Аннотация: В этой статье мы изучаем единственность целых функций относительно их разностного оператора и производных. Представление о целых и мероморфных функциях сильно зависит от этого направления. Рубель и Янг рассмотрели единственность целой функции и ее производных; они доказали, что если \(f(z)\) и \(f'(z)\) разделяют два значения \(a\), \(b\) с учетом кратностей, то \(f(z)\equiv f'(z)\). Позже Ли Пинг и Янг улучшили результат Рубеля и Янга: если \(f(z)\) - непостоянная целая функция, а \(a\) и \(b\) - два конечных различных комплексных значения, и если \(f(z)\) и \(f^{(k)}(z)\) разделяют \(a\) с учетом кратностей и \(b\) - без учета кратностей, то \(f(z)\equiv f^{(k)}(z)\). В последние годы проявляется значительный интерес к распределению значений мероморфных функций конечного порядка относительно разностного аналога. Заменив различные конечные комплексные значения многочленами, устанавливается следующий результат: пусть \(\Delta f(z)\) - трансцендентная целая функция конечного порядка, \(k\geq0\) - целое число, а \(P_{1}\) и \(P_{2}\) - два многочлена; если \(\Delta f(z)\) и \(f^{(k)}\) разделяют \(P_{1}\) с учетом кратностей и \(P_{2}\) игнорируя кратности, то \(\Delta f \equiv f^{(k) }\). Нетривиальное доказательства этого результата использует теорию распределения значений Неванлинны.
Образец цитирования: Rajeshwari S. and Sheebakousar B. Unicity on Entire Functions Concerning Their Difference Operators and Derivatives // Владикавк. мат. журн. 2023. Т. 25, № 1. C. 81-92 (in English).
DOI 10.46698/p5608-0614-8805-b
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