Bounds for certain harmonic sums

The monotonicity properties of the function Phi(n) = (pn + r + 1)(-1) + (pn + r + 2)(-1) + ... + (qn + s)(-1) are determined, where p, q, r, and s are fixed integers such that 0 < p < q and 0 less than or equal to p + r < q + s. The results extend earlier results of Adamovic and Taskovic (1969) and Simic (1979) for the cases r = s = 0 and r = 0, s = 1. We settle negatively a conjecture of Simic that Phi(n) is always monotonic when 0 less than or equal to r less than or equal to s. The results enable us to obtain sharp bounds for the function Phi(n), a problem initially raised, in the special case r = 0, s = 1, by Mitrinovic. The analysis uses properties of the psi function psi(x) = Gamma'(x)/Gamma(x). However, an elementary proof is also given for the main result of the above-mentioned authors (r = 0, s = 1). (C) 1997 Academic Press.