Approximation of homogeneous subharmonic functions

Let $u$ be a positive homogeneous subharmonic function, i.e.
$$ u(tz)=tu(z),\qquad t>0,\quad z\in\mathbf C, $$
and let $\mu$ be its associated measure. Let the function $\rho(z)$ be such that
$$ \mu(\{w\colon|w-z|<\rho(z)\})=1. $$
Then there exists an entire function $L$ for which
\begin{gather*} |L(z)|\leqslant\exp u(z),\qquad z\in\mathbf C,\\ |L'(a)|\leqslant\exp(u(a)-\ln\rho(a)+O(\ln^\frac45\rho(a)\ln\ln\rho(a))),\qquad L(a)=0. \end{gather*}

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