Extending two fixpoint theorems of Langley and Zheng (Q1781905)

Let \(f\) be a meromorphic function with order \(\rho(f)\), \(\infty\geq\rho(f)> \mu> 0\), for which the set of finite singular values of \(f^{-1}\) is bounded. Then, if \(Q\) is a rational function with a pole of multiplicity \(p>0\) at infinity, there exist infinitely many points \(z\) such that \(f(z)= Q(z)\) and \(|f'(z)|>|z|^{\mu/2+ p-1}\). A similar result for transcendental meromorphic functions involving \(\delta(\infty,f)\) is also given.