Sharp real-part theorems for high order derivatives

We obtain a representation for the sharp coefficient in an estimate of the modulus of the nth derivative of an analytic function in the upper half-plane ℂ ₊. It is assumed that the boundary value of the real part of the function on ∂ℂ ₊ belongs to L ^p. This representation is specified for p = 1 and p = 2. For p = ∞ and for derivatives of odd order, an explicit formula for the sharp coefficient is found. A limit relation for the sharp coefficient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. It is assumed that the boundary value of the real part of the function belongs to L ^p. The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in ℂ ₊. As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary. Bibliography: 8 titles.