Bounds for Certain Harmonic Sums

The monotonicity properties of the function y1 y1 y1 F n.s pnqrq1. q pnqrq2. q???q qnqs. are determined, where p, q, r, and s are fixed integers such that 0-p-q and 0Fpqr-qqs. The results extend earlier results of Adamovi´c and Taskovi´c 1969.and Simi´c 1979.for the cases rsss0 and rs0, ss1. We settle negatively a conjecture of Simi´c that F n. is always monotonic when 0FrFs. The results enable us to obtain sharp bounds for the function F n., a problem initially raised, in the special case rs0, ss1, by Mitrinovi´c. The analysis uses properties of the psi function c x.sG9 x.rG x.. However, an elementary proof is also given for the main result of the above-mentioned authors rs0, ss1..