An estimate for the subharmonic difference of subharmonic functions.~I

I. F. Krasichkov-Ternovskii

Sbornik. Mathematics, Tome 31 (1977) no. 2, pp. 191-218

Résumé

Let $u$, $v$ and $w=u-v$ be subharmonic functions in the half-plane $\Pi:\operatorname{Re}\omega>v$ and suppose that $u(\omega)$ and $v(\omega)$ are majorized by a positive function of the form $M(\omega)=\rho T(\rho,\tau)$, where $\rho=|\omega|$ and $\tau=1-\frac2\pi|\arg\omega|$. An inequality for the subharmonic difference $w=u-v$ is obtained in terms of the function $T(t,\tau)$, $0$, $0\tau1$, which then gives an estimate for the difference from above. This inequality is carried over by conformal mappings to a class of regions with cusps (horn regions). Bibliography: 12 titles.