Let f be a circumferentially mean p-valent function in the disk |z| < 1 with Montel’s normalization f(0) = 0, f(ω) = ω (0 < ω < 1). Under an additional assumption on the covering of concentric circles by f, sharp lower and upper bounds on the modulus |f(z)| for some z ∈ (−1, 0) are established. It is shown that for nontrivial bounds to hold, such an assumption is necessary. Bibliography: 15 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 429, 214, pp. 44–54.
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Dubinin, V.N. Inequalities for the Moduli of Circumferentially mean p-Valent Functions. J Math Sci 207, 832–838 (2015). https://doi.org/10.1007/s10958-015-2407-4
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DOI: https://doi.org/10.1007/s10958-015-2407-4