Skip to main content

Bounds for the second Hankel determinant of certain univalent functions

Abstract

The estimates for the second Hankel determinant a 2 a 4 a 3 2 of the analytic function f(z)=z+ a 2 z 2 + a 3 z 3 + , for which either z f (z)/f(z) or 1+z f (z)/ f (z) 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 denote the class of all analytic functions

f(z)=z+ a 2 z 2 + a 3 z 3 +
(1)

defined on the open unit disk D:={zC:|z|<1}. The Hankel determinants H q (n) (n=1,2, , q=1,2,) 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).

Hankel determinants are useful, for example, in showing that a function of bounded characteristic in D, 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,),

where β>1/4000 and K depends only on q. Later, Hayman [6] proved that | H 2 (n)|<A n 1 / 2 (n=1,2, ; 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 is the well-known Fekete-Szegö functional. For results related to this functional, see [28, 29]. The second Hankel determinant H 2 (2) is given by H 2 (2)= a 2 a 4 a 3 2 .

An analytic function f is subordinate to an analytic function g, written f(z)g(z), if there is an analytic function w:DD with w(0)=0 satisfying f(z)=g(w(z)). Ma and Minda [30] unified various subclasses of starlike ( S ) and convex functions (C) by requiring that either of the quantity z f (z)/f(z) or 1+z f (z)/ f (z) is subordinate to a function φ with a positive real part in the unit disk D, φ(0)=1, φ (0)>0, φ maps D 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 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 be the class of functions with positive real part consisting of all analytic functions p:DC satisfying p(0)=1 and Rep(z)>0. We need the following results about the functions belonging to the class P.

Lemma 1 [31]

If the function pP is given by the series

p(z)=1+ c 1 z+ c 2 z 2 + c 3 z 3 +,
(2)

then the following sharp estimate holds:

| c n |2(n=1,2,).
(3)

Lemma 2 [32]

If the function pP is given by the series (2), then

2 c 2 = c 1 2 +x ( 4 c 1 2 ) ,
(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,
(5)

for some x, z with |x|1 and |z|1.

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

Subclasses of starlike functions are characterized by the quantity z f (z)/f(z) lying in some domain in the right half-plane. For example, f is strongly starlike of order β if z f (z)/f(z) lies in a sector |argw|<βπ/2, while it is starlike of order α if z f (z)/f(z) lies in the half-plane Rew>α. 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:DC 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).
(6)

The class S (φ) of Ma-Minda starlike functions with respect to φ consists of functions fA satisfying the subordination

z f ( z ) f ( z ) φ(z).

For the function φ given by φ α (z):=(1+(12α)z)/(1z) , 0<α1, the class S (α):= S ( φ α ) 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 .

Then the class

S P := S ( φ PAR )= { f A : Re ( z f ( z ) f ( z ) ) > | z f ( z ) f ( z ) 1 | }

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<β1, the class

S β := S ( ( 1 + z 1 z ) β ) = { f A : | arg ( z f ( z ) f ( z ) ) | < β π 2 }

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 }

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

Theorem 1 Let the function f S (φ) be given by (1).

  1. 1.

    If B 1 , B 2 and B 3 satisfy the conditions

    | B 2 | B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | 3 B 1 2 0,

then the second Hankel determinant satisfies

| a 2 a 4 a 3 2 | B 1 2 4 .
  1. 2.

    If B 1 , B 2 and B 3 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,

or the conditions

| B 2 | B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | 3 B 1 2 0,

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 | .
  1. 3.

    If B 1 , B 2 and B 3 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,

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 ) .

Proof Since f S (φ), there exists an analytic function w with w(0)=0 and |w(z)|<1 in D such that

z f ( z ) f ( z ) =φ ( w ( z ) ) .
(7)

Define the functions p 1 by

p 1 (z):= 1 + w ( z ) 1 w ( z ) =1+ c 1 z+ c 2 z 2 +,

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 + ) .
(8)

Then p 1 is analytic in D with p 1 (0)=1 and has a positive real part in D. 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 +.
(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 +,
(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 ] .

