Coefficient, Distortion and Growth Inequalities for Certain Close-to-Convex Functions

In the present investigation, certain subclasses of close-to-convex functions are investigated. In particular, we obtain an estimate for the Fekete-Szeg\"o functional for functions belonging to the class, distortion, growth estimates and covering theorems.


Introduction
Let D := {z ∈ C : |z| < 1} be the open unit disk in the complex plane C. Let A be the class of analytic functions defined on D and normalized by the conditions f (0) = 0 and f ′ (0) = 1. Let S be the subclass of A consisting of univalent functions. Sakaguchi [7] introduced a class of functions called starlike functions with respect to symmetric points; they are the functions f ∈ A satisfying the condition These functions are close-to-convex functions. This can be easily seen by showing that the function (f (z) − f (−z))/2 is a starlike function in D. Motivated by the class of starlike functions with respect to symmetric points, Gao and Zhou [2] discussed a class K s of close-to-convex univalent functions. A function f ∈ K s if it satisfy the following inequality Re z 2 f ′ (z) g(z)g(−z) < 0 (z ∈ D) for some function g ∈ S * (1/2). The idea here is to replace the average of f (z) and −f (−z) by the corresponding product −g(z)g(−z) and the factor z is included to normalize the expression so that −z 2 f ′ (z)/(g(z)g(−z)) takes the value 1 at z = 0. To make the functions univalent, it is further assumed that g is starlike of order 1/2 so that the function −g(z)g(−z)/z is starlike which in turn implies the close-to-convexity of f . For some recent works on the problem, see [11,9,10,12]. In stead of requiring the quantity −z 2 f ′ (z)/(g(z)g(−z)) to lie in the right-half plane, we can consider more general regions. This could be done via subordination between analytic functions.
Let f and g be analytic in D. Then f is subordinate to g, written f ≺ g or f (z) ≺ g(z) (z ∈ D), if there is an analytic function w(z), with w(0) = 0 and |w(z)| < 1, such that f (z) = g(w(z)). In particular, if g is univalent in D, then f is subordinate to g, This work was completed during the third author's visit to Pukyong National University and the support and the hospitality of Prof. Cho is gratefully acknowledged.
if f (0) = g(0) and f (D) ⊆ g(D). In terms of subordination, a general class K s (ϕ) is introduced in the following definition. Definition 1. [11] For a function ϕ with positive real part, the class K s (ϕ) consists of functions f ∈ A satisfying for some function g ∈ S * (1/2).
This class was introduced by Wang, Gao and Yuan [11]. A special subclass K s (γ) := K s (ϕ) where ϕ(z) := (1 + (1 − 2γ)z)(1 − z), 0 ≤ γ < 1, was recently investigated by Kowalczyk and Leś-Bomba [5]. They proved the sharp distortion and growth estimates for functions in K s (γ) as well as some sufficient conditions in terms of the coefficient for function to be in this class K s (γ).
In the present investigation, we obtain a sharp estimate for the Fekete-Szegö functional for functions belonging to the class K s (ϕ). In addition, we also investigate the corresponding problem for the inverse functions for functions belonging to the class K s (ϕ). Also distortion, growth estimates as well as covering theorem are derived. Some connection with earlier works are also indicated.

Fekete-Szegö Inequality
In this section, we assume that the function ϕ(z) is an univalent analytic function with positive real part that maps the unit disk D onto a starlike region which symmetric with respect to real axis and is normalized by ϕ(0) = 1 and ϕ ′ (0) > 0. In such case, the function ϕ has an expansion of the form ϕ(z) = 1 + B 1 z + B 2 z 2 + · · · , B 1 > 0.
It is known that every univalent function f has an inverse f −1 , defined by Corollary 2.1. Let f ∈ K s (ϕ). Then the coefficients d 2 and d 3 of the inverse function f −1 (w) = w + d 2 w 2 + d 3 w 3 + · · · satisfy the inequality Proof. A calculation shows that the inverse function f −1 has the following Taylor's series expansion: From this expansion, it follows that d 2 = a 2 and d 3 = 2a 2 2 − a 3 and hence |d 3 − µd 2 2 | = |a 3 − (2 − µ)a 2 2 |. Our result follows at once from this identity and Theorem 2.1.

Distortion and growth theorems
The second coefficient of univalent function plays an important role in the theory of univalent function; for example, this leads to the distortion and growth estimates for univalent functions as well as the rotation theorem. In the next theorem, we derive the distortion and growth estimates for the functions in the class K s (ϕ). In particular, if we let r → 1− in the growth estimate, it gives the bound |a 2 | ≤ B 1 /2 for the second coefficient of functions in K s (ϕ). If f ∈ K s (ϕ), the following sharp inequalities holds: Proof. Since the function f ∈ K s (ϕ), there is a normalized analytic function g ∈ S * (1/2) such that Define the function G : D → C by the equation Then it is clear that G is odd starlike function in D and therefore r 1 + r 2 ≤ |G(z)| ≤ r 1 − r 2 (|z| = r < 1) Using the definition of subordination between analytic function, and the equation (3.1), we see that there is an analytic function w(z) with |w(z)| ≤ |z| such that or zf ′ (z) = G(z)ϕ(w(z)). Since w(D) ⊂ D, we have, by maximum principle for harmonic functions, The other inequality for |f ′ (z)| is similar. Since the function f is univalent, the inequality for |f (z)| follows from the corresponding inequalities for |f ′ (z)| by Privalov's Theorem [3, Theorem 7, p. 67].
To prove the sharpness of our results, we consider the functions Define the function g 0 and g 1 by g 0 (z) = z/(1 − z) and g 1 (z) = z/ √ 1 + z 2 . These functions are clearly starlike functions of order 1/2. Also a calculations shows that Thus the function f 0 satisfies the subordination (1.1) with g 0 while the function f 1 satisfies it with g 1 ; therefore, these functions belong to the class K s (ϕ). It is clear that the upper estimates for |f ′ (z)| and |f (z)| are sharp for the function f 0 given in (3.2) while the lower estimates are sharp for f 1 given in (3.2).
Remark 3.1. We note that Xu et al. [12] also obtained a similar estimates and our results differ from their in the hypothesis. Also we have shown that the results are sharp. Our hypothesis is same as the one assumed by Ma and Minda [6].

A Subordination Theorem
It is well-known [8] ∈ (0, δ), where δ is a positive real number; also the function is starlike with respect to symmetric points if (1 − t)f (z) + tf (−z) ≺ f (z). In the following theorem, we extend these results to the class K s . The proof of our result is based on the following version of a lemma of Stankiewicz [8].
The function F is analytic in D (of course, one has to redefine the function F at z = 0 where it has removable singularity.) Since all the hypothesis of Lemma 4.1 are satisfied, we have Re g(z)g(−z) z 2 f ′ (z) < 0.
Since a function p(z) has negative real part if and only if its reciprocal 1/p(z) has negative real part, we have Re Thus, f ∈ K s .