An improvement of Massart's inequality for the distribution of Smirnov's statistic

Let $F_n$ be the empirical distribution function for a sample of independent identically distributed random variables with distribution function $F.$ The main result is the inequality
\begin{equation*} \mathbb{P}\{\sqrt n\sup_{|x|<\infty}(F_n(x)-F(x))>\lambda\}\leq \exp\{-2\lambda^2-\lambda^4/36n\} \end{equation*}
for $n\geq 39, \min\{ \gamma n^{-1/6}, \sqrt{\ln 2/2}\}\leq\lambda\leq\sqrt n/2, \gamma=1.0841.$ It is also proved for the same $n$ and $\lambda \leq \sqrt{n}/2$ that
\begin{equation*} \mathbb{P}\{\sqrt n\sup_{|x|<\infty}(F_n(x)-F(x))>\lambda\}\leq 2\exp^{(\ln 2)^2/(144n)}\exp\{-2\lambda^2-\lambda^4/36n\}. \end{equation*}
In particular cases $n=2,3,4$ it is proved that
\begin{equation*} \mathbb{P}\{\sqrt n\sup_{|x|<\infty}(F_n(x)-F(x))>\lambda\}\leq \exp\{-2\lambda^2-4\lambda^4/9n\}. \end{equation*}