Meromorphic Solutions of Algebraic Differential Equations and α-Yoshida Functions

Abstract

A function f meromorphic on the complex plane c is called an α-Yosida function if \( {{f}^{\# }}(z) = O(|z{{|}^{\alpha }})\,as|z| \to \infty ,\alpha \geqslant - 1\;where\,{{f}^{\# }}(z) = |f'(z)|(/1 + |f(z){{|}^{2}}) \) We give a sufficient and necessary condition for a meromorphic function f to be not an α-Yosida function by proving a relationship between α-Yosida functions and normal families. We also give some integral characterizations of α-Yosida functions. Applying our results we know that the meromorphic solutions of some higher order algebraic differential equations are α-Yosida functions. Moreover, some estimates for the order of growth of these meromorphic solutions by studying α-Yosida functions are given.Keywordsα-Yosida functionsmeromorphic solutionsalgebraic differential equations