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Владикавказский математический журнал

Vladikavkaz Mathematical Journal

Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН)

On a New Class of Meromorphic Functions Associated with Mittag-Leffler Function

О новом классе мероморфных функций, ассоциированном с функцией Миттаг-Леффлера

10.46698/p1426-1765-3037-f

15762

03 2025

27 1 70 86

https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=15788&SECTION_ID=620

Муругусундарамурти

Г.

Murugusundaramoorthy

gmsmoorthy@yahoo.com

Виджая

К.

Vijaya

kvijaya@vit.ac.in

Технологический институт Веллора, Школа передовых наук, Индия, 632014, Теннесси, Веллор School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, TN, India

The Mittag-Leffler function arises naturally in solving differential and integral equations of fractional order and especially in the study of fractional generalization of kinetic equation, random walks, Levy flights, super-diffusive transport and in the study of complex systems. In the present investigation, the authors define a new class \(\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\) of meromorphic functions defined in the punctured unit disk \(\Delta^*:= \{z\in\mathbb{C}: 0<|z|<1\}\) based on Mittag-Leffler. We discuss extensively its characteristic properties like coefficient inequalities, growth and distortion inequalities, as well as closure results for \(f\in\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\). Properties of a certain integral operator and its inverse defined on the new class \(\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\) are also discussed. Coefficient inequalities, growth and distortion inequalities, as well as closure results are obtained. We also establish some results concerning neighborhoods and the partial sums of meromorphic functions in this new class. We also state some new subclasses and their characteristic properties by specializing the parameters which are new and not studied before in association with Mittag-Leffler functions.

Функция Миттаг-Леффлера естественным образом возникает при решении дифференциальных и интегральных уравнений дробного порядка, и особенно, при изучении дробного обобщения кинетического уравнения, случайных блужданий, полетов Леви, супердиффузионного переноса и при изучении сложных систем. В настоящем исследовании авторы определяют новый класс \(\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\) мероморфных функций, определенных в проколотом единичном круге \(\Delta^*:= \{z\in\mathbb{C}: 0<|z|<1\}\) на основе функции Миттаг-Леффлера. Подробно обсуждаются его характерные свойства, такие как коэффициентные неравенства, неравенства роста и искажения, а также результаты замыкания для \(f\in\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\). Рассматриваются свойства некоторого интегрального оператора и его обратного, определенного на классе \(\mathfrak{M}^{\tau,\kappa}_{\varsigma,\varrho}(\vartheta,\wp)\). Получены коэффициентные неравенства, неравенства роста и искажения, а также результаты замыкания. Установлены также некоторые результаты, касающиеся окрестностей и частичных сумм мероморфных функций в этом новом классе. Указаны некоторые новые подклассы и характеристические их свойства, специализируя параметры, которые являются новыми и не изучались ранее в связи с функциями Миттаг-Леффлера.

мероморфные функции звездообразная функция свертка положительные коэффициенты коэффициентные неравенства интегральный оператор функция Миттаг-Леффлера оператор Гильбертова пространства

meromorphic functions starlike function convolution positive coefficients coefficient inequalities integral operator Mittag-Leffler function Hilbert space operator

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