  We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1}
< sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For
the case r = 1 we give two proofs. The first one is by means of a general
numerical technique (maximal slope principle) for proving inequalities between
elementary functions. The second proof is fully analytical. Finally we prove a
general rearrangement theorem and apply it to the remaining case 0 < r < 1.
Some of these inequalities are needed for obtaining general sharp bounds for
the errors committed when applying the Riemann-Siegel expansion of Riemann's
zeta function.
