The lower order of functions of the class $\mathfrak B$

S. Yu. Favorov

Matematičeskie zametki, Tome 18 (1975) no. 3, pp. 445-452.

Résumé

The class of functions $\Phi(z,t)$ defined for $z\in C^n$ and $t\ge0$ such that the functions $\Phi(z,|w|)$, $w\in C$, are plurisubharmonic in $C^{n+1}$ is called the class $\mathfrak B$. A typical example of functions of the class $\mathfrak B$ are functions of the form $\ln M_g(z,t)=\ln\sup\limits_{|w|=t}|g(z,w)|$ where $g(z,w)$, $z\in C^n$, $w\in C$, is an entire function in $C^{n+1}$. In this note it is proved under certain restrictions on the function $\Phi(z,t)\in\mathfrak B$ that its lower order relative to the variable t is the same for all $z\in C^n$ except, possibly, for the points $z$ of a set of zero $\Gamma$ capacity.