Sharp estimate of approximation by the Szasz–Mirakjan operator in terms of the second modulus of continuity

Let $M_n$ be the Szasz-Mirakjan operator and $f : [0, \infty) \to \mathbb{R}$ be a function bounded on $[0, a]$ together with $M_nf$. Then
\begin{equation*} \|M_nf-f\|_{[0, a]} \le \omega_2\left(f, 4\cdot\sqrt{\frac{a}n}\right). \end{equation*}
The constant $1$ in front of $\omega_2$ is sharp for every $n in \mathbb{N}$. Until now a similar result was known only for the Bernstein operator.