Oscillation results of higher order linear differential equation

TL;DR

This paper investigates the oscillation behavior of solutions to higher order linear differential equations with transcendental entire coefficients, establishing conditions under which solutions have infinitely many zeros based on growth properties.

Contribution

It introduces new conditions linking the growth of coefficients to the zero distribution of solutions for higher order differential equations.

Findings

Solutions with certain growth constraints have infinitely many zeros.

The zero distribution is influenced by the order and type of the entire coefficient functions.

New oscillation criteria are established for higher order equations.

Abstract

We study higher order linear differential equation $y^{(k)}+A_1(z)y=0$ with $k\geq2$, where $A_1=A+h$, $A$ is a transcendental entire function of finite order with $\frac{1}{2}\leq \mu(A)<1$ and $h\neq0$ is an entire function with $\rho(h)<\mu(A)$. Then it is shown that, if $f^{(k)}+A(z)f=0$ has a solution $f$ with $\lambda(f)<\mu(A)$ then exponent of convergence of zeros of any non trivial solutions of $y^{(k)}+A_1(z)y=0$ is infinite.