The Zalcman conjecture for certain analytic and univalent functions

Abstract

Let A denote the class of analytic functions defined for z∈D, given by f(z)=z+∑n=2∞anzn, and let S denote the subclass of A consisting of univalent (i.e., one-to-one) functions. In 1960s, L. Zalcman conjectured that if f∈S, then |an2−a2n−1|≤(n−1)2 for n≥2, which implies the famous Bieberbach conjecture |an|≤n for n≥2. For f∈S, Ma [19] proposed a generalized Zalcman conjecture|anam−an+m−1|≤(n−1)(m−1) for n≥2,m≥2. Let U be the class of functions f∈A satisfying|f′(z)(zf(z))2−1|<1 for z∈D, and CR+ denote the class of functions f∈A satisfying Re(1−z)2f′(z)>0 for z∈D. In the present paper, we prove the Zalcman conjecture and the generalized Zalcman conjecture for the class U using extreme point theory. We also prove the generalized Zalcman conjecture for the class CR+ for the initial coefficients.