Certain properties of the class of univalent functions with real coefficients

TL;DR

This paper investigates properties of univalent functions with real positive coefficients, providing sharp coefficient estimates, verifying the Zalcman conjecture for this class, and analyzing Hankel determinants.

Contribution

It offers the first sharp estimates for initial coefficients and Hankel determinants for functions in ${ mf U}^+$, and confirms the Zalcman conjecture for this class.

Findings

Sharp bounds for initial coefficients and logarithmic coefficients.

Sharp estimates of second and third Hankel determinants.

Verification of the Zalcman conjecture for ${ mf U}^+$ functions.

Abstract

Let ${\mathcal U}^+$ be the class of analytic functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its inverse. In this paper we give sharp estimates of the initial coefficients and initial logarithmic coefficients for $f$, as well as, sharp estimates of the second and the third Hankel determinant for $f$ and $f^{-1}$. We also show that the Zalcman conjecture holds for functions $f$ from ${\mathcal U}^+$.