An Addition to the Tumura–Clunie Theorem

Abstract.

Exploiting Hua’s extension of the Tumura–Clunie theorem and some general results on differential polynomials, we show that if P is an arbitrary differential polynomial of degree at most n − 1 with constant coefficients, f is an entire function and ψ := f ⁿ + P[f] is nonvanishing in ${\mathbb{C}}$, then f itself has a Picard exceptional value and satisfies certain differential equations. Under additional assumptions, f has the form f(z) = e ^az+b. We give some counterexamples to show that these results are sharp in some sense.