A note on the product of meromorphic functions and its derivatives

It is shown that if f is an even or odd transcendental meromorphic function and if c is any even meromorphic function which does not vanish identically and satis®es TrY c oTrY f as r 3 y, then f f H À c has in®nitely many zeros.Later, he conjectured [6] that this remains valid for the cases n 1Y 2. In 1979, E. Mues [7] proved the case n 2 and the conjecture was proven by A. Eremenko and W. Bergweiler [2] in 1995 and independently by H. H. Chen and M. L. Fang [3].In 1994, Yik-Man Chiang asked W. Bergweiler whether f f H À c has in®nitely many zeros if f is a transcendental meromorphic function and if c is a meromorphic function which does not vanish identically and satis®es TrY c oTrY f as r 3 y.In [8], Q. D. Zhang studied the value distribution of jz f zf H z and obtained the following theorem.Theorem B. If f is a transcendental meromorphic function and j is a nonzero meromorphic function such that TrY j SrY f as r 3 y, thenSrY f X By this, we have 339 2000