Monotonicity Properties of the Hurwitz Zeta Function

• Autores: Horst Alzer
• Localización: Canadian mathematical bulletin, ISSN 0008-4395, Vol. 48, Nº 3, 2005, págs. 333-340
• Idioma: inglés
• DOI: 10.4153/cmb-2005-031-3
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• Resumen

  • 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