Journal Title
Title of Journal: Comput Methods Funct Theory
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Abbravation: Computational Methods and Function Theory
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Publisher
Springer Berlin Heidelberg
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Authors: Gershon Kresin
Publish Date: 2016/04/28
Volume: 16, Issue: 4, Pages: 637-652
Abstract
The representation for the sharp constant mathrmK n p in an estimate of the modulus of the nth derivative of an analytic function in the upper halfplane 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 Lp The representation for mathrmK n p implies an optimization problem for a parameter in some integral This problem is solved for p=2m+1/2m+1n nle 2m+1 and for some first derivatives of even order in the case p=infty The formula for mathrmK n 2m+1/2m+1n contains for instance the known expressions for mathrmK 2m+1 infty and mathrmK m 2 as particular cases Also a twosided estimate for mathrmK 2m infty is derived which leads to the asymptotic formula mathrmK 2m infty =22m12/pi + O2m12/2m1 as m rightarrow infty The lower and upper bounds of mathrmK 2m infty are compared with its value for the cases m=1 2 3 4 As applications some realpart theorems with explicit constants for high order derivatives of analytic functions in subdomains of the complex plane are described
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