Certain properties of a new subclass of close-to-convex functions

In the present paper we introduce and investigate an interesting subclass \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{K}_{s}^{(k)}(\lambda,h)}$$\end{document} of analytic and close-to-convex functions in the open unit disk \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{U}}$$\end{document} . For functions belonging to the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{K}_{s}^{(k)}(\lambda,h)}$$\end{document} , we derive several properties as the inclusion relationships and distortion theorems. The various results presented here would generalize many known recent results.


Introduction and preliminaries
Let S denote the class of functions of the form f (z) = z + ∞ n=2 a n z n , (1.1) In many earlier investigations, various interesting subclasses of the class S have been studied from a number of different view points. In particular, Gao and Zhou [2] introduced the next subclass K s of analytic functions, which is indeed a subclass of close-to-convex functions: Definition 1.1 [2] Let the function f be analytic in U and normalized by the condition (1.1). We say that f ∈ K s , if there exists a function g ∈ S * 1 2 such that In a very recent paper ofŞeker [6], it is introduced the following class K k s (γ ): Definition 1.2 [6] Let the function f be analytic in U and normalized by the condition (1.1). We say that f ∈ K (k) s (γ )(0 ≤ γ < 1), if there exists a function g ∈ S * k−1 k (k ∈ N is a fixed integer) such that Re where g k is defined by the equality For k = 2 we get the class K s (γ ) ≡ K (2) s (γ ), introduced and studied by Kowalczyk and Leś-Bomba [3]. Also, for k = 2 and γ = 0 we obtain the class K (2) s (0) ≡ K s (0) ≡ K s given in the Definition 1.1. Definition 1.3 (see, e.g. [5]) For two functions f and g analytic in U, we say that the function f is subordinate to g, and write f (z) ≺ g(z), if there exists a Schwarz function w, which (by definition) is analytic in U, with w(0) = 0, and |w(z)| < 1 for all z ∈ U, such that In particular, if the function g is univalent in U, then above subordination is equivalent to Motivated by the aforementioned works we now introduce the following subclass of analytic functions: Suppose also that the function h satisfies the following conditions for all r ∈ (0, 1): Let the function f be analytic in U and normalized by the condition (1.1). We say that f ∈ K (k) For λ = 0 and k = 2 we obtain the class K s (h) ≡ K (2) s (0, h), recently studied by Xu et al. [11]. (i) If we let then it is easy to verify that h is a convex function in U, and satisfies the hypothesis of Definition 1.4.
is a fixed integer), and this class will be denoted by K For k = 2 we have the class K s (λ, A, B) ≡ K (2) s (λ, A, B), recently studied by Wang and Chen [7].
, which consists of the functions f that are analytic in U and normalized by the condition (1.1), satisfying Re Also, for γ = 0 we obtain the new class K s (λ, γ ) which consists of the functions f that are analytic in U and normalized by the condition (1.1), satisfying Re (iii) Letting λ = 0 in (1.7) we get the class K (k) s (γ ) given in Definition 1.2.
In this work, by using the principle of subordination, we obtain inclusion theorem and distortion theorems for functions in the function class K (k) s (λ, h). Our results unify and extend the corresponding results obtained by Xu et al. [11], Wang and Chen [7], Wang et al. [8,9],Şeker [6], Kowalczyk and Leś-Bomba [3], and Gao and Zhou [2].

Main results
We assume throughout this section that k ∈ N is a fixed integer.
In order to prove our main results for the functions class K (k) s (λ, h), we first recall the following lemmas.
We also mention that the above lemma is a special case of Theorem 4 obtained by Wu [10]. We now state and prove the main results of our present investigation: where g k is given by (1.2).
Proof This result can be proven fairly easily by using the Definition 1.3 combined with the definition inequality (1.7).
In view of the Remark 1.1, if we set λ = 0 and in Theorem 2.1, we deduce the following corollary: (1.1).

Corollary 2.1 Let f be an analytic function in U and normalized by the condition
where g k is given by (1.2).
Note that Corollary 2.1 was proven byŞeker [6, Theorem 1]. However, by using Theorem 2.1 we are able to deduce this result as an easy consequence of the theorem.
s (λ, h) be an arbitrary function, and let define the corresponding functions F and G k by Then, the condition (2.2) can be written as By Lemma 2.1 we have G k ∈ S * , and from the above subordination combined with the fact that Re h(z) > 0 for all z ∈ U, we deduce that Now we will consider the following two cases: Denoting γ = 1/λ − 1, then Re γ ≥ 0, and by using Lemma 2.2 we conclude that f ∈ K, which complete the proof of our theorem. for all z ∈ U and θ ∈ [0, 2π), where the coefficients B n are given by (2.1).
Proof Since g ∈ S * k−1 k , then g is univalent in U, hence it follows that g k (z) From here, according to the definition of the subordination of two functions, there exists a function w, which is analytic in U, with w(0) = 0, and |w(z)| < 1, z ∈ U, such that and thus According to (2.4) and using the fact that h is univalent in U, the previous subordination is equivalent to If we denote  (λ, h).
For the special case when the function h is given by (1.6), from Theorem 2.3 we obtain the following result: Corollary 2.2 Suppose that g ∈ S * k−1 k , and g k is given by (1.2). If f is an analytic function in U of the form (1.1), such that for all z ∈ U and θ ∈ [0, 2π). Now, a simple computation combined with the assumption (2.8) shows that for all z ∈ U and θ ∈ [0, 2π), hence (2.9) holds.

Corollary 2.3
Suppose that g ∈ S * k−1 k , and g k is given by (1.2). If f is an analytic function in U of the form (1.1), such that where the coefficients B n are given by (2.1), then f ∈ K  3 we obtain the result given byŞeker [6]. (2.13) From the definition (1.5) combined with (1.4), we deduce that (2.14) Letting , and the inequality (2.14) may be written as which proves (2.10).
If denotes the closed line-segment that connects the points 0 and z = re iθ (0 ≤ r < 1), i.e. = 0, re iθ , then and from the right-hand side part of (2.17) we deduce that whereG k ∈ S * , and Re h(z) > 0 for all z ∈ U. Thus, we deduce that F ∈ K, hence the function F is univalent in U.