Explicit Real-Part Estimates for High Order Derivatives of Analytic Functions

Abstract

The representation for the sharp constant \(\mathrm{K}_{n, p}\) in an estimate of the modulus of the n-th derivative of an analytic function in the upper half-plane \({\mathbb C}_+\) is considered in this paper. 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 \(\mathrm{K}_{n, p}\) implies an optimization problem for a parameter in some integral. This problem is solved for \(p=2(m+1)/(2m+1-n)\), \(n\le 2m+1\), and for some first derivatives of even order in the case \(p=\infty \). The formula for \(\mathrm{K}_{n, 2(m+1)/(2m+1-n)}\) contains, for instance, the known expressions for \(\mathrm{K}_{2m+1, \infty }\) and \(\mathrm{K}_{m, 2}\) as particular cases. Also, a two-sided estimate for \(\mathrm{K}_{2m, \infty }\) is derived, which leads to the asymptotic formula \(\mathrm{K}_{2m, \infty }=2((2m-1))^2/\pi + O(((2m-1))^2/(2m-1))\) as \(m \rightarrow \infty \). The lower and upper bounds of \(\mathrm{K}_{2m, \infty }\) 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 the complex plane are described.