Uniqueness of Meromorphic Functions with Respect to Their Shifts Concerning Derivatives

Abstract

An example in the article shows that the first derivative of \(f(z)=\frac{2}{1-e^{-2z}}\) sharing \(0\) CM and \(1,\infty\) 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 \(k\)th derivatives. We use a different method from Qi and Yang [1] to improves entire function to meromorphic function, the first derivative to the \(k\)th derivatives, and also finite values to small functions. As for \(k=0\), we obtain: Let \(f(z)\) be a transcendental meromorphic function of \(\rho_{2}(f)<1\), let \(c\) be a nonzero finite value, and let \(a(z)\not\equiv\infty,b(z)\not\equiv\infty\in\hat{S}(f)\) be two distinct small functions of \(f(z)\) such that \(a(z)\) is a periodic function with period \(c\) and \(b(z)\) is any small function of \(f(z)\). If \(f(z)\) and \(f(z+c)\) share \(a(z),\infty\) CM, and share \(b(z)\) IM, then either \(f(z)\equiv f(z+c)\) or

$$e^{p(z)}\equiv\frac{f(z+c)-a(z+c)}{f(z)-a(z)}\equiv\frac{b(z+c)-a(z+c)}{b(z)-a(z)},$$

where \(p(z)\) is a nonconstant entire function of \(\rho(p)<1\) such that \(e^{p(z+c)}\equiv e^{p(z)}\).