Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form:
$$ \left|\zeta\left(\frac12+it\right)\right|\leqslant c_0|t|^{\varepsilon(|t|)} $$
where $\varepsilon(t)$ is a decreasing function, $\varepsilon(2t)\geqslant\frac12\varepsilon(t)$, $\varepsilon(t)\geqslant\frac1{\sqrt{\log t}}$. Then it is proved that for $|t|\geqslant T_0$ the $disk\left\{s: \left|s-\frac12-it\right|\leqslant v\right\}$ contains at most $20v\log|t|$ zeros of $\zeta(s)$ if $\frac12\geqslant v\geqslant\sqrt{\varepsilon(t)}$. There exists an absolute constant $A$ such that for $|t|\geqslant T_1$ the $disk\left\{s: \left|s-\frac12-it\right|\leqslant A\varepsilon^{1/3}(t)\right\}$ contains at least one zero of $\zeta(s)$.