Alzer, Horst A harmonic mean inequality for the gamma function. (English) Zbl 0888.33001 J. Comput. Appl. Math. 87, No. 2, 195-198 (1997). Summary: We prove that for all positive real numbers \(x\neq 1\), the harmonic mean of \((\Gamma (x))^2\) and \((\Gamma (1/x))^2\) is greater than 1. This refines a result of Gautschi (1974). Cited in 2 ReviewsCited in 7 Documents MSC: 33B15 Gamma, beta and polygamma functions Keywords:gamma function; inequalities; harmonic mean × Cite Format Result Cite Review PDF Full Text: DOI Digital Library of Mathematical Functions: §5.6(i) Real Variables ‣ §5.6 Inequalities ‣ Properties ‣ Chapter 5 Gamma Function References: [1] Bullen, P. S.; Mitrinović, D. S.; Vasić, P. M., Means and Their Inequalities (1988), Reidel,: Reidel, Dordrecht · Zbl 0687.26005 [2] Fichtenholz, G. M., Differential — und Integralrechnung II (1979), Dt. Verlag Wissensch,: Dt. Verlag Wissensch, Berlin · Zbl 0143.27002 [3] Gautschi, W., A harmonic mean inequality for the gamma function, SIM J. Math. Anal., 5, 278-281 (1974) · Zbl 0239.33002 [4] Gautschi, W., Some mean value inequalities for the gamma function, SIAM J. Math. Anal., 5, 282-292 (1974) · Zbl 0239.33003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.