Not so long before I asked this question, I encountered a question that asked me to prove a proposition, but I'm not sure how to go about it.
Prove that $\gamma'(n+1)\gamma(n+1)$ is incremental
How to prove that $\gamma'(n+1)\gamma(n+1)$ is incremental
3
$\begingroup$
analysis
-
1For what it's worth, the verb is _prove_, not _proof_. _Proof_ is a noun. (And _prove_ is pronounced "_proov_" for some reason.) – 2017-01-11
1 Answers
1
I assume that you are trying to show that $\Gamma'/\Gamma$ on $]1,\rightarrow]$ is increasing.
By Bohr-Mollerup Theorem, $\Gamma$ is logarithmically convex.
$\Gamma$ is meromorphic, so it is smooth on $]0,\rightarrow]$. Hence $\log \Gamma$ is differentiable, with $$ \mathrm D (\log \Gamma) = \frac{\Gamma'}{\Gamma} $$ Moreover, $\log \Gamma$ is convex, so its derivative is increasing. (After you modified your question, your proof would be complete after this step)
If you can evaluate $\log\Gamma$ at $1$, you would have $\mathrm D(\log \Gamma) \geq 0$ on $]1,\rightarrow]$, which completes the proof.