Explicit real-part estimates for high order derivatives of analytic functions

TL;DR

This paper derives explicit formulas and estimates for the sharp constants in bounds of high order derivatives of analytic functions in the upper half-plane, with applications to real-part theorems.

Contribution

It provides new explicit representations and asymptotic estimates for the sharp constants in derivative bounds, extending known results and solving related optimization problems.

Findings

Derived explicit formulas for ${ m K}_{n,p}$ in specific cases.

Established asymptotic behavior of ${ m K}_{2m,\infty}$ as m increases.

Compared bounds with exact values for small m.

Abstract

The representation for the sharp constant ${\rm K}_{n, p}$ in an estimate of the modulus of the $n$-th derivative of an analytic function in the upper half-plane ${\mathbb C}_+$ is considered. It is assumed that the boundary value of the real part of the function on $\partial{\mathbb C}_+$ belongs to $L^p$. The representation for ${\rm K}_{n, p}$ comprises an optimization problem by parameter inside of the integral. This problem is solved for $p=2(m+1)/(2m+1-n)$, $n\leq 2m+1$, and for some first derivatives of even order in the case $p=\infty$. The formula for ${\rm K}_{n,\; 2(m+1)/(2m+1-n)}$ contains, for instance, the known expressions for ${\rm K}_{2m+1, \infty}$ and ${\rm K}_{m, 2}$ as particular cases. Also, a two-sided estimate for ${\rm K}_{2m, \infty}$ is derived, which leads to the asymptotic formula ${\rm K}_{2m, \infty}=2\big ((2m-1)!!\big )^2/\pi + O\big ( \big ((2m-1)!!\big )^2 /(2m-1)\big )$ as $m \rightarrow \infty $. The lower and upper bounds of ${\rm K}_{2m, \infty}$ are compared with its value for the cases $m=1, 2, 3, 4$. As applications, some real-part theorems with explicit constants for high order derivatives of analytic functions in subdomains of complex plane are described.