On a New Subclass of p-Valent Close-to-Convex Mappings Defined by Two-Sided Inequality

for all z ∈ U. The concept of close-to-convex was introduced by Kaplan [1] in 1952. A number of results for close-to-convex functions in U have been obtained by several authors (see, e.g., [2–14]). A functionf ∈ A(p) is said to be in the classKp(α1, α2, β) if it satisfies the following two-sided inequality: −α2π 2 < arg{f 󸀠 (z) pzp−1 − β} < α1π 2 (z ∈ U) (3) for 0 < α1, α2 ≤ 1 and 0 ≤ β < 1. Note that if f ∈ Kp(α1, α2, β), then f is p-valent close-to-convex in U. Furthermore, if f ∈ Kp(α, α, β) (0 < α ≤ 1, 0 ≤ β < 1), then (3) becomes


Introduction
Let () be the class of functions of the form which are -valent analytic in the open unit disk  = { ∈ C : || < 1}.There and in the following, let N, C, and R be the sets of positive integers, complex numbers, and real numbers, respectively.A function  analytic in  is said to be close-toconvex if there is a convex function  such that Re {   ()   () } > 0 for all  ∈ .The concept of close-to-convex was introduced by Kaplan [1] in 1952.A number of results for close-to-convex functions in  have been obtained by several authors (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13][14]).
A function  ∈ () is said to be in the class   ( 1 ,  2 , ) if it satisfies the following two-sided inequality: for 0 <  1 ,  2 ≤ 1 and 0 ≤  < 1.Note that if becomes A function  ∈   (, , ) is called -valent close-to-convex of order  and type  in .

Journal of Function Spaces
Throughout this paper, we let In order to prove our main results, we need the following lemmas.
In this paper we shall derive some criteria for a function  ∈ () to be in the class   ( 1 ,  2 , ).

Main Results
Our first result is the following theorem. where The bounds  1 and  2 in ( 22) are sharp for the function  defined by (7).
The other conditions of Lemma 2 are also satisfied.Therefore, we conclude that and the function  is the best dominant of (20).The proof of the theorem is completed.
Furthermore, for the function  defined by ( 7), we have