Sharp real-part theorems in the upper half-plane and similar estimates for harmonic functions

Explicit formulas for sharp coefficients in estimates of the modulus of an analytic function and its derivative in the upper half-plane are found. It is assumed that the boundary values of the real part of the function are in L^p. As corollaries, sharp estimates for the modulus of the gradient of a harmonic function in ℝ²₊ are deduced. Besides, a representation for the best constant in the estimate of the modulus of the gradient of a harmonic function in ℝⁿ₊ by the L^p-norm of the boundary normal derivative is given, 1 ≤ p < ∞. This representation is formulated in terms of an optimization problem on the unit sphere which is solved for p ∈ [1, n]. Bibliography: 6 titles.