On estimation of the maximal probability for sums of lattice random variables

This paper deals with the estimation of the maximal probability for sums of independent unimodal symmetric lattice random variable $\xi_k$. The author proves the following inequality
$$ \sup_x\mathbf{P}(S_n=x)\le\sqrt{\frac6{\pi}}\frac{p_0}{\sqrt{n(1-p_0^2)}}\bigl(1+\frac{c}{\sqrt{n}}\bigr) $$
where $S_n=\xi_1+\dots+\xi_n, p_0=\sup_x\mathbf{P}(\xi_k-x)$ and $c$ is an absolute constant (one may take $c=2$).