On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Extremal Coefficients

Abstract

We consider that the linear differential equations $f^{(k)}+A_{k-\mathrm{1}}(z)f^{(k-\mathrm{1})}+\cdots +A_{\mathrm{1}}(z)f^{\mathrm{\prime }}+A_{\mathrm{0}}(z)f=\mathrm{0}$, where $A_j(j=\mathrm{0,1},\dots ,k-\mathrm{1})$, are entire functions. Assume that there exists $l\in \{\mathrm{1,2},\dots ,k-\mathrm{1}\}$, such that $A_l$ is extremal for Yang's inequality; then we will give some conditions on other coefficients which can guarantee that every solution $f(\not\equiv \mathrm{0})$ of the equation is of infinite order. More specifically, we estimate the lower bound of hyperorder of $f$ if every solution $f(\not\equiv \mathrm{0})$ of the equation is of infinite order.