Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L^p. This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L^p-norm of its boundary values is attained at the radial direction. This confirms D. Khavinson's conjecture for p = 1 and p = 2. Bibliography: 11 titles.