On a successive property of strongly starlikeness for multivalent functions

For f analytic in the unit disk D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}$$\end{document}, of the form f(z)=zp+⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(z)=z^p+\cdots $$\end{document}, we consider some consequences of strongly starlikeness of f(p-1)(z)/p!\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{(p-1)}(z)/p!$$\end{document}.


Introduction
We denote by H the class of functions f (z) which are holomorphic in the open unit disc D = {z ∈ C : |z| < 1}. Denote by A p , p ∈ N = {1, 2, . . .}, the class of functions f (z) ∈ H given by f (z) = z p + ∞ n= p+1 a n z n , (z ∈ D).
(1.1) Lemma 1.1 [2, Theorem 5] If f (z) ∈ A p , then for all z ∈ D, we have In this paper we consider a generalization of the above result. In Lemma 1.1 we have assumed that z f ( p) (z)/ f ( p−1) (z) lies in the right half-plane while in this paper we work with a sector.
Recall that if f (z) ∈ A p and 3) then f ( p−1) (z)/ p! is called a strongly starlike function of order γ and such functions we consider in the paper. This class for the case p = 1 was introduced by Brannan and Kirwan [1]. Also, if f (z) ∈ A p satisfies (1. 3), then f (z) is called p-valently strongly starlike function of order γ . For the proof of main result we need the following lemma.
This completes the proof.
Let us go to next step and define the function and applying the same method as the above, we have the following theorem. (2.8) Then we have where β p−2 we obtain from β p−1 using formula (2.1). Furthermore,
Applying the same step as the above and under the hypothesis of Theorem 2.1, we have the following theorem It is easy to see that Theorem 2.3 holds for the case β p = β p−1 = · · · = β 1 = 1 and then Theorem 2.3 becomes Lemma 1.1 and in this sense Theorem 2.3 improves Lemma 1.1.
(2.11) Therefore, from (2.11), we have From the properties of the sequence (2.1) we have following corollary.