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

Representations for the sharp coefficient in an estimate of the modulus of the n-th derivative of an analytic function in the unit disk D are obtained. It is assumed that the boundary value of the real part of the function on ∂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 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 ε (1, ∞), as the product of monotonic functions of |z|.