Given $x\in\mathbb{R}$ ,$\forall x\ge1$ seems to hold the following inequality: $$\Gamma(x)+\Gamma\left(\frac{1}{x}\right)\le\Gamma\left(1+x+\frac{1}{x}\right)$$ where the sign of equality holds only for $x=1.$ How can it be proven? Thanks.
2026-04-07 05:00:27.1775538027
An inequality involving the Gamma function
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A proof of this (rather non-trivial for $x\approx 1$) inequality was provided by Jameson in $2012$ in 'An inequality for the gamma function conjectured by D. Kershaw'.
Concerning the product a proof is in Giordano and Laforgia's paper.