Geodesic
Approximation of homogeneous subharmonic functions
Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 507-523 Cet article a éte moissonné depuis la source Math-Net.Ru

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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*} Bibliography: 6 titles. 
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R. S. Yulmukhametov. Approximation of homogeneous subharmonic functions. Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 507-523. http://geodesic.mathdoc.fr/item/SM_1989_62_2_a10/

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