) f ( z ) = 1 + a 2 z + [ 2 a 3 a 2 2 ] 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-2013-412/MediaObjects/13660_2013_Article_2378_Equq_HTML.gif
      Therefore
      a 2 = 1 4 p 1 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ6_HTML.gif
      (2.2)
      a 3 = 1 8 p 2 3 64 p 1 2 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ7_HTML.gif
      (2.3)
      a 4 = 1 12 p 3 7 96 p 1 p 2 + 13 768 p 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ8_HTML.gif
      (2.4)
      This implies that
      | a 3 μ a 2 2 | = 1 8 | p 2 1 8 ( 4 μ + 3 ) p 1 2 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equr_HTML.gif

      Now, using Lemma 1.1, we have the required result. □

      The results are sharp for the functions K i ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq40_HTML.gif, i = 1 , 2 , 3 , 4 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq41_HTML.gif, such that
      z K 1 ( z ) K 1 ( z ) = 1 + z if  μ < 3 4  or  μ > 5 4 , z K 2 ( z ) K 2 ( z ) = 1 + z 2 if  3 4 < μ < 5 4 , z K 3 ( z ) K 3 ( z ) = 1 + Φ ( z ) if  μ = 3 4 , z K 4 ( z ) K 4 ( z ) = 1 Φ ( z ) if  μ = 5 4 , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equs_HTML.gif

      where Φ ( z ) = z ( z + η ) 1 + η z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq42_HTML.gif with 0 η 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq43_HTML.gif.

      Theorem 2.2 Let f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif and of the form (1.1). Then for a complex number μ,
      | a 3 μ a 2 2 | 1 4 max { 1 ; | μ 1 4 | } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equt_HTML.gif
      Proof Since
      | a 3 μ a 2 2 | = 1 8 | p 2 1 8 ( 4 μ + 3 ) p 1 2 | , http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equu_HTML.gif
      therefore, using Lemma 1.2, we get the result. This result is sharp for the functions
      z f ( z ) f ( z ) = 1 + z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equv_HTML.gif
      or
      z f ( z ) f ( z ) = 1 + z 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equw_HTML.gif

       □

      For μ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq44_HTML.gif, we have H 2 ( 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq45_HTML.gif.

      Corollary 2.3 Let f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif and of the form (1.1). Then
      | a 3 a 2 2 | 1 4 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equx_HTML.gif
      Theorem 2.4 Let f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif and of the form (1.1). Then
      | a 2 a 4 a 3 2 | 1 16 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equy_HTML.gif
      Proof From (2.2), (2.3) and (2.4), we obtain
      a 2 a 4 a 3 2 = 1 48 ( p 1 p 3 7 8 p 1 2 p 2 + 13 64 p 1 4 ) ( 1 8 p 2 3 64 p 1 2 ) 2 = 1 48 p 1 p 3 1 64 p 2 2 5 768 p 1 2 p 2 + 25 12 , 288 p 1 4 = 1 12 , 288 ( 256 p 1 p 3 192 p 2 2 80 p 1 2 p 2 + 25 p 1 4 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equz_HTML.gif
      Putting the values of p 2 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq46_HTML.gif and p 3 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq47_HTML.gif from Lemma 1.3, we assume that p > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq48_HTML.gif, and taking p 1 = p [ 0 , 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq49_HTML.gif, we get
      | a 2 a 4 a 3 2 | = 1 12 , 288 | 64 p 1 { p 1 3 + 2 ( 4 p 1 2 ) p 1 x ( 4 p 1 2 ) p 1 x 2 + 2 ( 4 p 1 2 ) ( 1 | x | 2 ) z } 48 { p 1 2 + x ( 4 p 1 2 ) } 2 40 p 1 2 { p 1 2 + x ( 4 p 1 2 ) } + 25 p 1 4 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equaa_HTML.gif
      After simple calculations, we get
      | a 2 a 4 a 3 2 | = 1 12 , 288 | 41 p 4 8 ( 4 p 2 ) p 2 x 128 ( 4 p 2 ) ( 1 | x | 2 ) z + x 2 ( 4 p 2 ) ( 64 p 2 + 48 ) ( 4 p 2 ) | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equab_HTML.gif
      Now, applying the triangle inequality and replacing | x | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq50_HTML.gif by ρ, we obtain
      | a 2 a 4 a 3 2 | 1 12 , 288 [ 41 p 4 + 128 ( 4 p 2 ) + 8 ( 4 p 2 ) p 2 ρ + ρ 2 ( 4 p 2 ) ( 16 p 2 + 64 ) ] = F ( p , ρ ) (say). http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equac_HTML.gif
      Differentiating with respect to ρ, we have
      F ( p , ρ ) ρ = 1 12 , 288 [ 8 ( 4 p 2 ) p 2 + 2 ρ ( 4 p 2 ) ( 16 p 2 + 64 ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equad_HTML.gif
      It is clear that F ( p , ρ ) ρ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq51_HTML.gif, which shows that F ( p , ρ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq52_HTML.gif is an increasing function on the closed interval [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq53_HTML.gif. This implies that maximum occurs at ρ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq54_HTML.gif. Therefore max F ( p , ρ ) = F ( p , 1 ) = G ( p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq55_HTML.gif (say). Now
      G ( p ) = 1 12 , 288 [ 17 p 4 96 p 2 + 768 ] . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equae_HTML.gif
      Therefore
      G ( p ) = 1 12 , 288 [ 68 p 3 192 p ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equaf_HTML.gif
      and
      G ( p ) = 1 12 , 288 [ 204 p 2 192 ] < 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equag_HTML.gif
      for p = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq56_HTML.gif. This shows that maximum of G ( p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq57_HTML.gif occurs at p = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq56_HTML.gif. Hence, we obtain
      | a 2 a 4 a 3 2 | 768 12 , 288 = 1 16 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equah_HTML.gif
      This result is sharp for the functions
      z f ( z ) f ( z ) = 1 + z http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equai_HTML.gif
      or
      z f ( z ) f ( z ) = 1 + z 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equaj_HTML.gif

