The following is a text from Section 5.1 of Titchmarsh's book The Theory of the Riemann Zeta-Function:
For each $\sigma$ we define a number $\mu(\sigma)$ as the lower bound of numbers $\xi$ such that $\zeta(\sigma + it) = O (|t|^{\xi})$. It follows from the general theory of Dirichlet series that, as a function of $\sigma$, $\mu(\sigma)$ is continuous, non-increasing, and convex downwards in the sense that no arc of the curve $y = \mu(\sigma)$ has any point above its chord; also $\mu(\sigma)$ is never negative.
The author cites to his other book Theory of Functions for the proofs. This book was useful any other time when a subject was cited, but for this part I couldn't understand the proofs. A reference for proofs of the claims above about $\mu(\sigma)$ would be much appreciated, especially a proof for the convexity. I could not find some book or lecture notes for this.