We study the higher-order delay differential equation \(w(z+1)-w(z-1)+a(z)\frac{w^{(k)}(z)}{w^2(z)}=R(z,w(z)),\) where \(k\) is a positive integer, \(a(z)\) is a rational function and \(R(z,w)\) is rational in \(w\) with rational coefficients. We obtain necessary conditions on the degree of \(R(z, w)\) for this delay differential equation to admit subnormal transcendental meromorphic solutions. On the other hand, we obtain a reduced form of the equation above when \(R(z,w)\) becomes a rational function.
