The infinite growth of solutions of second order linear complex differential equations with completely regular growth coefficient

In this paper we discuss the classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all nontrivial solutions of f ′′ + A(z)f ′ + B(z)f = 0 are of infinite order. We assume A(z) is an entire function of completely regular growth and B(z) satisfies three different conditions, then we obtain three results respectively. The three conditions are (1) B(z) has a dynamical property with a multiply connected Fatou component, (2) B(z) satisfies T (r,B) ∼ logM(r,B) outside a set of finite logarithmic measure, (3) B(z) is extremal for Denjoy’s conjecture.

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