For $\theta=p+iq$ with $0\leq p\leq1$,
$$
|e^{\theta\log S}|=S^p.
$$
The elementary inequality $x^p\leq1+x$ for $x>0$ gives
$$
\mathbb E|e^{\theta\log S}|
=\mathbb E[S^p]\leq1+\mathbb ES=2.
$$
Therefore the <Mellin transform> $M(\theta)=\mathbb E[S^\theta]$ is absolutely well-defined throughout the strip and
$$
\boxed{|M(\theta)|\leq2.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 2 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-211/2/a/solution