Timeline for Possible limit involving the gamma function

7 events

when toggle format	what		by	license	comment
Jun 30, 2020 at 16:03	vote	accept	Clark Kimberling
Jun 30, 2020 at 0:45	comment	added	Iosif Pinelis		@GregMartin : The above proof will indeed hold if you write $h(x)$ and $xh(x)$ instead of $\Gamma(x)$ and $\Gamma(1+x)$, respectively, everywhere in the proof, assuming that $h(x)-1/x$ is bounded.
Jun 30, 2020 at 0:01	comment	added	Greg Martin		More generally, if $h(x)$ is any function on $(0,1]$ such that $h(x)-\frac1x$ is continuous at $x=0$, then $\int_0^1 h(x)^{n/(n+1)}\,dx - n \to \int_0^1 \big( h(x)-\frac1x \big)\,dx$. The proof of this more general statement along the lines of your solution is even cleaner (it shows that the functional equation for $\Gamma$ is irrelevant, for example).
Jun 29, 2020 at 15:21	history	edited	Iosif Pinelis	CC BY-SA 4.0	added 186 characters in body
Jun 29, 2020 at 15:14	history	edited	Iosif Pinelis	CC BY-SA 4.0	edited body
Jun 29, 2020 at 15:06	history	edited	Iosif Pinelis	CC BY-SA 4.0	edited body
Jun 29, 2020 at 15:00	history	answered	Iosif Pinelis	CC BY-SA 4.0