Some results related to a question of Hinkkanen (Q2385330)

\textit{G. Frank}, \textit{X. Hua} and \textit{R. Vaillancourt} [Can. J. Math. 56, No. 6, 1190--1227 (2004; Zbl 1065.30027)] solved Hinkkanen's problem [\textit{K. F. Barth}, \textit{D. A. Brannan} and \textit{W. K. Hayman}, Bull. Lond. Math. Soc. 16, 490--517 (1984; Zbl 0593.30001)] by showing that any meromorphic function is determined by its zeros and poles and the zeros of its first four derivatives. In fact, we assume that two functions \(f\) and \(g\) are meromorphic and have the same zeros and poles counting multiplicities, and moreover their derivatives \(f^{(k)}\) and \(g^{(k)}\) have the same zeros with their multiplicities for each \(k\) \((1\leq k\leq 4)\), respectively. Then \(f\) and \(g\) should belong to one of the following four cases: (i) \(f= cg\) \((c\neq 0)\), (ii) \(f(z)= e^{az+b}\), \(g(z)= e^{cz+d}\) \((a,c\neq 0)\), (iii) \(f(z)= a(1- be^cz)\), \(g(z)= d(e^{-cz}- b)\), \((a,b,c,d\neq 0)\), (iv) \(f(z)= a(1- be^{\alpha(z)})^{-1}\), \(g(z) = a(e^{-\alpha(z)}-b)^{-1}\) (\(a, b\neq 0\), \(\alpha(z)\) a nonconstant entire function), where \(a\), \(b\), \(c\) and \(d\) are constants. In this paper, the authors consider the sharing value problem in connection with Hinkkanen's problem regarding the hyper-order of meromorphic functions. They define the hyper-order for meromorphic function \(f\) as follows \[ \sigma_2(f)= \limsup_{r\to\infty}\,\log\log T(r, f)/\log r. \] They obtained the following result. Let \(f\) and \(g\) be nonconstant meromorphic functions of hyper-order less than one. If \(f^{(j)}\) and \(g^{(j)}\) share the value \(0\) and \(\infty\) CM for \(j= 0,1\), and if \(f''\) and \(g''\) share \(0\) IM, then \(f\) and \(g\) satisfying one of the forms (i) to (iv) above. In particular, \(\alpha(z)\) in the case (iv) is of order less than one.