Subharmonic addition to the Beurling-Malliavin multiplier theorem

We prove a version of the Beurling-Malliavin multiplier theorem. This version is formulated here in a simplified form. Let $u\equiv ̸-\mathrm{\infty }$ and $M\equiv ̸-\mathrm{\infty }$ be a pair of subharmonic functions on the complex plane $\mathbb{C}$ with positive parts $u^{+}:=\sup \{u,0\}$ and $M^{+}$ such that $$\mathrm{type}[u]:=\underset{z\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{u^{+}(z)}{\mathrm{\mid }z\mathrm{\mid }}<+\mathrm{\infty },\mathrm{type}[M]<+\mathrm{\infty },{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{u^{+}(x)+M^{+}(x)}{1+x^2}\mathrm{d}x<+\mathrm{\infty }\mathrm{.}$$ If $\mathrm{type}[u]<a<+\mathrm{\infty }$, $0<b<+\mathrm{\infty }$, and $\mathrm{type}[M]<c<+\mathrm{\infty }$, then there are an entire function $h\equiv ̸0$ with $\mathrm{type}[\log \mathrm{\mid }h\mathrm{\mid }]<c$ and a subset $iY$ in the imaginary axis $i\mathbb{R}$ of linear Lebesgue measure $<b$ such that the function $h$ is bounded on the real axis and $u(z)-M(z)+\log \mathrm{\mid }h(z)\mathrm{\mid }\le a\mathrm{\mid }\mathrm{\Im }z\mathrm{\mid }$ on each straight line parallel to the real axis and not intersecting $iY$.