Geodesic
Some estimates in the class of analytic functions of bounded type
Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 267-284 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the class $A_M$ of functions regular in the disk $|\zeta|<1$ which for any $r$, $0\leqslant r<1$, satisfy the condition $$ \int_0^{2\pi}\ln^+|f(re^{i\theta})|\,d\theta\leqslant2\pi M, $$ where $M$ does not depend on the function. Using a parametric representation of this class, the authors find exact estimates of the mean arithmetic value and the mean geometric value of the modulus of the function at equally spaced points of the circumference, estimates of the moduli and arguments of the function, the moduli of the derivatives and other values for the class $A_M$ and certain of its subclasses. The solution of these problems is based on variation formulas introduced earlier by one of the authors (RZhMat., 1967, 11B99). Bibliography: 14 titles. 
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V. P. Vazhdaev; S. A. Gel'fer. Some estimates in the class of analytic functions of bounded type. Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 267-284. http://geodesic.mathdoc.fr/item/SM_1971_13_2_a4/

[1] G. Ts. Tumarkin, S. Ya. Khavinson, Kachestvennye svoistva reshenii ekstremalnykh zadach nekotorykh tipov, Issledovaniya po sovremennym problemam teorii funktsii kompleksnogo peremennogo, Fizmatgiz, Moskva, 1960

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

[3] S. A. Gelfer, “Metod variatsii v teorii funktsii ogranichennogo vida”, Matem. sb., 69(111) (1966), 422–433 | MR | Zbl

[4] P. Montel, Normalnye semeistva analiticheskikh funktsii, ONTI, Moskva–Leningrad, 1936

[5] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, Moskva, 1966 | MR

[6] R. Nevanlinna, “Ober beschrankte Funktionen, die in gegebenen Punkten vorgeschiebene Werte annehmen”, Ann. Acad. Sci. Fenn., Ser. A, XIII:1 (1919)

[7] G. M. Goluzin, “Otsenka proizvodnoi dlya funktsii, regulyarnykh i ogranichennykh v kruge”, Matem. sb., 16(58):3 (1945), 295–306 | MR | Zbl

[8] G. V. Kuzmina, “Opredelenie naimenshego radiusa odnolistnosti dlya odnogo klassa analiticheskikh funktsii”, DAN SSSR, 117:5 (1957), 751–754 | MR

[9] L. V. Kresnyakova, “O regulyarnykh funktsiyakh s ogranichennym srednim modulem”, DAN SSSR, 147:2 (1962), 290–293 | Zbl

[10] S. Ya. Khavinson, “O radiusakh odnolistnosti, zvezdoobraznosti i vypuklosti odnogo klassa analiticheskikh funktsii v mnogosvyaznykh oblastyakh”, Izv. VUZ'ov. Matematika, 1958, no. 3, 233–240 | Zbl

[11] Yu. E. Alenitsyn, S. Ya. Khavinson, “O radiuse $p$-listnosti dlya ogranichennykh analiticheskikh funktsii v mnogosvyaznykh oblastyakh”, Matem. sb., 52(94) (1960), 653–657 | Zbl

[12] S. A. Gelfer, L. V. Kresnyakova, “Metod variatsii v teorii analiticheskikh funktsii s ogranichennym srednim modulem”, Matem. sb., 67(109) (1965), 570–585 | MR | Zbl

[13] I. M. Galperin, “Nekotorye otsenki dlya ogranichennykh v edinichnom kruge funktsii”, Uspekhi matem. nauk, XX:1(121) (1965), 197–202 | MR

[14] A. A. Goldberg, “Ob odnom neravenstve, svyazannom s funktsiyami, vypuklymi otnositelno logarifma”, DAN USSR, 1957, no. 3, 227–230