On entire solutions of a certain type of nonlinear differential equation

Abstract

In this note, we shall study, via Nevanlinna's value distribution theory, the uniqueness of transcendental entire solutions of the following type of nonlinear differential equation: (*) L (f (z)) – p (z) fn(z) = h (z), where L (f) denotes a linear differential polynomial in f with polynomials as its co-efficients, p (z) a polynomial (≢ 0), h an entire function, and n an integer ≥ 3. We show that if the equation (*) has a finite order transcendental entire solution, then it must be unique, unless L (f) ≡ 0.