On the kth derivative of meromorphic functions with zeros of multiplicity at least k +1

In this paper, we prove the following Theorem. Let f (z) be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k+1 (k 2), except possibly finitely many, and all of whose poles are multiple, except possibly finitely many, and let the function a(z) = P(z) exp(Q (z)) ≡ 0, where P and Q are polynomials such that limr→∞( T (r,a) T (r, f ) + T (r, f ) T (r,a) )=∞. Then the function f (k)(z) − a(z) has infinitely many zeros.