On Ramanujan's Double Inequality for the Gamma Function |
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Authors: | Alzer Horst |
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Institution: | Morsbacher Str. 10, D-51545 Waldbröl, Germany alzerhorst{at}freenet.de |
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Abstract: | Ramanujan presented (without proof) the following double inequalityfor the gamma function:
.Recently, Karatsuba established that these inequalities holdfor x 1. We show that this can be slightly improved: the inequalitieshold for all x 0, even if we replace 1/100 by
where f(x) = (1/)3(x + 1)(e/x)x]68x34x2x.Moreover, and 1/30 are the best possible constant terms. 2000Mathematics Subject Classification 33B15 (primary); 26D15, 26D20(secondary). |
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