Bounds for the second Hankel determinant of certain univalent functions

  • See Keong Lee
  • V Ravichandran
  • Shamani Supramaniam
Open Access
Research

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

Cite this article as:
Lee, S.K., Ravichandran, V. & Supramaniam, S. J Inequal Appl (2013) 2013: 281. doi:10.1186/1029-242X-2013-281
Part of the following topical collections:
  1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

Abstract

The estimates for the second Hankel determinant a 2 a 4 a 3 2 Open image in new window of the analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + Open image in new window , for which either z f ( z ) / f ( z ) Open image in new window or 1 + z f ( z ) / f ( z ) Open image in new window 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 Open image in new window denote the class of all analytic functions
f ( z ) = z + a 2 z 2 + a 3 z 3 + Open image in new window
(1)
defined on the open unit disk D : = { z C : | z | < 1 } Open image in new window. The Hankel determinants H q ( n ) Open image in new window ( n = 1 , 2 , Open image in new window , q = 1 , 2 , Open image in new window) 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 ) . Open image in new window
Hankel determinants are useful, for example, in showing that a function of bounded characteristic in D Open image in new window, 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 , ) , Open image in new window

where β > 1 / 4000 Open image in new window and K depends only on q. Later, Hayman [6] proved that | H 2 ( n ) | < A n 1 / 2 Open image in new window ( n = 1 , 2 , Open image in new window ; A an absolute constant) for areally mean univalent functions. In [7, 8, 9], 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 [12, 13, 14]. Noor studied the Hankel determinant of Bazilevic functions in [15] and of functions with bounded boundary rotation in [16, 17, 18, 19]. In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [20, 21, 22, 23, 24, 25, 26, 27]. The Hankel determinant H 2 ( 1 ) = a 3 a 2 2 Open image in new window is the well-known Fekete-Szegö functional. For results related to this functional, see [28, 29]. The second Hankel determinant H 2 ( 2 ) Open image in new window is given by H 2 ( 2 ) = a 2 a 4 a 3 2 Open image in new window.

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

Lemma 1 [31]

If the function p P Open image in new windowis given by the series
p ( z ) = 1 + c 1 z + c 2 z 2 + c 3 z 3 + , Open image in new window
(2)
then the following sharp estimate holds:
| c n | 2 ( n = 1 , 2 , ) . Open image in new window
(3)

Lemma 2 [32]

If the function p P Open image in new windowis given by the series (2), then
2 c 2 = c 1 2 + x ( 4 c 1 2 ) , Open image in new window
(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 , Open image in new window
(5)

for somex, zwith | x | 1 Open image in new windowand | z | 1 Open image in new window.

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

Subclasses of starlike functions are characterized by the quantity z f ( z ) / f ( z ) Open image in new window lying in some domain in the right half-plane. For example, f is strongly starlike of order β if z f ( z ) / f ( z ) Open image in new window lies in a sector | arg w | < β π / 2 Open image in new window, while it is starlike of order α if z f ( z ) / f ( z ) Open image in new window lies in the half-plane Re w > α Open image in new window. 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 Open image in new window 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 ) . Open image in new window
(6)
The class S ( φ ) Open image in new window of Ma-Minda starlike functions with respect toφ consists of functions f A Open image in new window satisfying the subordination
z f ( z ) f ( z ) φ ( z ) . Open image in new window
For the function φ given by φ α ( z ) : = ( 1 + ( 1 2 α ) z ) / ( 1 z ) Open image in new window , 0 < α 1 Open image in new window, the class S ( α ) : = S ( φ α ) Open image in new window 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 . Open image in new window
Then the class
S P : = S ( φ PAR ) = { f A : Re ( z f ( z ) f ( z ) ) > | z f ( z ) f ( z ) 1 | } Open image in new window
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 Open image in new window, the class
S β : = S ( ( 1 + z 1 z ) β ) = { f A : | arg ( z f ( z ) f ( z ) ) | < β π 2 } Open image in new window
is the familiar class of strongly starlike functions of orderβ. The class
S L : = S ( 1 + z ) = { f A : | ( z f ( z ) f ( z ) ) 2 1 | < 1 } Open image in new window

