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