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 ] .

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 .
(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 | .
(12)

Since the function p( e i θ z) (θR) is in the class P for any pP, there is no loss of generality in assuming c 1 >0. Write c 1 =c, c[0,2]. Substituting the values of c 2 and c 3 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 | .

Replacing |x| by μ and substituting the values of d 1 , d 2 , d 3 and d 4 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 ) .
(13)

Note that for (c,μ)[0,2]×[0,1], differentiating F(c,μ) 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 ) ] .
(14)

Then, for 0<μ<1 and for any fixed c with 0<c<2, it is clear from (14) that F μ >0, that is, F(c,μ) is an increasing function of μ. Hence, for fixed c[0,2], the maximum of F(c,μ) occurs at μ=1, and

maxF(c,μ)=F(c,1)G(c).

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 ] .

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 .
(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 ,
(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 ,

where P, Q, R are given by (15). □

Remark 1 When B 1 = B 2 = B 3 =2, Theorem 1 reduces to [[24], Theorem 3.1].

Corollary 1

  1. 1.

    If f S (α), then | a 2 a 4 a 3 2 | ( 1 α ) 2 .

  2. 2.

    If f S L , then | a 2 a 4 a 3 2 |1/16=0.0625.

  3. 3.

    If f S P , then | a 2 a 4 a 3 2 |16/ π 4 0.164255.

  4. 4.

    If f S β , then | a 2 a 4 a 3 2 | β 2 .

Definition 2 Let φ:DC be analytic, and let φ(z) be given as in (6). The class C(φ) of Ma-Minda convex functions with respect to φ consists of functions f satisfying the subordination

1+ z f ( z ) f ( z ) φ(z).

Theorem 2 Let the function fC(φ) be given by (1).

  1. 1.

    If B 1 , B 2 and B 3 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,

then the second Hankel determinant satisfies

| a 2 a 4 a 3 2 | B 1 2 36 .
  1. 2.

    If B 1 , B 2 and B 3 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,

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,

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 | .
  1. 3.

    If B 1 , B 2 and B 3 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,

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 ) .

Proof Since fC(φ), there exists an analytic function w with w(0)=0 and |w(z)|<1 in D such that

1+ z f ( z ) f ( z ) =φ ( w ( z ) ) .
(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 +,
(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 ] .

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 ] .

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 ,
(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 | .
(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 | .

Replacing |x| by μ and then substituting the values of d 1 , d 2 , d 3 and d 4 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 ) .
(21)

Again, differentiating F(c,μ) 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 ) ] .
(22)

It is clear from (22) that F μ >0. Thus F(c,μ) is an increasing function of μ for 0<μ<1 and for any fixed c with 0<c<2. So, the maximum of F(c,μ) occurs at μ=1 and

maxF(c,μ)=F(c,1)G(c).

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 ] .

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 .
(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 ,

where P, Q, R are given in (23). □

Remark 2 For the choice of φ(z)=(1+z)/(1z), Theorem 2 reduces to [[24], Theorem 3.2].

3 Further results on the second Hankel determinant

Definition 3 Let φ:DC be analytic, and let φ(z) be as given in (6). Let 0γ1 and τC{0}. A function fA is in the class R γ τ (φ) if it satisfies the following subordination:

1+ 1 τ ( f ( z ) + γ z f ( z ) 1 ) φ(z).

Theorem 3 Let 0γ1, τC{0}, and let the function f as in (1) be in the class R γ τ (φ). Also, let

p= 8 9 ( 1 + γ ) ( 1 + 3 γ ) ( 1 + 2 γ ) 2 .
  1. 1.

    If B 1 , B 2 and B 3 satisfy the conditions

    2| B 2 |(1p)+ B 1 (12p)0, | B 1 B 3 p B 2 2 | p B 1 2 0,

then the second Hankel determinant satisfies

| a 2 a 4 a 3 2 | | τ | 2 B 1 2 9 ( 1 + 2 γ ) 2 .
  1. 2.

