Entire minorizing functions: an experience of application of Matsaev–Ostrovskii–Sodin's estimates

Let $q$ be a positive function on the positive semi-axis of the complex plane $\mathbb C$. Special estimates for positive subharmonic canonical integral of genus $1$ and for their Riesz measures from recent work of V. I. Matsaev, I. V. Ostrovskii, and M. I. Sodin are applied to the proof of existence an entire function $f (z)\not\equiv 0$, $z\in\mathbb C$, with certain restriction of growth of $|f|$ on $\mathbb C$, such that $|f(x)|\le e^{-q(|x|)} $ at all $x\in\mathbb R$.