Radii of univalence, starlikeness, and convexity

Abstract

Let a function be regular in the disk |z| < 1. The radius of univalence 0.164 … of the family of f with |an| ≤ n (n ≥ 2) is, actually, the radius of star-likeness. The radius of univalence 1 - [k/(l+K)]½ of the family of f with |an| ≤ k (n ≥ 2), where K > 0 is a constant, is, actually, the radius of starlikeness. The radii of convexity of the two families are estimated from below.

Radii of starlikeness and convexity of analytic functions satisfying certain coefficient inequalities

“…For 0 ≤ α < 1, the sharp radii of starlikeness and convexity of order α are obtained for functions f ∈ A b satisfying the condition |a n | ≤ n or |a n | ≤ M (M > 0) for n ≥ 3. Special case (α = 0) of the results shows that the lower bounds for the radii of convexity obtained by Yamashita [10] are indeed sharp. The coefficient inequalities are natural in the sense that the inequality |a n | ≤ n is satisfied by univalent functions and while the inequality |a n | ≤ M is satisfied by functions which are bounded by M. For a function p(z) = 1 + c 1 z + c 2 z 2 + · · · with positive real part, it is well-known that |c n | ≤ 2 and so if f ∈ A and Re f ′ (z) > 0, then |a n | ≤ 2/n.…”

Radius problems for holomorphic mappings on the unit ball in Cn

“…In the case of one complex variable, compare with [3] and [35]. We also obtain estimates for r * (F n ) and r c (F n ) (respectively r * (G n ) and r c (G n )).…”

“…In the last section we also give a generalization of the previous sections to the case of complex Hilbert spaces. For related results in one complex variable, the reader may consult [3], [4], [21]- [23], [30] and [35]. In the case of several complex variables, see [7] and [26].…”

On the density of zeros of linear combinations of Euler products for σ > 1

“…Let F be any subclass of A, then we write r c (F ) for the largest r, with 0 < r ≤ 1, such that f ({|z| < r}) is convex. In [39], Yamashita proved the following result.…”