    If B 1 , B 2 and B 3 satisfy the conditions

    2| B 2 |(1p)+ B 1 (12p)0,2 | B 1 B 3 p B 2 2 | 2(1p) B 1 | B 2 | B 1 0,

or the conditions

2| B 2 |(1p)+ B 1 (12p)0, | B 1 B 3 p B 2 2 | B 1 2 0,

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 | .
  1. 3.

    If B 1 , B 2 and B 3 satisfy the conditions

    2| B 2 |(1p)+ B 1 (12p)>0,2 | B 1 B 3 p B 2 2 | 2(1p) B 1 | B 2 | B 1 2 0,

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 ) ) .

Proof For f R γ τ (φ), there exists an analytic function w with w(0)=0 and |w(z)|<1 in D such that

1+ 1 τ ( f ( z ) + γ z f ( z ) 1 ) =φ ( w ( z ) ) .
(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 + .
(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 ] .

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 ) ] } ,

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 ) ] | ,
(26)

where

T= | τ | 2 B 1 2 128 ( 1 + γ ) ( 1 + 3 γ ) andp= 8 9 ( 1 + γ ) ( 1 + 3 γ ) ( 1 + 2 γ ) 2 .

It can be easily verified that p[ 64 81 , 8 9 ] for 0γ1.

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 .
(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 | .
(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 | .

Application of the triangle inequality, replacement of |x| by μ and substituting the values of d 1 , d 2 , d 3 and d 4 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 ) ,
(29)

where α=2, β=2p/(1p)>2.

Similarly as in the previous proofs, it can be shown that F(c,μ) is an increasing function of μ for 0<μ<1. So, for fixed c[0,2], let

maxF(c,μ)=F(c,1)G(c),

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 } .

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 .
(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 ,

where P, Q, R are given in (30). □

Remark 3 For the choice φ(z):=(1+Az)/(1+Bz) with 1B<A1, Theorem 3 reduces to [[36], Theorem 2.1].

Definition 4 Let φ:DC be analytic, and let φ(z) be as given in (6). For a fixed real number α, the function fA is in the class G α (φ) if it satisfies the following subordination:

(1α) f (z)+α ( 1 + z f ( z ) f ( z ) ) φ(z).

Al-Amiri and Reade [37] introduced the class G α := G α ((1+z)/(1z)) and they showed that G α S for α<0. Univalence of the functions in the class G α was also investigated in [38, 39]. Singh et al. also obtained the bound for the second Hankel determinant of functions in G α . The following theorem provides a bound for the second Hankel determinant of the functions in the class G α (φ).

Theorem 4 Let the function f given by (1) be in the class G α (φ), 0α1. Also, let

p= 8 9 ( 1 + 2 α ) ( 1 + α ) .
  1. 1.

    If B 1 , B 2 and B 3 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 ,

then the second Hankel determinant satisfies

| a 2 a 4 a 3 2 | B 1 2 9 ( 1 + α ) 2 .
  1. 2.

    If B 1 , B 2 and B 3 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 ,

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 ,

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 ) .
  1. 3.

    If B 1 , B 2 and B 3 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 ,

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 ) ] .

Proof For f G α (φ), 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 ) ] | ,
(31)

where

T= B 1 128 ( 1 + α ) ( 1 + 2 α ) andp= 8 9 ( 1 + 2 α ) ( 1 + α ) .

It can be easily verified that for 0α1, p[ 8 9 , 4 3 ]. 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 .
(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 | .
(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 ) ,
(34)

and for fixed c[0,2], maxF(c,μ)=F(c,1)G(c) 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 ] .

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 .
(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 ,

where P, Q, R are given in (35). □

Remark 4 For α=1, Theorem 4 reduces to Theorem 2. For 0α<1, let φ(z):=(1+(12α)z)/(1z). For this function φ, B 1 = B 2 = B 3 =2(1α). In this case, Theorem 4 reduces to [[40], Theorem 3.1].

