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Is the function $x\mapsto (\Gamma(x+1))^{1/x}$ concave?

Here Gamma is the Euler Gamma function.

Some papers say that this is true and proved in:
"J. Sándor; Sur La Fonction Gamma; Publ. C. Rech. Math. Pures Neuchâtel, Série I, 21 (1989), 4-7"
which is impossible to find online and not even listed in MathSciNet.

If it is true, it would imply a (positive) solution to question
Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

Carlo Mantegazza
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  • @Carlo Mantegazza Have you tried contacting the author of the cited publication? – njuffa Feb 27 '25 at 20:04
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    I found the referenced publication online here.

    It does not state concavity but rather log-concavity! (See Theorem 11, page 209)

     – epartow Feb 27 '25 at 20:28
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    Concavity holds for $x\geq 7$. A proof is in the reference provided by partini on page 207. – Mittens Feb 27 '25 at 21:27
  • For njuffa: Yes, I wrote to him but he did not answered, up to now. For partini: Thanks, but log-concavity is not enough. For Mittens: Thanks a lot, I will check it carefully. – Carlo Mantegazza Feb 27 '25 at 22:43
  • @CarloMantegazza: Integration methods (Holder inequality for example) shows convexity and long-convexity of $\Gamma(x)$ and $\frac1x\log(\Gamma(x)$. The the of properties you mentioned seem to require deeper analysis of the digamma function and its derivatives. The paper of Sandor does that kind of analysis. – Mittens Feb 27 '25 at 22:59
  • Yes I guess so. What I am asking seems to be absolutely non-trivial. I will check if everything is correct, since I found subsequent (in time) papers showing the log-concavity of the function I wrote, not mentioning the result of Sandor (which is clearly stronger). – Carlo Mantegazza Feb 27 '25 at 23:23

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