Approximation of subharmonic functions by logarithms of moduli of entire functions in integral metrics (Q2782544)

Let \(u\) be a subharmonic function on \(\mathbb{C}\) and let \(n(r)\) denote the mass assigned to \(\{|z|\leq r\}\) by the associated Riesz measure. Also, let \(\|\cdot\|_q\) denote the \(L^q\)-norm on \((0,2\pi)\), where \(q\geq 1\). It is shown that, if \(u\) is of finite order, then there is an entire function \(f\) such that NEWLINE\[NEWLINE\|u(re^{i\phi})- \log|f(re^{i\phi})|\|_q= O(\log r)NEWLINE\]NEWLINE as \(r\to\infty\). On the other hand, if \(u\) is of infinite order, then there is an entire function \(f\) such that the above \(q\)-norm is \(O(\log r+\log n(r))\) as \(r\to\infty\), possibly omitting a set of finite measure. This result, which is shown to be sharp, is proved using ideas of \textit{R. S. Yulmukhametov} [Anal. Math. 11, 257-282 (1985; Zbl 0594.31005)] together with some new ingredients.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].