Polynomial growth of L-function

Question

Let $f$ be a newform, $L(f,s)$ the related L-function with Ramanujan-Petersson conjunction $|\lambda(n)|\leq \sigma_0(n)$ (divisor counting function). How can I see that it grows only polynomially in vertical direction in any strip, especially the critical strip $0\leq \Re(s)\leq 1$? My idea was to use the Mellin transform and get $$|L(f,s)|=\int_0^\infty |L(f,t)|t^{s-1} \,dt\leq \int_0^\infty \bigg|\sum_{n\geq1}\sigma_0(n)n^{-t}\bigg|t^{\Re(s)-1} \,dt.$$

Answer 1

The prototypical way to bound polynomial growth in the critical strip is to use the Phragmén–Lindelöf Principle. In many other places, this is referred to as the Convexity Principle, and bounds coming from it are called Convexity bounds.

Now that you know the names of the methods, it's straightforward to find several various proofs using these methods for these convexity bounds.