Note on the second derivative of bounded analytic functions

TL;DR

This paper characterizes the possible values of the second derivative of bounded analytic functions within the unit disk, providing sharper bounds and new methods for existing inequalities.

Contribution

It determines the value region of the second derivative in terms of key function parameters and introduces a novel approach to derive sharp bounds for these derivatives.

Findings

Derived a smaller sharp upper bound for |g''(z_0)| than Ruscheweyh's inequality.

Obtained a sharp upper bound for |g''(z_0)| depending only on |z_0| using a new method.

Identified the form of extremal functions for the second derivative bounds.

Abstract

Assume $z_0$ lies in the open unit disk $\mathbb{D}$ and $g$ is an analytic self-map of $\mathbb{D}$. We will determine the region of values of $g''(z_0)$ in terms of $z_0$, $g(z_0)$ and the hyperbolic derivative of $g$ at $z_0$, and give the form of all the extremal functions. In particular, we obtain a smaller sharp upper bound for $|g''(z_0)|$ than Ruscheweyh's inequality for the case of the second derivative. Moreover, we use a different method to obtain Sz{\'a}sz's inequality, which provides a sharp upper bound for $|g''(z_0)|$ depending only on $|z_0|$.