A Malmquist Type Theorem for a Class of Delay Differential Equations

We show that if the following delay differential equation of rational coefficients wk(z) n-ary sumation mu=1se mu(z)w(z+c mu)+a(z)w(n)(z)w(z)= n-ary sumation i=0pai(z)wi n-ary sumation j=0qbj(z)wj w<^>k(z)\sum _{\mu =1}<^>se_\mu (z)w(z+c_\mu )+a(z)\frac{w<^>{(n)}(z)}{w(z)}= \frac{\sum _{i=0}<^>pa_i(z)w<^>i}{\sum _{j=0}<^>qb_j(z)w<^>j} \end{aligned}$$\end{document}admits a transcendental entire solution w of hyper-order less than one, then it reduces into a delay differential equation of rational coefficients wk(z) n-ary sumation mu=1se mu(z)w(z+c mu)+a(z)w(n)(z)w(z)=1w(z) n-ary sumation i=0k+2Ai(z)wi(z), w<^>k(z)\sum _{\mu =1}<^>se_\mu (z) w(z+c_\mu )+a(z)\frac{w<^>{(n)}(z)}{w(z)}=\frac{1}{w(z)}\sum _{i=0}<^>{k+2}A_{i}(z)w<^>i(z), \end{aligned}$$\end{document}which improves some known theorems obtained most recently by Zhang and Huang. Some examples are constructed to show that our results are accurate.