Infinite growth of solutions of second order complex differential equations with entire coefficient having dynamical property

Abstract

In this paper we discuss the classical problem of finding conditions that the entire coefficients A(z) and B(z) should satisfy to ensure all nontrivial solutions of f′′+A(z)f′+B(z)f=0 are of infinite order. Two different approaches are used. In the first approach the entire coefficient B(z) has dynamical property with a multiply connected Fatou component, and A(z) is extremal for Yang’s inequality or a nontrivial solution of w′′+P(z)w=0, where P(z) is a polynomial. In the second approach B(z) satisfies T(r,B)∼logM(r,B) outside a set of finite logarithmic measure, and A(z) has the same properties as in the first approach.