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

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 ${\rm K}_{2m, \infty}=2\big ((2m-1)!!\big )^2/π+ O\big ( \big ((2m-1)!!\big )^2 /(2m-1)\big )$ 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.