References

  1. Cantor DG: Power series with integral coefficients. Bull. Am. Math. Soc. 1963, 69: 362–366. 10.1090/S0002-9904-1963-10923-4

    Article  MathSciNet  MATH  Google Scholar 

  2. Wilson R: Determinantal criteria for meromorphic functions. Proc. Lond. Math. Soc. 1954, 4: 357–374.

    Article  MathSciNet  MATH  Google Scholar 

  3. Vein R, Dale P Applied Mathematical Sciences 134. In Determinants and Their Applications in Mathematical Physics. Springer, New York; 1999.

    Google Scholar 

  4. Pommerenke C: On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc. 1966, 41: 111–122.

    Article  MathSciNet  MATH  Google Scholar 

  5. Pommerenke C: On the Hankel determinants of univalent functions. Mathematika 1967, 14: 108–112. 10.1112/S002557930000807X

    Article  MathSciNet  MATH  Google Scholar 

  6. Hayman WK: On the second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc. 1968, 18: 77–94.

    Article  MathSciNet  MATH  Google Scholar 

  7. Noonan JW, Thomas DK: On the Hankel determinants of areally mean p -valent functions. Proc. Lond. Math. Soc. 1972, 25: 503–524.

    Article  MathSciNet  MATH  Google Scholar 

  8. Noonan JW: Coefficient differences and Hankel determinants of areally mean p -valent functions. Proc. Am. Math. Soc. 1974, 46: 29–37.

    MathSciNet  MATH  Google Scholar 

  9. Noonan JW, Thomas DK: On the second Hankel determinant of areally mean p -valent functions. Trans. Am. Math. Soc. 1976, 223: 337–346.

    MathSciNet  MATH  Google Scholar 

  10. Elhosh MM: On the second Hankel determinant of univalent functions. Bull. Malays. Math. Soc. 1986, 9(1):23–25.

    MathSciNet  MATH  Google Scholar 

  11. Elhosh MM: On the second Hankel determinant of close-to-convex functions. Bull. Malays. Math. Soc. 1986, 9(2):67–68.

    MathSciNet  MATH  Google Scholar 

  12. Noor KI: Higher order close-to-convex functions. Math. Jpn. 1992, 37(1):1–8.

    MathSciNet  MATH  Google Scholar 

  13. Noor KI: On the Hankel determinant problem for strongly close-to-convex functions. J. Nat. Geom. 1997, 11(1):29–34.

    MathSciNet  MATH  Google Scholar 

  14. Noor KI: On certain analytic functions related with strongly close-to-convex functions. Appl. Math. Comput. 2008, 197(1):149–157. 10.1016/j.amc.2007.07.039

    Article  MathSciNet  MATH  Google Scholar 

  15. Noor KI, Al-Bany SA: On Bazilevic functions. Int. J. Math. Math. Sci. 1987, 10(1):79–88. 10.1155/S0161171287000103

    Article  MathSciNet  MATH  Google Scholar 

  16. Noor KI: On analytic functions related with functions of bounded boundary rotation. Comment. Math. Univ. St. Pauli 1981, 30(2):113–118.

    MathSciNet  MATH  Google Scholar 

  17. Noor KI: On meromorphic functions of bounded boundary rotation. Caribb. J. Math. 1982, 1(3):95–103.

    MathSciNet  MATH  Google Scholar 

  18. Noor KI: Hankel determinant problem for the class of functions with bounded boundary rotation. Rev. Roum. Math. Pures Appl. 1983, 28(8):731–739.

    MathSciNet  MATH  Google Scholar 

  19. Noor KI, Al-Naggar IMA: On the Hankel determinant problem. J. Nat. Geom. 1998, 14(2):133–140.

    MathSciNet  MATH  Google Scholar 

  20. Arif M, Noor KI, Raza M: Hankel determinant problem of a subclass of analytic functions. J. Inequal. Appl. 2012., 2012: Article ID 22

    Google Scholar 

  21. Hayami T, Owa S: Generalized Hankel determinant for certain classes. Int. J. Math. Anal. 2010, 4(49–52):2573–2585.

    MathSciNet  MATH  Google Scholar 

  22. Hayami T, Owa S: Applications of Hankel determinant for p -valently starlike and convex functions of order α . Far East J. Appl. Math. 2010, 46(1):1–23.

