On certain sufficient conditions for close-to-convexity

Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
https://doi.org/10.1016/j.amc.2006.08.135Get rights and content

Abstract

In this paper, we derive certain sufficient conditions for close-to-convexity of analytic functions by using the techniques of differential subordinations. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

Introduction

Let Ap denote the class of functions of the formf(z)=zp+k=1ap+kzp+k(pN={1,2,})which are analytic in the open unit disk U={zC:|z|<1}. We write A1=A. Let Sp(α) and Kp(α) be the subclasses of Ap which are, respectively, p-valently starlike of order α and p-valently convex of order α (0  α < p). We denote by Sp and Kp, the subclasses of Ap consisting of functions which are p-valently starlike and p-valently convex in U.

A function fAp is said to be p-valently close-to-convex of order α (0  α < p), if there exists a p-valently starlike function gAp such thatRzf(z)g(z)>α(zU).It is well known that a p-valently close-to-convex function is p-valent in U [2, Theorem 1].

For real or complex numbers a,b, and c  0, −1, −2, …, the Gauss hypergeometric function is defined by2F1(a,b;c;z)=1+abcz1!+a(a+1)b(b+1)c(c+1)z22!+.We note that the above series converges absolutely for zU and hence represents an analytic function in U (see, for details, [11, Chapter 14]).

Suppose that f and g are analytic in U. We say that the function f is subordinate to g, written f  g or f(z)g(z)(zU), if there exists an analytic function ω in U with ω(0) = 0 and ∣ω(z)∣ < 1 for all zU, such that f(z) = g(ω(z)). In case g is univalent in U, then f  g is equivalent to f(0) = g(0) and f(U)g(U).

In this paper, we use the method of differential subordinations. The general theory of differential subordinations introduced by Miller and Mocanu is given in [3]. Namely, if Ψ:ΩC(whereΩC2,C is the complex plane) is an analytic function, h is univalent in U, and if ϕ is analytic in U with (ϕ(z), ′(z))  Ω when zU, then we say that ϕ satisfies a first-order differential subordination provided thatΨ(ϕ(z),zϕ(z))h(z)(zU)andΨ(ϕ(0),0)=h(0).

We say that a univalent function q is a dominant of the differential subordination (1.2), if ϕ  q for all analytic function ϕ satisfying (1.2). A dominant q˜ is called the best dominant of the differential subordination (1.2) if q˜q for all dominant q of (1.2).

The object of the present paper is to derive certain sufficient conditions for close-to-convexity of analytic functions by using the techniques of differential subordination. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

Section snippets

Preliminaries

To establish our results, we shall need the following lemmas.

Lemma 1

[1]. Let h be a convex (univalent) function in U with h(0) = 1. Also, letϕ(z)=1+c1z+c2z2+be analytic in U. Ifϕ(z)+zϕ(z)γh(z)(zU)for some complex number γ  0 with R(γ)0, thenϕ(z)ψ(z)=γz-γ0ztγ-1h(t)dth(z)(zU),and the function ψ is the best dominant.

Lemma 2

[4]. Let q be analytic and univalent in U, and let θ and ψ be analytic in a domain Ω containing q(U), with ψ(w)  0 when wq(U). Set Q(z) = zq(z)ψ(q(z)), h(z) = θ(q(z)) + Q(z), and suppose that

Main results and consequences

Unless otherwise mentioned, we assume through the sequel that −1  B < A  1, 0  α < p.

Theorem 1

Let 0 < λ < 1 and (1  A)  λ(1  B) > 0. If fAp satisfies the following differential subordination:(1-λ)f(z)zp-1+λ1+zf(z)f(z)p(1+Az)1+Bz(zU),thenf(z)zp-1λ(1-λ)Q(z)=q1(z)p1+A-λB1-λz1+Bz(zU),whereQ(z)=0ztpλ-p-11+Btz1+Bzp(A-B)λBdt(B0),0ztpλ-p-1exppλA(t-1)zdt(B=0),and q1 is the best dominant of (3.2). Furthermore, ifAmin1-λ(1-B),-λ(1-p)Bpwith-1B<0,thenRf(z)zp-1>pρ(zU),whereρ=2F11,p(B-A)λB;pλ-p+1;BB-1-1.The bound ρ

Acknowledgement

The authors would like to express their gratitude to Professor H.M. Srivastava of the University of Victoria for many valuable suggestions in the preparation of this paper. The present investigation was supported, in part, by the University Grants Commission of India under its DRS Financial Assistance Program and, in part, by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-521-C00008).

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