On certain sufficient conditions for close-to-convexity
Introduction
Let denote the class of functions of the formwhich are analytic in the open unit disk . We write . Let and be the subclasses of which are, respectively, p-valently starlike of order α and p-valently convex of order α (0 ≦ α < p). We denote by and , the subclasses of consisting of functions which are p-valently starlike and p-valently convex in .
A function is said to be p-valently close-to-convex of order α (0 ≦ α < p), if there exists a p-valently starlike function such thatIt is well known that a p-valently close-to-convex function is p-valent in [2, Theorem 1].
For real or complex numbers a,b, and c ≠ 0, −1, −2, …, the Gauss hypergeometric function is defined byWe note that the above series converges absolutely for and hence represents an analytic function in (see, for details, [11, Chapter 14]).
Suppose that f and g are analytic in . We say that the function f is subordinate to g, written f ≺ g or , if there exists an analytic function ω in with ω(0) = 0 and ∣ω(z)∣ < 1 for all , such that f(z) = g(ω(z)). In case g is univalent in , then f ≺ g is equivalent to f(0) = g(0) and .
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 is the complex plane) is an analytic function, h is univalent in , and if ϕ is analytic in with (ϕ(z), zϕ′(z)) ∈ Ω when , then we say that ϕ satisfies a first-order differential subordination provided that
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 is called the best dominant of the differential subordination (1.2) if 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 with h(0) = 1. Also, letbe analytic in . Iffor some complex number γ ≠ 0 with , thenand the function ψ is the best dominant. Lemma 2 [4]. Let q be analytic and univalent in , and let θ and ψ be analytic in a domain Ω containing , with ψ(w) ≠ 0 when . 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 satisfies the following differential subordination:thenwhereand q1 is the best dominant of (3.2). Furthermore, ifthenwhereThe 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).
References (11)
- et al.
Univalent solutions of Briot–Bouquet differential subordinations
J. Differ. Equations
(1985) - et al.
Some subclasses of multivalent functions involving a certain linear operator
J. Math. Anal. Appl.
(2005) - et al.
A certain connection between starlike and convex functions
Appl. Math. Lett.
(2003) - et al.
Subordination by convex functions
Proc. Amer. Math. Soc.
(1975) p-Valent close-to-convex functions
Trans. Amer. Math. Soc.
(1965)
Cited by (4)
On certain p-valent close-to-convex functions of order β and type α
2019, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: MatematicasOn a New Subclass of p-Valent Close-to-Convex Mappings Defined by Two-Sided Inequality
2018, Journal of Function SpacesOn Certain p-Valent Analytic Functions Involving a Linear Operator and Majorization Problems
2014, Vietnam Journal of MathematicsA general theorem associated with the Briot–Bouquet differential subordination
2014, Journal of Computational Analysis and Applications