Estimates on the Growth of Meromorphic Solutions of Linear Differential Equations with density conditions

We give an alternative and simpler method for getting pointwise estimate of meromorphic solutions of homogeneous linear differential equations with coefficients meromorphic in a finite disk or in the open plane originally obtained by Hayman and the author. In particular, our estimates generally give better upper bounds for higher order derivatives of the meromorphic solutions under consideration, are valid, however, outside an exceptional set of finite logarithmic density. The estimates again show that the growth of meromorphic solutions with a positive deficiency at infinity can be estimated in terms of initial conditions of the solution at or near the origin and the characteristic functions of the coefficients.