I am reading the following growth estimate of the reciprocal of the gamma function from Stein and Shakarchi's Complex Analysis.
In this proposition, the bounding constants $c_1$ and $c_2$ seem to be independent of $s$. However, from the proof, I am not sure how to get such bounds. Firstly, how do we get that for every $\epsilon>0$, there is a bound $c(\epsilon)$ so that $|1/\Gamma(s)| \le c(\epsilon) e^{c_2 |s|^{1+\epsilon}}$ from the bound $c_1 e^{c_2}|s|\log |s|$?
Next is the proof of the proposition. Here $\sigma=Re(s)$, and they bound the second term by $ce^{(|s|+1)\log(|s|+1)}e^{\pi |s|}$. And say this itself is majorized by $c_1 e^{c_2 |s| \log |s|}$. But I cannot figure out how to get the constants $c_1$ and $c_2$ independent of $s$.
Finally they deal with the first term as follows. In this case, they get a bound $ce^{\pi |s|}$. But we need a bound of the form $c_1 e^{c_2 |s| \log |s|}$. Again, how do we bound $ce^{\pi |s|}$ by $c_1 e^{c_2 |s| \log |s|}$ independent of $s$?
I would greatly appreciate some help in establishing these bounds.
Best Answer
For any $\varepsilon>0$, the function $x \mapsto x\log x - x^{1 + \varepsilon }$ will eventually tend to $-\infty$ because the $\log$ eventually grows slower than any positive power. Also, it tends to $0$ as $x\to0+$. Hence, there is a $k(\varepsilon)>0$, such that $x \mapsto x\log x - x^{1 + \varepsilon } <k(\varepsilon)$ for all $x>0$. Consequently $$ c_1 e^{c_2 \left| s \right|\log \left| s \right|} \le (c_1 e^{c_2 k(\varepsilon)}) e^{c_2\left| s \right|^{1 + \varepsilon } } . $$ For your second question, note that $$ (\left| s \right| + 1)\log (\left| s \right| + 1) + \pi \left| s \right| < (5\left| s \right|\log \left| s \right|) + 5 $$ for $|s|>0$. Thus, $$ ce^{(\left| s \right| + 1)\log (\left| s \right| + 1)} e^{\pi \left| s \right|} < (ce^5 )e^{5\left| s \right|\log \left| s \right|} . $$ From this, it is also obvious that $$ ce^{\pi \left| s \right|} < (ce^5 )e^{5\left| s \right|\log \left| s \right|} . $$