  In this paper, we investigate the delay differential equations of Malmquist
type of form \begin{equation*} w(z+1)-w(z-1)+a(z)\frac{w'(z)}{w(z)}=R(z,
w(z)),~~~~~~~~~~~~~~(*) \end{equation*} where $R(z, w(z))$ is an irreducible
rational function in $w(z)$ with rational coefficients and $a(z)$ is a rational
function. We characterize all reduced forms when the equation $(*)$ admits a
transcendental entire solutions with hyper-order less than one. When we compare
with the results obtained by Halburd and Korhonen[Proc.Amer.Math.Soc.,
forcoming],we obtain the reduced forms without the assumptions that the
denominator of rational function $R(z,w(z))$ has roots that are nonzero
rational functions in $z$. The growth order and value distribution of
transcendental entire solutions for the reduced forms are also investigated.