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

Theorem 1Let the function f S ( φ ) Open image in new windowbe given by (1).
  1. 1.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy the conditions
    | B 2 | B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | 3 B 1 2 0 , Open image in new window
     
then the second Hankel determinant satisfies
| a 2 a 4 a 3 2 | B 1 2 4 . Open image in new window
  1. 2.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
or the conditions
| B 2 | B 1 , | 4 B 1 B 3 B 1 4 3 B 2 2 | 3 B 1 2 0 , Open image in new window
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 | . Open image in new window
  1. 3.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
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 ) . Open image in new window
Proof Since f S ( φ ) Open image in new window, there exists an analytic function w with w ( 0 ) = 0 Open image in new window and | w ( z ) | < 1 Open image in new window in  D Open image in new window such that
z f ( z ) f ( z ) = φ ( w ( z ) ) . Open image in new window
(7)
Define the functions p 1 Open image in new window by
p 1 ( z ) : = 1 + w ( z ) 1 w ( z ) = 1 + c 1 z + c 2 z 2 + , Open image in new window
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 + ) . Open image in new window
(8)
Then p 1 Open image in new window is analytic in D Open image in new window with p 1 ( 0 ) = 1 Open image in new window and has a positive real part in D Open image in new window. 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 + . Open image in new window
(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 + , Open image in new window
(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 ] . Open image in new window
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 ] . Open image in new window
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 . Open image in new window
(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 | . Open image in new window
(12)
Since the function p ( e i θ z ) Open image in new window ( θ R Open image in new window) is in the class P Open image in new window for any p P Open image in new window, there is no loss of generality in assuming c 1 > 0 Open image in new window. Write c 1 = c Open image in new window, c [ 0 , 2 ] Open image in new window. Substituting the values of c 2 Open image in new window and c 3 Open image in new window 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 | . Open image in new window
Replacing | x | Open image in new window by μ and substituting the values of d 1 Open image in new window, d 2 Open image in new window, d 3 Open image in new window and d 4 Open image in new window 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 ) . Open image in new window
(13)
Note that for ( c , μ ) [ 0 , 2 ] × [ 0 , 1 ] Open image in new window, differentiating F ( c , μ ) Open image in new window 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 ) ] . Open image in new window
(14)
Then, for 0 < μ < 1 Open image in new window and for any fixed c with 0 < c < 2 Open image in new window, it is clear from (14) that F μ > 0 Open image in new window, that is, F ( c , μ ) Open image in new window is an increasing function of μ. Hence, for fixed c [ 0 , 2 ] Open image in new window, the maximum of F ( c , μ ) Open image in new window occurs at μ = 1 Open image in new window, and
max F ( c , μ ) = F ( c , 1 ) G ( c ) . Open image in new window
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 ] . Open image in new window
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 . Open image in new window
(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 , Open image in new window
(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 , Open image in new window

where P, Q, R are given by (15). □

Remark 1 When B 1 = B 2 = B 3 = 2 Open image in new window, Theorem 1 reduces to [[24], Theorem 3.1].

Corollary 1
  1. 1.

    If f S ( α ) Open image in new window, then | a 2 a 4 a 3 2 | ( 1 α ) 2 Open image in new window.

     
  2. 2.

    If f S L Open image in new window, then | a 2 a 4 a 3 2 | 1 / 16 = 0.0625 Open image in new window.

     
  3. 3.

    If f S P Open image in new window, then | a 2 a 4 a 3 2 | 16 / π 4 0.164255 Open image in new window.

     
  4. 4.

    If f S β Open image in new window, then | a 2 a 4 a 3 2 | β 2 Open image in new window.

     
Definition 2 Let φ : D C Open image in new window be analytic, and let φ ( z ) Open image in new window be given as in (6). The class C ( φ ) Open image in new window of Ma-Minda convex functions with respect toφ consists of functions f satisfying the subordination
1 + z f ( z ) f ( z ) φ ( z ) . Open image in new window
Theorem 2Let the function f C ( φ ) Open image in new windowbe given by (1).
  1. 1.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
then the second Hankel determinant satisfies
| a 2 a 4 a 3 2 | B 1 2 36 . Open image in new window
  1. 2.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
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 , Open image in new window
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 | . Open image in new window
  1. 3.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
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 ) . Open image in new window
Proof Since f C ( φ ) Open image in new window, there exists an analytic function w with w ( 0 ) = 0 Open image in new window and | w ( z ) | < 1 Open image in new window in D Open image in new window such that
1 + z f ( z ) f ( z ) = φ ( w ( z ) ) . Open image in new window
(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 + , Open image in new window
(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 ] . Open image in new window
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 ] . Open image in new window
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 , Open image in new window
(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 | . Open image in new window
(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 | . Open image in new window
Replacing | x | Open image in new window by μ and then substituting the values of d 1 Open image in new window, d 2 Open image in new window, d 3 Open image in new window and d 4 Open image in new window 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 ) . Open image in new window
(21)
Again, differentiating F ( c , μ ) Open image in new window 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 ) ] . Open image in new window
(22)
It is clear from (22) that F μ > 0 Open image in new window. Thus F ( c , μ ) Open image in new window is an increasing function of μ for 0 < μ < 1 Open image in new window and for any fixed c with 0 < c < 2 Open image in new window. So, the maximum of F ( c , μ ) Open image in new window occurs at μ = 1 Open image in new window and
max F ( c , μ ) = F ( c , 1 ) G ( c ) . Open image in new window
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 ] . Open image in new window
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 . Open image in new window
(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 , Open image in new window

