Sharp and maximized real-part estimates for derivatives of analytic functions in the disk

Abstract

Representations for the sharp coefficient in an estimate of the modulus of the $n$-th derivative of an analytic function in the unit disk ${\mathbb D}$ are obtained. It is assumed that the boundary value of the real part of the function on $\partial{\mathbb D}$ belongs to $L^p$. The maximum of a bounded factor in the representation of the sharp coefficient is found. Thereby, a pointwise estimate of the modulus of the $n$-th derivative of an analytic function in ${\mathbb D}$ with a best constant is obtained. The sharp coefficient in the estimate of the modulus of the first derivative in the explicit form is found. This coefficient is represented, for $p\in (1, \infty)$, as the product of monotonic functions of $|z|$.