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5 votes
3 answers
383 views

The exact constant in a bound on ratios of Gamma functions

The answer to another question (Upper bound of the fraction of Gamma functions) gave an asymptotic upper bound for an expression with Gamma functions: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
1 vote
1 answer
507 views

Upper bound of the fraction of Gamma functions

Is there a simple upper bound of the following fraction of Gamma functions for any $a,b\geq1/2$: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$ An upper bound in the following form is ...
5 votes
2 answers
243 views

Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?

This has received no full solution at StackExchange. As per https://dlmf.nist.gov/8.10#E13 we have $$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma% \left(n,n-1\...