Uniqueness of Meromorphic Functions with Respect to Their Shifts Concerning Derivatives

An example in the article shows that the first derivative of f(z) = 2/1-e-(2z )sharing 0 CM and 1, infinity IM with its shift pi i cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its kth derivatives. We use a different method from Qi and Yang [1] to improves entire function to meromorphic function, the first derivative to the kth derivatives, and also finite values to small functions. As fork=0,weobtain: Letf(z)be a transcendental meromorphic function of rho 2(f)<1, let c be a nonzero finite value, and leta(z)equivalent to infinity,b(z)equivalent to infinity is an element of S(f)be two distinct small functions off(z)such that a(z) is a periodic function with period c and b(z)is any small function off(z).Iff(z) and f(z+c) share a(z),infinity CM, and share b(z)IM, then either f(z)equivalent to f(z+c)or e(p(z) )equivalent to f(z+c)-a(z+c)/f(z)-a(z) equivalent to b(z+c)-a(z+c)/b(z)-a(z) where p(z) is a nonconstant entire function of rho(p)<1 such that e(p(z+c) )equivalent to e(p(z))