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 5 Documents MSC: 33B15 Gamma, beta and polygamma functions Keywords:gamma function; inequalities; harmonic mean PDF BibTeX XML Cite \textit{H. Alzer}, J. Comput. Appl. Math. 87, No. 2, 195--198 (1997; Zbl 0888.33001) Full Text: DOI References: [1] Bullen, P.S.; Mitrinović, D.S.; Vasić, P.M., Means and their inequalities, (1988), Reidel, Dordrecht · Zbl 0687.26005 [2] Fichtenholz, G.M., Differential — und integralrechnung II, (1979), 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.