An addition to the Tumura-Clunie theorem
Let \(P\) be an arbitrary differential polynomial of degree at most \(n-1\), with constant coefficients and let \(f\) be an entire function. In the paper under review, the author exploits \textit{X.-H. Hua}'s extension [Complex Variables, Theory Appl. 16, No. 1, 69--77 (1991; Zbl 0744.30023)] of the Tumura-Clunie theorem [c.f. \textit{J. Clunie}, J. Lond. Math. Soc. 37, 17--27 (1962; Zbl 0104.29504)], and uses some general results about differential polynomials, to show that if the entire function \(\psi:=f^n+P[f]\) does not vanish in \(\mathbb C\), then \(f\) itself has a Picard exceptional value and satisfies certain differential equations. Furthermore, under additional assumptions (cf. Theorem 1.6), \(f\) has the form \(f(z)=\exp(az+b)\). The author also shows by means of examples that a similar result need not hold for meromorphic functions.
scientific article; zbMATH DE number 5320903