Meromorphic functions sharing the same zeros and poles (Q1876847)

This paper is an announcement that the authors have solved Hinkkanen's problem (Question~0.3/1.1) completely by showing that any meromorphic function in the plane is determined by its zeros and poles and the zeros of its first \textit{four} derivatives. In fact, Theorem~0.4/1.3 states the following: Assume that two functions \(f\) and \(g\) are meromorphic in the plane 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) \(f(z)=a(1-b e^{cz}), g(z)=d(e^{-cz}-b)\), \((a,b,c,d\neq 0)\), (iv) \(f(z)=a\bigl(1-be^{\beta(z)}\bigr)^{-1}\), \(g(z)=a\bigl(e^{-\beta(z)}-b\bigr)^{-1}\) (\(a,b\neq 0\), \(\beta(z)\) a nonconstant entire function), where \(a\), \(b\), \(c\) and \(d\) are constants. A. Hinkkanen himself asked in Problem 2.65 [\textit{K. F. Barth, D. A. Brannan} and \textit{W. K. Hayman}, Bull. Lond. Math. Soc. 16, 490--517 (1984; Zbl 0593.30001)] whether there exists some positive integer \(n\) such that \(f\) and \(g\) satisfy one of the expressions (i), (ii), (iii), (iv), when \(f^{(k)}\) and \(g^{(k)}\) have the same zeros and poles counting multiplicities for \(k=0,1, \cdots, n\). Note that every pair \((f,g)\) in these four cases satisfies all the assumptions for `\(n=\infty\)'. Hinkkanen's example \(f(z)=(e^{2z}-1)\exp(-ie^z)\), \(g(z)=(1-e^{-2z})\exp(ie^{-z})\) shows that \(n\geq 3\). Furthermore, as in Example~2, the number \(n\) cannot be less than four. A general counter-example for \(n=3\) is a pair (v) \(f(z)=c\exp \bigl(e^{az+b}\bigr)\), \(g(z)=d\exp \bigl(e^{-az-b}\bigr)\), \(a, c, d\neq 0\), for which \(f^{(k)}/g^{(k)}\) are zero-free entire functions for \(k=0,1,2,3\) (but it has infinitely many zeros and poles for \(k=4\)). Hence the above theorem gives the best lower bound for the values of \(n\) that Hinkkanen asked about. \textit{L. Köhler} [Complex Variables, Theory Appl. 11, No. 1, 39--46 (1989; Zbl 0637.30029) and Meromorphe Funktionen mit gleichen Null- und Polstellen und jeweils gleichen Nullstellen einiger Ableitungen (1987; Zbl 0643.30023)] showed that the number \(n\) can be \(6\) in general and that the best lower bound is \(n=2\) under the assumption that both \(f\) and \(g\) have finite order. (See \textit{J. K. Langley} [Kodai Math. J. 19, No. 1, 52--61 (1996; Zbl 0853.30020)] for some generalizations in the case of finite order.) The authors of the paper under review seem to reform the method of Köhler extensively and estimate the value distribution of the functions \(H_j=f^{(j+1)}/f^{(j)} - g^{(j+1)}/g^{(j)}\) \((0\leq j\leq 4)\) very carefully. This makes them possible to remove the assumption for \(k=5,6\) in Köhler's result. Their whole discussion will appear in [\textit{G. Frank, X. H. Hua} and \textit{R. Vaillancourt}, Meromorphic functions sharing the same zeros and poles, Can. J. Math.]. One sees that the study of the functions \(H_j\) is a key to the solution, since all the \(H_j\) reduce to constants (possibly zero) for any \(j\geq 1\) in (i)-(iv). Reviewer's note: One might ask whether (v) is the only possibility besides these four cases when \(n=3\). It seems to be only known that this is the case if \(H_0\) is of order less than 2 as stated in [Kodai Math. J. 13, No. 1, 101--120 (1990; Zbl 0707.30023)].