Open Access

Hankel determinant problem of a subclass of analytic functions

Journal of Inequalities and Applications20122012:22

https://doi.org/10.1186/1029-242X-2012-22

Received: 2 October 2011

Accepted: 8 February 2012

Published: 8 February 2012

Abstract

In this article, we study the Hankel determinant problem of a subclass of analytic functions introduced recently by Arif et al.

2010 Mathematics Subject Classification: 30C45; 30C10.

Keywords

Robertson function strongly Bazilevic functions bounded boundary rotations Hankel determinant

1 Introduction

Let A be the class of analytic function satisfying the condition f (0) = 0, f'(0) - 1 = 0 in the open unit disc E = { z : z < 1 } . By S , S * , C , and K we means the well-known subclasses of A which consist of univalent, starlike, convex, and close-to-convex functions, respectively.

Let V k λ ( σ ) , k 2 , 0 σ < 1 , λ real , λ < π 2 , denote the class of functions f1(z) analytic and locally univalent in , f1(0) = 0, f 1 ( 0 ) = 1 and satisfying
0 2 π Re e i λ ( z f 1 ( z ) ) f 1 ( z ) - σ cos λ ( 1 - σ ) d θ k π cos λ , z = r e i θ .
(1.1)
This class was introduced and studied in details by Moulis [1]. For λ = 0, we obtain the class V k ( σ ) of analytic functions with bounded boundary rotations of order σ studied by Padmanabhan et al. [2] and when σ = 0 and λ = 0, we get the class V k discussed by Paatero [3], see also [48]. Also it can easily be shown that f 1 ( z ) V k λ ( σ ) if and only if there exists f 2 ( z ) V k such that
f 1 ( z ) = ( f 2 ( z ) ) ( 1 - σ ) e - i λ cos λ .
(1.2)

We now consider a class of analytic functions defined by Arif et al. [9] as follows:

Definition 1.1. Let f ( z ) A in E. Then f ( z ) B ̃ k ( λ , σ , β , γ ) , if for k ≥ 2, 0 ≤ β ≤ 1, 0 ≤ γ ≤ 1, λ is real with λ < π 2 there exists a function f 1 ( z ) V k λ ( σ ) , 0 ≤ σ < 1, such that
| arg { z 1 γ f ( z ) f ( z ) ( f ( z ) f 1 ( z ) ) γ } | β π 2 , z E .

By giving specific values to the parameters k, σ, λ, β, and γ in B ̃ k ( λ , σ , β , γ ) , we obtain many important subclasses studied by various authors in earlier articles, see [1016].

Using (1.1) and (1.2), we have
z f ( z ) = z γ ( f ( z ) ) 1 - γ ( f 1 ( z ) ) γ p β ( z ) ,
(1.3)

where f 1 ( z ) V k λ ( σ ) and p(z) belongs to the class P of functions whose real part is positive.

Throughout in this article, we shall assume, unless otherwise stated, that k ≥ 2, 0 ≤ β ≤ 1, 0 < γ ≤ 1, λ is real with λ < π 2 , 0 ≤ σ < 1.

In [17], the q th Hankel determinant H q ( n ) , q ≥ 1, n ≥ 1, for a function f ( z ) A is stated by Noonan and Thomas as:

Definition 1.2. Let f ( z ) A . Then the q th Hankel determinant of f (z) is defined for q ≥ 1, n ≥ 1 by
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 .
(1.4)

The Hankel determinant plays an important role, for instance, in the study of the singularities by Hadamard, see [[18], p. 329], Edrei [19] and in the study of power series with integral coefficients by Polya [[20], p. 323], Cantor [21], and many others.

In this article, we shall determine the rate of growth of the Hankel determinant H q ( n ) for f ( z ) B ̃ k ( λ , σ , β , γ ) with 0 < β < 2, as n → ∞. This determinant has been considered by several authors. That is, Noor [22] determined the rate of growth of H q ( n ) as n → ∞ for a function f(z) belongs to the class V k . Pommerenke in [23] studied the Hankel determinant for starlike functions. The Hankel determinant problem for other interesting classes of analytic functions were discussed by Noor [11, 12, 24].

