On solutions of a certain nonlinear differential-difference functional equation

We investigate all the possible finite order entire solutions of the Fermat-type differential-difference functional equation $(Af(z))^2+R^2(z)(Bf^{(m)}(z+c)+Cf^{(n)}(z))^2=Q(z)$, where $m,n\in\mathbb{N}$, $A,B,C\in\mathbb{C}\setminus\{0\}$ and $R(z)$, $Q(z)$ are nonzero polynomials. The results significantly improve some earlier findings, especially the results due to A. Banerjee and T. Biswas (2021). We also show that the equation does not have any non-entire meromorphic solution. We provide some examples to support the results.