Explicit real-part estimates for high order derivatives of analytic functions (Q523311)

Summary: 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 \(|z|\).