Oscillation results of higher order linear differential equation

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.