Existence of zero-order meromorphic solutions of certain q-difference equations

In this paper, we consider the q-difference equation $$ \bigl(f(qz)+f(z)\bigr) \bigl(f(z)+f(z/q)\bigr)=R(z,f), $$ where \(R(z,f)\) is rational in f and meromorphic in z. It shows that if the above equation assumes an admissible zero-order meromorphic solution \(f(z)\), then either \(f(z)\) is a solution of a q-difference Riccati equation or the coefficients satisfy some conditions.