Geodesic
An estimate for the subharmonic difference of subharmonic functions. I
Sbornik. Mathematics, Tome 31 (1977) no. 2, pp. 191-218 
I. F. Krasichkov-Ternovskii. An estimate for the subharmonic difference of subharmonic functions. I. Sbornik. Mathematics, Tome 31 (1977) no. 2, pp. 191-218. http://geodesic.mathdoc.fr/item/SM_1977_31_2_a4/
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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, $0<\tau<1$, 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). Bibliography: 12 titles.

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[6] V. I. Matsaev, “Faktorizatsiya v nekotorykh klassakh funktsii”, Tezisy dokl. Vsesoyuzn. konf. po TFKP, Kharkov, 1971, 138–139

[7] L. Ahlfors, “Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen”, Acta Soc. Scient. fennica, ser. A, 1:9 (1930), 2–40 | Zbl

[8] S. E. Varshavskii, “Konformnye otobrazheniya beskonechnykh polos”, Matematika, 2:4 (1958), 67–116

[9] E. Kolligvud, A. Lovater, Teoriya predelnykh mnozhestv, izd-vo «Mir», Moskva, 1971 | MR

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[11] R. Nevanlinna, Odnoznachnye analiticheskie funktsii, OGIZ, Moskva–Leningrad, 1941

[12] I. I. Privalov, Granichnye svoistva analiticheskikh funktsii, Gostekhizdat, Moskva–Leningrad, 1950