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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, 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
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