Normal families and growth of meromorphic functions with their \(k\)th derivatives (Q1644407)

By means of the theory of normal families, the authors consider uniqueness problems for meromorphic functions in connection with the order and the hyper order of growth. One of their results is the following. Let \(f\) be a nonconstant meromorphic function with finitely many poles, and let \(\alpha=R_1e^{\gamma}\) and \(\beta=R_2e^{\gamma}\), where \(R_1\), \(R_2\) (\(R_1\not\equiv R_2\)) are two rational functions and \(\gamma\) is an entire function. Let \(k\) be a positive integer. If all zeros of \(f-\alpha\) have multiplicity at least \(k\) and \(f(z)=\alpha(z)\Rightarrow f^{(k)}(z)=\alpha(z)\), \(f(z)=\beta(z)\Leftrightarrow f^{(k)}(z)=\beta(z)\), then \(\rho_2(f)\leq \rho_2(\alpha)=\rho(\gamma)\). Some examples are given to show the sharpness of the result above, which are constructed from exponential functions. The results in this paper are improvements of the investigations in [\textit{F. Lü} and \textit{H. Yi}, J. Korean Math. Soc. 48, No. 3, 499--512 (2011; Zbl 1232.30024)].