       □

      Theorem 2.5 Let f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif and of the form (1.1). Then
      | a 2 a 3 a 4 | 1 6 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equak_HTML.gif
      Proof Since
      a 2 = 1 4 p 1 , a 3 = 1 8 p 2 3 64 p 1 2 , a 4 = 1 12 p 3 7 96 p 1 p 2 + 13 768 p 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equal_HTML.gif
      Therefore, by using Lemma 1.3, we can obtain
      | a 2 a 3 a 4 | 1 768 { 7 p 3 + 8 p ρ ( 4 p 2 ) + 32 ( 4 p 2 ) + 16 ρ 2 ( p 2 ) ( 4 p 2 ) } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equam_HTML.gif
      Let
      F 1 ( p , ρ ) = 1 768 { 7 p 3 + 8 p ρ ( 4 p 2 ) + 32 ( 4 p 2 ) + 16 ρ 2 ( p 2 ) ( 4 p 2 ) } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equ9_HTML.gif
      (2.5)
      We assume that the upper bound occurs at the interior point of the rectangle [ 0 , 2 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq58_HTML.gif. Differentiating (2.5) with respect to ρ, we get
      F 1 ρ = 1 768 { 8 p ( 4 p 2 ) + 32 ρ ( p 2 ) ( 4 p 2 ) } . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equan_HTML.gif
      For 0 < ρ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq59_HTML.gif and fixed p ( 0 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq60_HTML.gif, it can easily be seen that F 1 ρ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq61_HTML.gif. This shows that F 1 ( p , ρ ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq62_HTML.gif is a decreasing function of ρ, which contradicts our assumption; therefore, max F 1 ( p , ρ ) = F 1 ( p , 0 ) = G 1 ( p ) http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq63_HTML.gif. This implies that
      G 1 ( p ) = 1 768 { 21 p 2 64 p } http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equao_HTML.gif
      and
      G 1 ′′ ( p ) = 1 768 { 42 p 64 } < 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equap_HTML.gif

      for p = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq56_HTML.gif. Therefore p = 0 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq56_HTML.gif is a point of maximum. Hence, we get the required result. □

      Lemma 2.6 If the function f ( z ) = n = 1 a n z n http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq64_HTML.gif belongs to the class SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq11_HTML.gif, then
      | a 2 | 1 / 2 , | a 3 | 1 / 4 , | a 4 | 1 / 6 , | a 5 | 1 / 8 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equaq_HTML.gif

      These estimations are sharp. The first three bounds were obtained by Sokół [3]and the bound for | a 5 | http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq65_HTML.gif can be obtained in a similar way.

      Theorem 2.7 Let f SL http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq12_HTML.gif and of the form (1.1). Then
      | H 3 ( 1 ) | 43 576 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equar_HTML.gif
      Proof Since
      H 3 ( 1 ) = a 3 ( a 2 a 4 a 3 2 ) a 4 ( a 4 a 2 a 3 ) + a 5 ( a 1 a 3 a 2 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equas_HTML.gif
      Now, using the triangle inequality, we obtain
      | H 3 ( 1 ) | | a 3 | | a 2 a 4 a 3 2 | + | a 4 | | a 2 a 3 a 4 | + | a 5 | | a 1 a 3 a 2 2 | . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equat_HTML.gif
      Using the fact that a 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_IEq66_HTML.gif with the results of Corollary 2.3, Theorem 2.4, Theorem 2.5 and Lemma 2.6, we obtain
      | H 3 ( 1 ) | 43 576 . http://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-412/MediaObjects/13660_2013_Article_2378_Equau_HTML.gif

       □

      Declarations

      Acknowledgements

      The authors would like to thank the referees for helpful comments and suggestions which improved the presentation of the paper.

      Authors’ Affiliations

      (1)
      Department of Mathematics, G.C. University Faisalabad
      (2)
      Department of Mathematics, COMSATS Institute of Information Technology

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      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.