A monotonicity property of Euler's gamma function

• Autores: José Antonio Adell Pascual, Horst Alzer
• Localización: Publicationes Mathematicae Debrecen, ISSN 0033-3883, Tomus 78, Fasc. 2, 2011, págs. 443-448
• Idioma: inglés
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• Resumen

  • Let ¥Ä(x) = log ¥Ã(x + 1) x (.1 < x .= 0); ¥Ä(0) = .:

    For all n = 0; 1; 2; : : : and x > .1, we show that (.1)n¥Ä(n+1)(x) = (n + 1)! ¡ò 1 0 un+1(n + 2; xu + 1) du;

    where  denotes the Hurwitz zeta function. This representation implies that ¥Ä¡Ç is completely monotonic on (.1;¡Ä). This extends a result published in 1996 by Grab- ner, Tichy, and Zimmermann, who proved that ¥Ä is increasing and concave on (.1;¡Ä).