  This paper deals with an extremal problem for harmonic functions in the unit
ball of $\mathbf{R}^n$. We are concerned with the pointwise sharp estimates for
the gradient of real--valued bounded harmonic functions. Our main result may be
formulated as follows. The sharp constants in the estimates for the absolute
value of the radial derivative and the modulus of the gradient of a bounded
harmonic function coincide near the boundary of the unit ball. This result
partially confirms a conjecture posed by D. Khavinson.
