This is a doubt in the Littlewood's first estimate(in which it is assumed that Riemann hypothesis is true) given in this book, page 433.
Let $s = \sigma + it$ and $\delta $ such that $\frac12 + \delta \le \sigma \le 1,$ and $t$ is sufficiently large.
We apply Borel Caratheodory theorem to $\log \zeta(s)$ with circles $|z - (2+it)| = \frac32 -\frac{\delta}{2}$ and $|z - (2+it)| = \frac32 -\delta$
It has been already proved that $|\zeta(s)| \ll_A t^A$ on the outer circle where $A > 0$. So by Borel Caratheodory theorem we get $$|\log \zeta(s) | \le \frac{2(3-2\delta)}{\delta} A \log t + \frac{6-3\delta}{\delta}|\log \zeta(2+it)|. $$ Then the author writes the inequality $$ \frac{2(3-2\delta)}{\delta} A \log t + \frac{6-3\delta}{\delta}|\log \zeta(2+it)| < \frac{A}{\delta} \log t$$
I do not understand how to bound $|\log \zeta(2+it)|,$ and get the above inequality.
Suppose I accept the inequality and for $\delta \ne 0$ I get $$(5-4\delta) A \log t + (6-3\delta)|\log \zeta(2+it)| < 0 $$
Since, $\frac12 + \delta \le 1$ we have $\delta \le 1/2.$ So, in the above inequality the left hand side is positive which is incorrect.
I have attached the screen shot of the equation in the book.
Note : In this book The Theory of the Riemann Zeta-function by Edward Charles Titchmarsh, D. R. Heath-Brown, page 336 the same equation is written, so I am not sure If I am missing something obvious.
Notice that
$$ \log\zeta(s)=\sum_{n\ge2}{\Lambda(n)\over\log n}\cdot{1\over n^s}, $$
for $\Re(s)>1$, so we have
$$ |\log\zeta(2+it)|\le\sum_{n\ge2}{\Lambda(n)\over\log n}\cdot{1\over n^2}=O(1). $$