Refinement of Gautschi's harmonic mean inequality for the gamma function

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Abstract

In 1974, W. Gautschi proved that

1 < \frac{2}{1/Γ ( x ) + 1/Γ ( 1/ x )} for 0 < x \neq = 1.

Here, we present the following refinement:

1 < Γ (\frac{2}{x + 1/ x}) < \frac{2}{1/Γ ( x ) + 1/Γ ( 1/ x )}, 0 < x \neq = 1.

Horst Alzer, Refinement of Gautschi's harmonic mean inequality for the gamma function. Rend. Sem. Mat. Univ. Padova 153 (2025), pp. 211–219