Fixed points of meromorphic functions and of their differences and shifts

Let f(z) be a finite order transcendental meromorphic function such that lambda(1/f(z)) < sigma(f(z)), and let c is an element of C \ {0} be a constant such that f(z + c) not equivalent to f(z) + c. We mainly prove that max{tau(f(z)), tau(Delta(c)f(z))} = max{tau(f(z)), tau(f(z + c))} = max{tau(Delta(c)f(z)), tau(f(z + c))} = sigma(f(z)), where tau(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and sigma(g(z)) denotes the order of growth of g(z).