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A harmonic mean inequality for the gamma function. (English) Zbl 0888.33001

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).

MSC:

33B15 Gamma, beta and polygamma functions
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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
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