The Zalcman conjecture for certain analytic and univalent functions

[摘要] Let A denote the class of analytic functions defined for z is an element of D, given by f(z) = z + Sigma(infinity)(n=2) a(n)z(n), and let Sdenote the subclass of Aconsisting of univalent (i.e., one-to-one) functions. In 1960s, L. Zalcman conjectured that if f is an element of S, then vertical bar a(n)(2) - a(2n-1)vertical bar <= (n - 1)(2) for n >= 2, which implies the famous Bieberbach conjecture vertical bar a(n)vertical bar <= n for n >= 2. For f is an element of S, Ma [19] proposed a generalized Zalcman conjecture vertical bar a(n)a(m) - a(n)vertical bar(m-1)vertical bar <= (n - 1)(m - 1) for n >= 2, m >= 2. Let U be the class of functions f is an element of A satisfying vertical bar f'(z)(z/f(z)(2) -1 vertical bar < 1 for z is an element of D, and CR+ denote the class of functions f is an element of Asatisfying Re(1 - z)(2)f'(z) > 0 for z is an element of D. In the present paper, we prove the Zalcman conjecture and the generalized Zalcman conjecture for the class Uusing extreme point theory. We also prove the generalized Zalcman conjecture for the class CR+ for the initial coefficients. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-12-15

[关键词] Univalent;Starlike;Convex functions;Extreme point;Subordination;Zalcman conjecture