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 pr
oblem initially raised, in the special case r = 0, s = 1, by Mitrinovi
c. The analysis uses properties of the psi function psi(x) = Gamma'(x)
/Gamma(x). However, an elementary proof is also given for the main res
ult of the above-mentioned authors (r = 0, s = 1). (C) 1997 Academic P
ress.