Uniqueness of entire functions and fixed points

Let f and g be entire functions, n, k and m be positive integers, and lambda, mu be complex numbers with vertical bar lambda vertical bar + vertical bar mu vertical bar not equal 0. We prove that (f(n)(z)(lambda f(m)(z) + mu))((k)) must have infinitely many fixed points if n >= k + 2, furthermore, if (f(n)(z)(lambda f(m)(z) + mu))((k)) and (g(n)(z)(lambda g(m)(z) + mu))((k)) have the same fixed points with the same multiplicities, then either f equivalent to cg for a constant c, or f and g assume certain forms provided that n > 2k + m* + 4, where m* is an integer that depends only on lambda.