Radii of convexity for certain classes of univalent analytic functions

O. P. Juneja and M. L. Mogra

Vol. 78 (1978), No. 2, 359–368

DOI: 10.2140/pjm.1978.78.359

Abstract

Let P(α,β) denote the class of functions p(z) = 1 + b₁z + ⋯ which are analytic and satisfy the inequality |(p(z) − 1)∕{2β(p(z) − α) − (p(z) − 1)}| < 1 for some α, β (0 ≦ α < 1,0 < β ≦ 1) and all z ∈ E = {z : |z| < 1}. Also, let P_b(α,β) = {p ∈ P(α,β) : p′(0) = 2bβ(1 − α),0 ≦ b ≦ 1}. In the present paper, we determine sharp estimates for the radii of convexity for functions in the classes R_a(α,β) and S_a^∗(α,β) where R_a(α,β) = {f(z) = z + aβ(1 −α)z² + ⋯ : f′∈ P_a(α,β),0 ≦ a ≦ 1}, S_a^∗(α,β) = {g(z) = z + 2aβ(1 −α)z² + ⋯ : zg′∕g ∈ P_a(α,β),0 ≦ a ≦ 1}. The results thus obtained not only sharpen and generalize the various known results but also give rise to several new results.

Mathematical Subject Classification 2000

Primary: 30C45

Milestones

Received: 26 September 1977

Revised: 17 March 1978

Published: 1 October 1978