On delay-differential equations with subnormal transcendental meromorphic solutions
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The necessary conditions on the degree of $R(z,\omega(z))$ for the delay-differential equations \begin{align*} \omega(z+1)\omega(z-1)+a(z)\frac{\omega^{(k)}(z)}{\omega(z)}=R(z,\omega(z)) \end{align*} and \begin{align*} (\omega(z+1)+\omega(z))(\omega(z-1)+\omega(z))+a(z)\frac{\omega^{(k)}(z)}{\omega(z)}=R(z,\omega(z)) \end{align*} to admit a subnormal transcendental meromorphic solution are obtained by using Nevanlinna theory, where $k$ is a positive integer, $a(z)$ is a rational function and $R(z,\omega)$ is rational in $\omega$ with rational coefficients, which extends and generalizes some previous results. Some examples are given to illustrate these results.
Abstract
The necessary conditions on the degree of $R(z,\omega(z))$ for the delay-differential equations \begin{align*} \omega(z+1)\omega(z-1)+a(z)\frac{\omega^{(k)}(z)}{\omega(z)}=R(z,\omega(z)) \end{align*} and \begin{align*} (\omega(z+1)+\omega(z))(\omega(z-1)+\omega(z))+a(z)\frac{\omega^{(k)}(z)}{\omega(z)}=R(z,\omega(z)) \end{align*} to admit a subnormal transcendental meromorphic solution are obtained by using Nevanlinna theory, where $k$ is a positive integer, $a(z)$ is a rational function and $R(z,\omega)$ is rational in $\omega$ with rational coefficients, which extends and generalizes some previous results. Some examples are given to illustrate these results.