Bounding $\zeta (s)$ in the critical strip

Assuming the Riemann Hypothesis, we make use of the recently discovered \cite{CLV} extremal majorants and minorants of prescribed exponential type for the function $\log \left(\frac{4+x^2}{(\alpha -1\mathrm{/}2{)}^2+x^2}\right)$ to find upper and lower bounds with explicit constants for $\log \mathrm{\mid }\zeta (\alpha +it)\mathrm{\mid }$ in the critical strip, extending the work of Chandee and Soundararajan \cite{CS}.