where P, Q, R are given in (23). □

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

3 Further results on the second Hankel determinant

Definition 3 Let φ : D C Open image in new window be analytic, and let φ ( z ) Open image in new window be as given in (6). Let 0 γ 1 Open image in new window and τ C { 0 } Open image in new window. A function f A Open image in new window is in the class R γ τ ( φ ) Open image in new window if it satisfies the following subordination:
1 + 1 τ ( f ( z ) + γ z f ( z ) 1 ) φ ( z ) . Open image in new window
Theorem 3Let 0 γ 1 Open image in new window, τ C { 0 } Open image in new window, and let the functionfas in (1) be in the class R γ τ ( φ ) Open image in new window. Also, let
p = 8 9 ( 1 + γ ) ( 1 + 3 γ ) ( 1 + 2 γ ) 2 . Open image in new window
  1. 1.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
then the second Hankel determinant satisfies
| a 2 a 4 a 3 2 | | τ | 2 B 1 2 9 ( 1 + 2 γ ) 2 . Open image in new window
  1. 2.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
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 , Open image in new window
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 | . Open image in new window
  1. 3.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
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 ) ) . Open image in new window
Proof For f R γ τ ( φ ) Open image in new window, there exists an analytic function w with w ( 0 ) = 0 Open image in new window and | w ( z ) | < 1 Open image in new window in D Open image in new window such that
1 + 1 τ ( f ( z ) + γ z f ( z ) 1 ) = φ ( w ( z ) ) . Open image in new window
(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 + . Open image in new window
(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 ] . Open image in new window
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 ) ] } , Open image in new window
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 ) ] | , Open image in new window
(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 . Open image in new window

It can be easily verified that p [ 64 81 , 8 9 ] Open image in new window for 0 γ 1 Open image in new window.

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 . Open image in new window
(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 | . Open image in new window
(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 | . Open image in new window
Application of the triangle inequality, replacement of | x | Open image in new window by μ and substituting the values of d 1 Open image in new window, d 2 Open image in new window, d 3 Open image in new window and d 4 Open image in new window 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 ) , Open image in new window
(29)

where α = 2 Open image in new window, β = 2 p / ( 1 p ) > 2 Open image in new window.

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

where P, Q, R are given in (30). □

Remark 3 For the choice φ ( z ) : = ( 1 + A z ) / ( 1 + B z ) Open image in new window with 1 B < A 1 Open image in new window, Theorem 3 reduces to [[36], Theorem 2.1].

Definition 4 Let φ : D C Open image in new window be analytic, and let φ ( z ) Open image in new window be as given in (6). For a fixed real number α, the function f A Open image in new window is in the class G α ( φ ) Open image in new window if it satisfies the following subordination:
( 1 α ) f ( z ) + α ( 1 + z f ( z ) f ( z ) ) φ ( z ) . Open image in new window

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

Theorem 4Let the functionfgiven by (1) be in the class G α ( φ ) Open image in new window, 0 α 1 Open image in new window. Also, let
p = 8 9 ( 1 + 2 α ) ( 1 + α ) . Open image in new window
  1. 1.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
then the second Hankel determinant satisfies
| a 2 a 4 a 3 2 | B 1 2 9 ( 1 + α ) 2 . Open image in new window
  1. 2.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
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 , Open image in new window
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 ) . Open image in new window
  1. 3.
    If B 1 Open image in new window, B 2 Open image in new windowand B 3 Open image in new windowsatisfy 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 , Open image in new window
     
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 ) ] . Open image in new window
Proof For f G α ( φ ) Open image in new window, 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 ) ] | , Open image in new window
(31)
where
T = B 1 128 ( 1 + α ) ( 1 + 2 α ) and p = 8 9 ( 1 + 2 α ) ( 1 + α ) . Open image in new window
It can be easily verified that for 0 α 1 Open image in new window, p [ 8 9 , 4 3 ] Open image in new window. 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 . Open image in new window
(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 | . Open image in new window
(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 ) , Open image in new window
(34)
and for fixed c [ 0 , 2 ] Open image in new window, max F ( c , μ ) = F ( c , 1 ) G ( c ) Open image in new window 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 ] . Open image in new window
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 . Open image in new window
(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 , Open image in new window

where P, Q, R are given in (35). □

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

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.

Copyright information

© Lee et al.; licensee Springer 2013

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.

Authors and Affiliations

  • See Keong Lee
    • 1
  • V Ravichandran
    • 2
  • Shamani Supramaniam
    • 1
  1. 1.School of Mathematical SciencesUniversiti Sains MalaysiaPenangMalaysia
  2. 2.Department of MathematicsUniversity of DelhiDelhiIndia

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