Coefficient bounds for some classes of starlike functions

Roger W. Barnard and John Lawson Lewis

Vol. 56 (1975), No. 2, 325–331

DOI: 10.2140/pjm.1975.56.325

Abstract

Let t be given, 1∕4 ≦ t ≦∞, and let S(t) denote the class of normalized starlike univalent functions f in |z| < 1 satisfying (i) |f(z)∕z|≧ t, |z| < 1, if 1∕4 ≦ t ≦ 1, (ii) |f(z)∕z|≦ t, |z| < 1, if 1 < t ≦∞. If f(z) = z + Σ_k=2^xa_kz^k ∈ S(t) and n is a fixed positive integer, then the authors obtain sharp coefficient bounds for |a_n| when t is sufficiently large or sufficiently near 1∕4. In particular a sharp bound is found for |a₃| when 1∕4 ≦ t ≦ 1 and 5 ≦ t ≦∞. Also a sharp bound for |0₄| is found when 1∕4 ≦ t ≦ 1 or 12.259 ≦ t ≦∞.

Mathematical Subject Classification

Primary: 30A34

Milestones

Received: 4 December 1973

Published: 1 February 1975