Linear Differential Equations and Logarithmic Derivative Estimates

Abstract

We prove two sharp inequalities for the growth of solutions of certain linear differential equations in the unit disk. For the proofs of these inequalities, we use the method of successive approximations and sharp estimates for the logarithmic derivatives of finite order meromorphic functions in the unit disk. These techniques can also be used to give an alternate proof of a well-known result in the plane. The sharp logarithmic derivative estimates are a corollary of general estimates, and all these estimates have independent interest. 2000 Mathematics Subject Classification 34M10, 30D35.