Order and Hyper-order of Solutions of Second Order Linear Differential Equations

TL;DR

This paper investigates conditions on the coefficients of second-order linear differential equations that ensure all non-trivial solutions are of infinite order and determines their hyper-order under specific conditions.

Contribution

It provides new criteria for the coefficients of second-order linear differential equations to guarantee solutions of infinite order and analyzes their hyper-order.

Findings

Solutions are of infinite order under certain coefficient conditions.

Hyper-order of solutions is characterized when (z) and B(z) satisfy specific conditions.

Conditions on B(z) influence the growth order of solutions.

Abstract

We have discussed the problem of finding the condition on coefficients of $f''+A(z)f'+B(z)f=0, \quad B(z)(\not \equiv 0)$ so that all non-trivial solutions are of infinite order. The hyper-order of these non-trivial solutions of infinite order is also found when $\lambda(A)<\rho(A)$ and $B(z)$ is a transcendental entire function satisfying some conditions.