Is the lower incomplete gamma function (https://en.wikipedia.org/wiki/Incomplete_gamma_function#Definition) bounded by the gamma function in the right half plane or in a strip parallel to the imaginary axis? In the sense that there exists a $\,C>0\,$ so that for all $\,x>0\,$ and all $\,s\,$ with $\,0<a\leq\Re(s)\leq b\,$ holds
$$|\gamma(s,x)|\leq C|\Gamma(s)| \,.$$
I have found the solution myself. The incomplete gamma function can't be bounded by the gamma function in a strip. Assume there exists a $\,C_\sigma>0\,$, such that for all $\,x,t\in\mathbb{R}\,$ holds $$|\gamma(\sigma+it,x)|\leq C_\sigma|\Gamma(\sigma+it)|\,.$$ Then we see from Stirling's formula for the gamma function that the decay of the gamma function for $\,|t|\rightarrow\infty\,$ is like $\,\exp(-\frac{\pi}{2}|t|)\,$. Hence the decay of the incomplete gamma function must be exponentially in imaginary directions too. But from the recurrence relation $$\gamma(\sigma+1+it,x) = (\sigma+it)\gamma(\sigma+it,x)-x^{\sigma+it}e^{-x}$$ would follow $$\lim_{|t|\rightarrow\infty}x^{\sigma+it}e^{-x} = 0\,,$$ what is obviously nonsens.