# a proof for the following Gamma function inequality

Could you please provide or point me to a proof of inequality 5.6.8 found at this site? That is,

$\left|\frac{\Gamma(z+a)}{\Gamma(z+b)}\right| \leq \frac{1}{|z|^{b-a}}$

for $z\in \mathbb{C}$, $a,b\in\mathbb{R}$, and $a≥0, b≥a+1, Re(z)>0$.

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At that link it has additional assumptions which you left out: $a \ge 0$, $b \ge a + 1$, $\text{Re}(z) > 0$. –  Robert Israel May 4 '12 at 4:03
The (i) link to the right of the inequality gives a reference to Paris and Kaminski (2001): dlmf.nist.gov/bib/P#bib1827 –  Robert Israel May 4 '12 at 4:08
Also you got the left side wrong: it should be $\left| \dfrac{\Gamma(z+a)}{\Gamma(z+b)}\right|$ –  Robert Israel May 4 '12 at 4:31
Ah thanks for that! –  yep May 4 '12 at 5:11
The book is available at Google Books, but p. 34, to which the citation points, isn't shown (for me). –  joriki May 4 '12 at 5:37

For real $x>0$, the strict log-convexity of $\Gamma$ (see the end of this answer) implies that for $a\ge0$, $b\ge1$ and $b\ge a$, $$\frac{\log\Gamma(x+1)-\log\Gamma(x)}{1}\le\frac{\log\Gamma(x+b)-\log\Gamma(x+a)}{b-a}\tag{1}$$ Which translates to $$\frac{\Gamma(x+a)}{\Gamma(x+b)}\le\frac{1}{x^{b-a}}\tag{2}$$ Simpler, but not as general.