Bounds and behaviour of the quantities $P(x)$, $\Delta(x)$ on short intervals

We study the dependence of upper bounds for the quantity $|P(n)|$ on certain properties of the behaviour of $|P(x)|$ in a neighbourhood of the point $x=n$. In particular, it is proved that, if $n$ is a point of local maximum of the quantity $|P(x)|$, where $|P(n)|>Cn^{1/4}$ and the maximum is broad ($|P(x)-P(n)|<B|P(n)|$, $B<1$, if $|x-n|<Cn^{1/2-\varepsilon}$), then $|P(n)|>Cn^{1/4+\varepsilon}$.