Lemma 1.1. Let f ( z ) A . Let the q th Hankel determinant of f (z) for q ≥ 1, n ≥ 1 be defined by (1.4). Then, writting Δ j (n) = Δ j (n, z1, f(z)), we have
H q ( n ) = Δ 2 q - 2 ( n ) Δ 2 q - 3 ( n + 1 ) Δ q - 1 ( n + q - 1 ) Δ 2 q - 3 ( n + 1 ) Δ 2 q - 4 ( n + 2 ) Δ q - 2 ( n + q - 2 ) Δ q - 1 ( n + q - 1 ) Δ q - 2 ( n + q - 2 ) Δ q ( n + 2 q - 2 ) ,
where with Δ0(n) = a n , we define for j ≥ 1,
Δ j ( n , z 1 , f ( z ) ) = Δ j - 1 ( n , z 1 , f ( z ) ) - Δ j - 1 ( n + 1 , z 1 , f ( z ) ) .
(1.5)
Lemma 1.2. With z 1 = n n + 1 y and v ≥ 0 any integer,
Δ j ( n + v , z 1 , z f ( z ) ) = m = 0 j j m y m ( v - ( m - 1 ) n ) ( n + 1 ) m Δ j - m ( n + m + v , f ( z ) ) .

Lemmas 1.1 and 1.2 are due to Noonan and Thomas [17].

Lemma 1.3. Let h1(z) be starlike univalent function in . Then
  1. (i)

    there exists a z1 with |z1| = r such that for all z, |z| = r

     
z - z 1 h 1 ( z ) 2 r 2 1 - r 2 ,
   see [25]
(ii)
r ( 1 + r ) 2 h 1 ( z ) r ( 1 + r ) 2 ,

   see[26].

2 Hankel determinant problem

Theorem 2.1. Let f ( z ) B ̃ k ( λ , σ , β , γ ) with 0 < β < 2 and let the q th Hankel determinant H q ( n ) of f(z) be defined as in (1.4). Then
H q ( n ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 2 , q = 1 n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 1 q - q 2 , q 2 , k 8 ( q - 1 ) ( 1 - σ ) γ cos 2 λ - 2 ,

where k > 4 j + 2 - β ( 1 - σ ) γ cos 2 λ - 2 and O(1) is a constant depending on k, λ, β, σ, γ, and j only.

Proof. It is well known [1] that for starlike functions h1(z) and h2(z)
f 1 ( z ) = ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ .
(2.1)
Using (2.1) in (1.3), we have
z f ( z ) = z γ ( f ( z ) ) 1 - γ ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ p β ( z ) .
(2.2)

where p ( z ) P .

