Integrals with a Meromorphic Function or the Difference of Subharmonic Functions over Discs and Planar Small Sets

Хабибуллин, Б. Н.

doi:10.1134/s1995080221060111

Lobachevskii J Math

2021

DOI: 10.1134/s1995080221060111

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Integrals with a Meromorphic Function or the Difference of Subharmonic Functions over Discs and Planar Small Sets

Б. Н. Хабибуллин

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2022

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Cited by 2 publications

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Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic

Хабибуллин

2022

Sb. Math.

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Integral inequalities for integrals of differences of subharmonic functions against Borel measures on balls in multidimensional Euclidean spaces are obtained. These integrals are estimated from above in terms of the product of the Nevanlinna characteristic of the function and various characteristics of the Borel measure and its support. The main theorem, which is a criterion concerning such estimates, presents several equivalent statements of different character. All results are new for the logarithms of the moduli of meromorphic functions on discs in the complex plane. They cover all preceding results, which go back to the classical small arcs lemma of Edrei and Fuchs, as special cases. Integrals against Borel measures with support on fractal sets are also allowed; in this case estimates are in terms of the Hausdorff measure and Hausdorff content of the support of the measure. Special cases of functions on the whole complex plane or space, or in the unit disc or ball, which are important for applications are distinguished, as well as cases involving integration against arc length over subsets of Lipschitz curves and against surface area over subsets of Lipschitz hypersurfaces. Bibliography: 42 titles.

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Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic

Хабибуллин

2022

Sb. Math.

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Интегралы От Разности Субгармонических Функций По Мерам И Характеристика Неванлинны

Хабибуллин¹

2022

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Получены интегральные неравенства для интегралов от разностей субгармонических функций по мерам Бореля на шарах в многомерном евклидовом пространстве. Эти интегралы оцениваются сверху через произведение характеристики Неванлинны функции на различные характеристики меры Бореля и ее носителя. Основная теорема - критерий о таких оценках - дается с несколькими эквивалентными утверждениями различной природы. Все результаты новые для логарифмов модулей мероморфных функций на кругах в комплексной плоскости. Они содержат в себе как частные случаи все предшествующие результаты, восходящие к классической лемме Эдрея-Фукса о малых дугах. Допускается интегрирование по мерам Бореля с носителем на фрактальных множествах, а оценки в этих случаях даются через меру и обхваты Хаусдорфа носителя меры Бореля. Отдельно отмечены важные в применениях частные случаи функций во всей комплексной плоскости и пространстве, в единичном круге или шаре, а также интегрирования по длине на подмножествах липшицевых кривых и по площади на подмножествах липшицевых гиперповерхностей. Библиография: 42 названия.

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Integrals with a Meromorphic Function or the Difference of Subharmonic Functions over Discs and Planar Small Sets

Cited by 2 publications

References 7 publications

Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic

Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic

Интегралы От Разности Субгармонических Функций По Мерам И Характеристика Неванлинны

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