Uniqueness of Q-Shift Difference Polynomials of a Meromorphic (Entire) Functions of Zero Order
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Abstract
In this article, with the notion of the weighted sharing of values, we have examined the uniqueness results for -shift difference polynomials associated with transcendental meromorphic (entire) functions of zero order in the complex plane . Especially, we have examined the uniqueness of and share a small function with finite weight. We have proved three theorems that provide a new perspective, enhancing, generalizing, and improving recent results of Harina P.W. and Vijayalakshmi S.B. (New Trends in Mathematical Sciences 7, no. 3 (2019): 328-341, [11]), and we have also given some examples that support our results.
Introduction: Nevanlinna theory, a corestone of complex analysis, studies the distribution of values of meromorphic functions, with applications extending to difference polynomials. This distribution explores the theory’s connection to shifted difference polynomials, utilizing the concept of weighted sharing to analyze their uniqueness and value distribution.
Objectives: This theory provides a frame work for understanding the distribution of values of meromorphic functions by relating the distribution of zeros and poles to the growth of the function. The objective of using Nevanlinna theory with q-shift operators in uniqueness problems is to extend the traditional Nevanlinnas theory tools for studying the value distribution of merormorphic functions to the q shift operators, allowing for the investigation of uniqueness properties of meromorphic functions based on theory q-shift values.
Methods: The Second funcdamental Theorem combined with weighted sharing conditions and lemmas like Clunie’s Lemma and the Logarithmic Derivative Lemma form the core techniques used in Nevanlinna theory to prove the uniqueness of entire and meromorphic functions under weighted sharing.
Results: We have proved the results for uniquenss of meromorphic (entire) functions of zero order for the q-shift with differnec polynomials by using the concept of weighted sharing. The results are imporved by proving the conditions for uniqueness.
- tg for a constant such that , where .
- and satisfy the algebraic equations where
- , where is a complex
Abstract: In this article, with the notion of the weighted sharing of values, we have examined the uniqueness results for -shift difference polynomials associated with transcendental meromorphic (entire) functions of zero order in the complex plane . Especially, we have examined the uniqueness of and share a small function with finite weight. We have proved three theorems that provide a new perspective, enhancing, generalizing, and improving recent results of Harina P.W. and Vijayalakshmi S.B. (New Trends in Mathematical Sciences 7, no. 3 (2019): 328-341, [11]), and we have also given some examples that support our results.
Introduction: Nevanlinna theory, a corestone of complex analysis, studies the distribution of values of meromorphic functions, with applications extending to difference polynomials. This distribution explores the theory’s connection to shifted difference polynomials, utilizing the concept of weighted sharing to analyze their uniqueness and value distribution.
Objectives: This theory provides a frame work for understanding the distribution of values of meromorphic functions by relating the distribution of zeros and poles to the growth of the function. The objective of using Nevanlinna theory with q-shift operators in uniqueness problems is to extend the traditional Nevanlinnas theory tools for studying the value distribution of merormorphic functions to the q shift operators, allowing for the investigation of uniqueness properties of meromorphic functions based on theory q-shift values.
Methods: The Second funcdamental Theorem combined with weighted sharing conditions and lemmas like Clunie’s Lemma and the Logarithmic Derivative Lemma form the core techniques used in Nevanlinna theory to prove the uniqueness of entire and meromorphic functions under weighted sharing.
Results: We have proved the results for uniquenss of meromorphic (entire) functions of zero order for the q-shift with differnec polynomials by using the concept of weighted sharing. The results are imporved by proving the conditions for uniqueness.
- tg for a constant such that , where .
- and satisfy the algebraic equations where
- , where is a complex constant satisfying .
Conclusions: By considering the q-shift difference polynomial in the functions of the form and , along with weighted sharing concept in Theorem 3, we prove important analogous results for transcendental meromorphic (resp. entire) functions of zero order which improves the previous results.
constant satisfying .
Conclusions: By considering the q-shift difference polynomial in the functions of the form and , along with weighted sharing concept in Theorem 3, we prove important analogous results for transcendental meromorphic (resp. entire) functions of zero order which improves the previous results.