Coefﬁcient Estimates for a Subclass of Starlike Functions

: In this note, we consider a subclass H 3/2 ( p ) of starlike functions f with f (cid:48)(cid:48) ( 0 ) = p for a prescribed p ∈ [ 0, 2 ] . Usually, in the study of univalent functions, estimates on the Taylor coefﬁcients, Fekete–Szegö functional or Hankel determinats are given. Another coefﬁcient problem which has attracted considerable attention is to estimate the moduli of successive coefﬁcients | a n + 1 | − | a n | . Recently, the related functional | a n + 1 − a n | for the initial successive coefﬁcients has been investigated for several classes of univalent functions. We continue this study and for functions f ( z ) = z + ∑ ∞ n = 2 a n z n ∈ H 3/2 ( p ) , we investigate upper bounds of initial coefﬁcients and the difference of moduli of successive coefﬁcients | a 3 − a 2 | and | a 4 − a 3 | . Estimates of the functionals | a 2 a 4 − a 23 | and | a 4 − a 2 a 3 | are also derived. The obtained results expand the scope of the theoretical results related with the functional | a n + 1 − a n | for various subclasses of univalent functions.


Introduction
As usual, denote by A the family of all normalized analytic functions f (z) = z + ∞ ∑ n=2 a n z n (1) defined in the open unit disk U = {z ∈ C : |z| < 1} and let S be the subset of univalent functions in A. Let be the class of starlike functions of order α (see [1]). The family S * (0) = S * is the well-known class of starlike functions in U. Denote by K the class of convex functions in U, i.e., In 1997, Silverman [2] investigated the properties of a subclass of A, defined in terms of the quotient (1 + z f (z) f (z) )/ z f (z) f (z) . More precisely, for 0 < b ≤ 1, Silverman's class G b is defined as follows In [2], Silverman proved that all functions in G b are starlike of order 2/(1 + √ 1 + 8b). Lately, Obradović and Tuneski [3] improved the results of Silverman and obtained new starlike criteria for the class G b . Among others, they obtained the next result.
Starting from the above result, we consider the following subclass of S * : It is not difficult to show that for |a| < 1/2, the function f (z) = z 1+az ∈ H 3/2 . During the years, great attention has been given to the difference of moduli of successive coefficients ||a n+1 | − |a n || of a function in S * . In 1963, Hayman [4] proved that ||a n+1 | − |a n || ≤ A (A ≥ 1) for f ∈ S * . Further, Leung [5] proved Pommerenke's [6] conjecture ||a n+1 | − |a n || ≤ 1 for f ∈ S * . Estimates of the difference of moduli of successive coefficients, for certain subclasses of S * , were also obtained by Z. Ye [7,8], and others (see, for example [9]). Moreover, since ||a n+1 | − |a n || < |a n+1 − a n |, the study of the functional |a n+1 − a n | has been also considered. For all functions f ∈ K, Robertson [10] obtained the inequality |a n+1 − a n | ≤ 2n+1 3 |a 2 − 1| and proved that the factor (2n + 1)/3 cannot be replaced by any smaller number independent of f . Recently, Li and Sugawa [11] investigated the problem of maximizing the functionals |a 3 − a 2 | and |a 4 − a 3 | for a refined subclass of K, K(p) = { f ∈ K : f (0) = p, p ∈ [0, 2]}. The upper bounds of the same funtionals |a 3 − a 2 | and |a 4 − a 3 | for various subclasses of univalent functions were obtained by Peng and Obradović [12] and L. Shi et al. [13].
Motivated by the results given in [11][12][13], in the present paper we obtain upper bounds of the initial coefficients and upper bounds of |a 3 − a 2 | and |a 4 − a 3 | for a refined subclass of H 3/2 , defined by where p is a given number satisfying −2 ≤ p ≤ 2. Moreover, upper bounds for functionals |a 2 a 4 − a 2 3 | and |a 4 − a 2 a 3 | for the same subclass H 3/2 (p) are also derived. The first functional is known as the second Hankel determinant, studied in many papers (see [14][15][16][17]). The second functional is a particular case of the generalized Zalcman functional, investigated by Ma [18], Efraimidis and Vukotić [19] and many others (see [20][21][22][23]).

Preliminary Results
Let P be the class of analytic functions p with a positive real part in U, satisfying the condition p(0) = 1. A member p ∈ P is called a Carathéodory function and has the Taylor series expansion It is known that |p n | ≤ 2 for p ∈ P and n = 1, 2, . . . (see [1]). In order to prove our main results, the following two lemmas will be used. The first is due to Libera and Złotkiewicz [24,25]. Lemma 1. Let −2 ≤ p 1 ≤ 2 and p 2 , p 3 ∈ C. Then there exists a function p ∈ P of the form (7) such that and for some x, y ∈ C with |x| ≤ 1 and |y| ≤ 1.
The second lemma is a special case of a more general result due to Ohno and Sugawa [26] (see also [11]).

Lemma 2.
For some given real numbers a, b, c, let If ac ≥ 0, then If ac < 0, then where

Main Results
We begin this section by finding the absolute values of the first three initial coefficients in the function class H 3/2 (p). Theorem 2. Let 0 ≤ p ≤ 2 and let f , given by (1), be in the class H 3/2 (p). Then Proof.

Denote by
Then, by using (15) and (16), a simple computation shows that and sup where a 3 ( f ) and a 4 ( f ) are the coefficients of f .
The upper bounds for the difference of the initial coefficients for the class H 3/2 (p) are given in the next result. Theorem 3. Let 0 ≤ p ≤ 2 and f (z) = z + ∑ ∞ n=2 a n z n ∈ H 3/2 (p). Then, and Proof. Proceeding as in the proof of Theorem 2 and making use of (22), we obtain Now, we shall find the estimate of |a 4 − a 3 |. For this, using (22) and (23), we have where Y(a, b, c) is given by (10) and Since 0 ≤ p ≤ 2, we have a > 0. Note also that for p ∈ [0, 2] the inequality |b| ≥ 2(1 − |c|) is equivalent to Making use of Lemma 2, a computation gives Therefore, we get In view of the estimates (25) and (26), we deduce that where H 3/2 (+) is given by (24) and a 2 ( f ), a 3 ( f ), 4 2 ( f ) are the coefficients of f .
In the next result, we obtain the estimates of the functionals |a 2 a 4 − a 2 3 | and |a 4 − a 2 a 3 |.