For a subharmonic function u ≢ −∞ defined in the closure of a domain we prove lower estimates at points of the domain in terms of the maximum of the function u on the boundary. The result is new even for plane domains and shows that the key role in such estimates is played by the Harnack distance. We obtain estimates for the Harnack distance of two types. The estimates of the first type are based on the geometric concept of the entropy of arcwise connectedness, which was already used by the authors to study the distribution of zeros of holomorphic functions in weighted classes. The estimates of the second type are based on the new notion of the separatedness of a subset of the domain, introduced in the present paper for the first time.
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Translated from Problemy Matematicheskogo Analiza 113, 2022, pp. 113-126.
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Khabibullin, B.N., Taipova, E.U. Lower Estimates for Subhramonic Functions and the Harnack Distance. J Math Sci 260, 833–849 (2022). https://doi.org/10.1007/s10958-022-05731-0
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DOI: https://doi.org/10.1007/s10958-022-05731-0