Generalized Zalcman conjecture for starlike and typically real functions
Let \(S\) be the class of normalized univalent functions, i.e. those functions of the form \(f(z) = z + \sum_{n=1}^\infty a_n z^n\) which are analytic and univalent in the unit disc. Zalcman conjectured that if \(f \in S\) then \(|a_n^2 - a_{2n-1}|\leq (n -1)^2\) for all \(n \geq 2\). This is clearly difficult to prove since it implies the Bieberbach conjecture. (The proof of the latter does not seem to offer any help in the proof of the Zalcman conjecture.) The Zalcman conjecture has been proved true for starlike functions, for functions in \(S\) with real coefficients, for typically real functions, and for close-to-convex functions when \(n \geq 4\). In this paper, the author offers the generalized Zalcman conjecture: \(f \in S\) implies \(|a_n a_m - a_{n+m-1}|\leq (n-1)(m-1)\) for all \(m,n \geq 2\). This is proved true for starlike functions and for functions with real coefficients. The latter result is proved with the help of the result, also proved in the paper, that the generalized Zalcman conjecture holds for typically real functions when \(n,m \geq 4\).
scientific article; zbMATH DE number 1309062