Resumen de Monotonicity Properties of the Hurwitz Zeta Function

Horst Alzer

• Let ?(s,x) = ?n = 08 1/(n + x)s (s > 1, x > 0) be the Hurwitz zeta function and let Q(x) = Q(x ; a, ß; a, b) = {(?(a, x))a}/{(?(ß, x))b}, where a, &beta, > 1 and a, b > 0 are real numbers. We prove:

  (i) The function Q is decreasing on (0, 8) iff a a - ß b = max(a - b, 0).

  (ii) Q is increasing on (0, 8) iff a a - ß b = min(a - b, 0).

  An application of part (i) reveals that for all x > 0 the function s \mapsto [(s - 1) ?(s,x)]1/(s-1) is decreasing on (1, 8). This settles a conjecture of Bastien and Rogalski