Let F (z) = zf'(z). Then for j ≥ 1, z1 any non-zero complex and z = re , consider Δ j (n, z1, F(z)) as defined by (1.5). Then
Δ j ( n , z 1 , F ( z ) ) = 1 2 π r n + j 0 2 π z - z 1 j F ( z ) e i ( n + j ) θ d θ ,
and by using (2.2), we have
Δ j ( n , z 1 , F ( z ) ) 1 2 π r n + j 0 2 π z - z 1 j z γ f ( z ) 1 - γ ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ p ( z ) β d θ ( M ( r ) ) 1 - γ 2 π r n + j 0 2 π z - z 1 j ( h 1 ( z ) z ) k 4 + 1 2 ( h 2 ( z ) z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ p ( z ) β d θ .
(2.3)
Since
( h 1 ( z ) / z ) k 4 + 1 2 ( h 2 ( z ) / z ) k 4 - 1 2 ( 1 - σ ) γ e - i λ cos λ ( h 1 ( z ) / z ) k 4 + 1 2 ( h 2 ( z ) / z ) k 4 - 1 2 γ ( 1 - σ ) cos 2 λ e γ ( 1 - σ ) γ π 2 sin 2 λ 2 = ( h 1 ( z ) / z ) k 4 + 1 2 γ ( 1 - σ ) cos 2 λ ( h 2 ( z ) / z ) k 4 - 1 2 γ ( 1 - σ ) cos 2 λ c 1 , say
therefore (2.3) becomes
Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 π r n + j 0 2 π z - z 1 j ( h 1 ( z ) / z ) k 4 + 1 2 γ ( 1 - σ ) cos 2 λ ( h 2 ( z ) / z ) k 4 - 1 2 γ ( 1 - σ ) cos 2 λ p ( z ) β d θ .
Now using Lemma 1.3, we have
Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 π r n + j ( 1 + r ) 2 r k 4 - 1 2 γ ( 1 - σ ) cos 2 λ 1 r γ ( 1 - σ ) cos 2 λ 2 r 2 1 - r 2 j 0 2 π ( h 1 ( z ) ) k 4 + 1 2 γ ( 1 - σ ) cos 2 λ - j p ( z ) β d θ .
The well-known Holder's inequality will give us
Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 k 4 - 1 2 γ ( 1 - σ ) cos 2 λ + j r n - j + k 4 + 1 2 γ ( 1 - σ ) cos 2 λ 1 1 - r j 1 2 π 0 2 π p ( z ) 2 d θ β 2 1 2 π 0 2 π | h 1 ( z ) | k 2 + 1 γ ( 1 - σ ) cos 2 λ - 2 j 2 - β d θ 2 - β 2 .
(2.4)
Also, it is known [15] that, for p ( z ) P , z E,
1 2 π 0 2 π p ( z ) 2 d θ 1 + 3 r 2 1 - r 2 .
(2.5)
Using (2.5) in (2.4), we obtain
Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 k 4 - 1 2 γ ( 1 - σ ) cos 2 λ + j r n - j + k 4 + 1 2 γ ( 1 - σ ) cos 2 λ 1 1 - r j 1 + 3 r 2 1 - r 2 β 2 1 2 π 0 2 π | ( h 1 ( z ) ) | k 4 + 1 γ ( 1 - σ ) cos 2 λ - 2 j 2 - β d θ 2 - β 2 .
Therefore, we can write
Δ j ( n , z 1 , F ( z ) ) c 1 ( M ( r ) ) 1 - γ 2 k 4 - 1 2 γ ( 1 - σ ) cos 2 λ + β + j r n - j + k 4 + 1 2 γ ( 1 - σ ) cos 2 λ 1 1 - r j + β 2 r k 4 + 1 2 γ ( 1 - σ ) cos 2 λ - j 1 2 π 0 2 π d θ 1 - r e i θ ( k + 2 ) γ ( 1 - σ ) cos 2 λ - 4 j 2 - β 2 - β 2 .
Now using a subordination result for starlike functions, we have
Δ j ( n , z 1 , F ( z ) ) c 2 ( M ( r ) ) 1 - γ r n 1 1 - r j + β 2 1 1 - r γ ( k + 2 ) ( 1 - σ ) cos 2 λ - 4 j 2 - β - 1 2 - β 2 , = c 2 ( M ( r ) ) 1 - γ r n 1 1 - r γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 1 - j ,

where c2 is a constant depending on k, λ, β, σ, γ, j only and γ(k + 2) (1 - σ) cos2 λ - 4j > 2 - β.

Applying Lemma 1.2 and putting z 1 = n n + 1 e i θ n , ( n ) , r = 1 - 1 n , we have for k 4 j - β + 2 ( 1 - σ ) γ cos 2 λ - 2 ,
Δ j n , e i θ n , f ( z ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - j - 2 ,

where O(1) is a constant depending on k, λ, β, σ, γ, and j only.

We now estimate the rate of growth of H q ( n ) .

For q = 1, H q ( n ) = a n = Δ 0 ( n ) and
H 1 ( n ) = a n = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 2 .
For q ≥ 2, we use similar argument due to Noonan and Thomas [17] together with Lemma 1.1 to have
H q ( n ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 ( 1 - σ ) cos 2 λ + β - 1 q - q 2 , k 8 ( q - 1 ) ( 1 - σ ) γ cos 2 λ - 2

and O(1) depends only on k, λ, β, σ, γ, and j.

For choosing different values of λ, β, σ, and γ in Theorem 2.1, we obtain the following results discussed by Noor [10, 11] and Noor et al. [14].

Corollary 2.1. For λ = 0, β = 1, σ = 0, f ( z ) B k ( γ ) , where the class B k ( γ ) was introduced by Noor et al. [14] and
H q ( n ) = O ( 1 ) ( M ( r ) ) 1 - γ n γ k 2 + 1 - 1 , q = 1 n γ k 2 + 1 q - q 2 , q 2 , k 8 ( q - 1 ) γ - 2

where O(1) is a constant depending only on k and γ.

Corollary 2.2. For λ = 0 , γ = 1 , f ( z ) T ̃ k ( β , σ ) and
H q ( n ) = O ( 1 ) n ( 1 - σ ) k 2 + 1 + β - 2 , q = 1 n ( 1 - σ ) k 2 + 1 + β - 1 q - q 2 , q 2 , k 8 ( q - 1 ) ( 1 - σ ) - 2

where O(1) is a constant depends on k, σ, and β only.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Abdul Wali Khan University
(2)
COMSATS Institute of Information Technology

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© Arif et al; licensee Springer. 2012

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