Criteria for univalence, starlikeness and convexity

Abstract

Abstract. Let A denote the class of all normalized analytic functions f (f (0) = 0 = f (0) − 1) in the open unit disc ∆. For 0 < λ ≤ 1, defineRecently, the problem of finding the starlikeness of these classes has been considered by Obradović and Ponnusamy, and later by Obradović et al. In this paper, the authors consider the problem of finding the order of starlikeness and of convexity of U (λ) and P(2λ), respectively. In particular, for f ∈ A with f (0) = 0, we find conditions on λ, β * (λ) and β(λ) so that U (λ) S * (β * (λ)) and P(2λ) K(β(λ)). Here, S * (β) and K(β) (β < 1) denote the classes of functions in A that are starlike of order β and convex of order β, respectively. In addition to these results, we also provide a coefficient condition for functions to be in K(β). Finally, we propose a conjecture that each function f ∈ U (λ) with f (0) = 0 is convex at least when 0 < λ ≤ 3 − 2 √ 2.

On the radius of convexity for a class of conformal maps

“…Theorem 2.3 implies that the conjecture proposed in [5] is not true. Indeed, if it were, even for a single λ with 0 < λ ≤ 1, then for each fixed r < 1 we would have…”

“…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. In [6], Vasundhra also obtained a lower bound for the radius of convexity of the families U (λ) and U 0 (λ).…”

Two Parameter Families of Close-to-Convex Functions and Convolution Theorems

“…The class U{ 1) ξ U together with its various generalizations and subclasses have been discussed by M. Obradovic and S. Ponnusamy [9] and later by a number of authors (see [10,11,15]). In fact, Krzyz [6] has shown that function in U{λ) admits a Q-quasiconformal extension to the whole complex plane with Q = --γ whenever 0 < Λ < 1.…”

Two Parameter Families of Close-to-Convex Functions and Convolution Theorems

“…then the Hadamard product or convolution, denoted by / * g. is defined bv"X il*fl)(z) = Σ ":'>•>'" 11=0and that /»9 ξ W. Next, we consider the class U{\) (0 < λ < 1 ) which is defined as follows:U(A) = |/ e A : j(ypy) /'( = ) -lj < A. ; € ΔThe class U{ 1) ξ U together with its various generalizations and subclasses have been discussed by M. Obradovic and S. Ponnusamy ¡9! and later by a number of authors (see[10,11,15]). In fact.Krzyz ;6] has shown that function in U( A) admits a Q-quasiconformal extension to the whole complex plane with Q = --γ whenever 0 < A < 1.…”