On Subharmonic Functions of the First Order with Restrictions on the Real Axis - Poedintseva, I. V. (Журнал математичної фізики, аналізу, геометрії, 2008)
Subharmonic functions $v$ of the first proximate order $ρ(r)$ with the integral $\displaystyle\int_0^R \frac{t^{1-ρ(t)}(v(t)+v(-t))}{1+t^2}dt$ bounded with respect to $R$ are studied. This is an extension of a result by N.I. Akhiezer, who studied the case $ρ(r)\equiv 1$, $v(z)=\ln |f(z)|$, where $f(z)$ is an entire function. Mathematics Subject Classification: 31A05.