Approximation of homogeneous subharmonic functions

R. S. Yulmukhametov

Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 507-523

Résumé

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.