Two results related to a question of Hinkkanen
In 1984 A. Hinkkanen proposed the following problem: Suppose that \(f\) and \(g \) are meromorphic in the plane such that \(f^{(k)}\) and \(g^{(k)}\) share 0 and \(\infty\) \(CM\), for \( 0 \leq k \leq n\). How large need \(n\) be in order to determine some relationship between \(f\) and \(g \)? For meromorphic functions \(f\) and \(g \not \equiv\text{const}\) of finite order the author proves: If \(f/g\) and \(f'/g'\) have only finitely many zeros and poles, then one of the following holds: (a) \(f/g\) is constant, (b) \(f'/f\) and \(g'/g\) are both rational (c) There exist a rational function \(S\) and a nonconstant polynomial \(Q\) and constants \(c_j\) and \(d\) such that \[ f = c_1 (se^Q - 1)^d,\;g = c_2 (1 - e^{- Q}/S)^d, \] (d) There exist nonconstant polynomials \(Q\) and \(P\) such that \[ f'/f = {P(Q)Q' \over 1 - e^{-Q}},\;g'/g = {P(Q)Q' \over e^{-Q} - 1}. \] Furthermore he shows that if, in addition, \(f''/g''\) has only finitely many zeros and poles then we have either (a), (b) or (c) with \(d = \pm 1\). In a second result he answers this question in the case that \(f\) and \(g\) share \(\infty\) \(IM\) only.
scientific article; zbMATH DE number 895930
scientific article