ratio of gamma function How can we show that $f(x)=\Gamma(x+s)/\Gamma(x)$ is an increasing function for $x>1$ and $0<s<1$. I have checked by plotting in sage or wolframalpha. So the result is true for sure.
I started proving by showing derivative of $f$ positive for $x>1$. But then we have $$f'(x) = f(x)(\psi(x+s)-\psi(x)),$$ where $\psi$ is digamma function as defined in the literature. But now I need to show $\psi$ is an increasing function, which I again know is true, though i cannot figure out how to show it formally. Any other ways of proving $f$ is an increasing function are welcome.   
 A: $$y=\frac{\Gamma (x+s)}{\Gamma (x)}\implies \log(y)=\log (\Gamma (x+s))-\log (\Gamma (x))$$
Now, using Stirling approximation
$$\log (\Gamma (t))=t (\log (t)-1)+\frac{1}{2} \left(\log (2 \pi
   )-\log \left({t}\right)\right)+\frac{1}{12 t}+O\left(\frac{1}{t^2}\right)$$ apply it twice and continue with Taylor series to get
$$\log(y)=s \left(\log(x)-\frac{1}{2 x}-\frac{1}{12 x^2}\right)+O\left(s^2\right)$$
$$y=e^{\log(y)}=1+s \left(\log(x)-\frac{1}{2 x}-\frac{1}{12 x^2}\right)+O\left(s^2\right)$$ and the quantity inside brackets is an increasing function of $x$ and it is positive as soon as $x >1.5$.
Edit
Using Stirling approximation up to $O\left(\frac{1}{t^{21}}\right)$, expanding $\log(y)$ up to $O\left(s^{20}\right)$ and making $s=1$ what is obtained is 
$$\lim_{s\to 1} \, \log \left(\frac{\Gamma (x+s)}{\Gamma (x)}\right)=\log (x)+\frac{1}{420 x^{20}}+O\left(\frac{1}{x^{21}}\right)$$ that is to say
$$\lim_{s\to 1} \, \frac{\Gamma (x+s)}{\Gamma (x)}=x+\frac{1}{420 x^{19}}+O\left(\frac{1}{x^{20}}\right)$$ instead of $x$.