    MathSciNet  MATH  Google Scholar 

  23. Hayami T, Owa S: Hankel determinant for p -valently starlike and convex functions of order α . Gen. Math. 2009, 17(4):29–44.

    MathSciNet  MATH  Google Scholar 

  24. Janteng A, Halim SA, Darus M: Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 2007, 1(13–16):619–625.

    MathSciNet  MATH  Google Scholar 

  25. Mishra AK, Gochhayat P: Second Hankel determinant for a class of analytic functions defined by fractional derivative. Int. J. Math. Math. Sci. 2008., 2008: Article ID 153280

    Google Scholar 

  26. Mohamed N, Mohamad D, Cik Soh S: Second Hankel determinant for certain generalized classes of analytic functions. Int. J. Math. Anal. 2012, 6(17–20):807–812.

    MathSciNet  MATH  Google Scholar 

  27. Murugusundaramoorthy G, Magesh N: Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant. Bull. Math. Anal. Appl. 2009, 1(3):85–89.

    MathSciNet  MATH  Google Scholar 

  28. Ali RM, Lee SK, Ravichandran V, Supramaniam S: The Fekete-Szegö coefficient functional for transforms of analytic functions. Bull. Iran. Math. Soc. 2009, 35(2):119–142. 276

    MathSciNet  MATH  Google Scholar 

  29. Ali RM, Ravichandran V, Seenivasagan N: Coefficient bounds for p -valent functions. Appl. Math. Comput. 2007, 187(1):35–46. 10.1016/j.amc.2006.08.100

    Article  MathSciNet  MATH  Google Scholar 

  30. Ma WC, Minda D: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, 1992. Conf. Proc. Lecture Notes Anal., vol. I, pp. 157-169. International Press, Cambridge (1922)

    Google Scholar 

  31. Duren PL Grundlehren der Mathematischen Wissenschaften 259. In Univalent Functions. Springer, New York; 1983.

    Google Scholar 

  32. Grenander U, Szegö G California Monographs in Mathematical Sciences. In Toeplitz Forms and Their Applications. University of California Press, Berkeley; 1958.

    Google Scholar 

  33. Rønning F: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 1993, 118(1):189–196.

    Article  MathSciNet  MATH  Google Scholar 

  34. Ali RM, Ravichandran V: Uniformly convex and uniformly starlike functions. Math. News Lett. 2011, 21(1):16–30.

    Google Scholar 

  35. Sokół J, Stankiewicz J: Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 1996, 19: 101–105.

    MathSciNet  MATH  Google Scholar 

  36. Bansal D: Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett. 2013, 26(1):103–107. 10.1016/j.aml.2012.04.002

    Article  MathSciNet  MATH  Google Scholar 

  37. Al-Amiri HS, Reade MO: On a linear combination of some expressions in the theory of the univalent functions. Monatshefte Math. 1975, 80(4):257–264. 10.1007/BF01472573

    Article  MathSciNet  MATH  Google Scholar 

  38. Singh S, Gupta S, Singh S: On a problem of univalence of functions satisfying a differential inequality. Math. Inequal. Appl. 2007, 10(1):95–98.

    MathSciNet  MATH  Google Scholar 

  39. Singh V, Singh S, Gupta S: A problem in the theory of univalent functions. Integral Transforms Spec. Funct. 2005, 16(2):179–186. 10.1080/10652460412331270571

    Article  MathSciNet  MATH  Google Scholar 

  40. Verma S, Gupta S, Singh S: Bounds of Hankel determinant for a class of univalent functions. Int. J. Math. Math. Sci. 2012., 2012: Article ID 147842

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to See Keong Lee.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors jointly worked on the results and they read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Lee, S.K., Ravichandran, V. & Supramaniam, S. Bounds for the second Hankel determinant of certain univalent functions. J Inequal Appl 2013, 281 (2013). https://doi.org/10.1186/1029-242X-2013-281

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-281

Keywords