An estimate for the subharmonic difference of subharmonic functions. I

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 $T(t,\tau)$, $0<t<\infty$, $0<\tau<1$, 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).
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