New criteria and distortion theorems for univalent functions

Abstract

Let P, d p ( X ) denote the family of functionsf, normalized by f(0) = 0 = f' (0) -1, that are analytic in the open unit disc A = {z : Izi < 11, f(z)#O for z E A\{0) and such that for some ru # 0, E C and g E A with g(z) # 0 in 0 < Izj < 1. The main object of this paper is to study this class and to find conditions on a , and on the function g such that each function f in P,,,:,(X) belongs to a family which is contained in the family of univalent functions in the unit disc A. We also find the exact value of max{Iz/f (z) -1 / } in the class P(X) := P,,o,2(X) for fixed z E A. Further, we also determine condition on X for functions f in P(X) to be in the class of strongly starlike functions, or in the class of functions whose derivative lies in a sector of angle less than or equal to ~y / 2 with y E (O,l]. Finally, we also obtain a sufficient condition for an analytic function f to satisfy the analytic univalence criteria of Noshiro-Warschawski. Several examples are stated in support of the sharpness of our criteria.

Univalence and starlikeness of certain transforms defined by convolution of analytic functions

“…The first part of Lemma 3 is from [7,8] whereas the second part is obtained from [16,Theorem 1]. At this place it is important to present the following example: Consider the function…”

Criteria for univalence, starlikeness and convexity

“…This result was originally stated as a conjecture in [3] and was proved in [4]. In this article, we discuss the relationship between U(λ) and S * (δ), as well as between P(2λ) and K(δ).…”

“…Functions of this form have been studied, for example, in [7,3]. In [3], Obradović and Ponnusamy obtained various coefficient conditions in terms of b k 's for the corresponding f of the above form to be univalent, strongly starlike etc. In particular, the following results are known: Theorem 1.6.…”

“…According to a result due to Ozaki and Nunokawa [6], we have the inclusion U(λ) ⊂ S for 0 < λ ≤ 1, and from [3], we also have the inclusion P(2λ) ⊂ U(λ). In [4], the authors have shown that certain results obtained in [3] also hold if P(2λ) is replaced by U(λ). In this connection, we recall the following result from [4].…”

On the radius of convexity for a class of conformal maps

“…The properties of functions in U (λ) and U 0 (λ) have been studied in detail in the literature (see [2]- [5]). Recently, Ponnusamy and Vasundhra [5] proposed the conjecture that f ∈ U 0 (λ) is convex at least when 0 < λ ≤ 3 − 2 √ 2.…”