Generalized Zalcman conjecture for some classes of analytic functions

TL;DR

This paper establishes sharp bounds for a generalized Zalcman coefficient functional across various subclasses of normalized analytic functions, resolving an open problem and extending previous results to starlike, convex, and other function classes.

Contribution

It provides the first sharp upper bounds for the generalized Zalcman coefficient functional for functions with Re f'(z) > β, covering multiple subclasses and parameter ranges.

Findings

Sharp bounds for the functional when Re f'(z) > β.

Estimates for starlike and convex functions of order α with λ ≤ 0.

Results for typically real, univalent with real coefficients, and close-to-convex functions.

Abstract

For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $\phi(f,n,m;\lambda):=|\lambda a_n a_m -a_{n+m-1}|$. For all real parameters $\lambda$ and $ \beta<1$, we provide the sharp upper bound of $\phi(f,n,m;\lambda)$ for functions $f$ satisfying $\operatorname{Re}f'(z) > \beta$ and hence settles the open problem of estimating $\phi(f,n,m;\lambda)$ recently proposed by Agrawal and Sahoo [S. Agrawal and S. k. Sahoo, On coefficient functionals associated with the Zalcman conjecture, arXiv preprint, 2016]. It is worth mentioning that the sharp estimations of $\phi(f,n,m;\lambda)$ follow for starlike and convex functions of order $\alpha$ $(\alpha <1)$ when $\lambda \leq 0$. Moreover, for certain positive $\lambda$, the sharp estimation of $\phi(f,n,m;\lambda)$ is given when $f$ is a typically real function or a univalent function with real coefficients or is in some subclasses of close-to-convex functions.