On the Conjecture of Bruck for Solutions of Linear Differential Equations

Abstract

In this thesis, we use Nevanlinna theory to investigate Br¨uck’s conjecture for solutions of both homogeneous and non-homogeneous linear differential equations with meromorphic coefficients. We begin by examining Br¨uck’s conjecture for solutions of second-order homogeneous linear differential equations. Subsequently, we apply alternative methods to study the conjecture for solutions of higher-order homogeneous linear differential equations. We also consider the nonhomogeneous case. Finally, we generalize certain previous results on small function sharing between meromorphic functions and their linear differential polynomials, using these results to confirm Br¨uck’s conjecture for entire functions under certain conditions. 2020 Mathematics Subject Classification (MSC2020): 30D35