A Milloux-type inequality for good linear operators

We establish a Milloux-type inequality for iterates Fk=L∘k(f) of a good linear operator L. We first show that every iterate of a good linear operator is again good. We then introduce the associated difference operator and derive the main inequality. This method is applied to a Briot-Bouquet type of differential equation, namely the last case of Eremenko's conjecture to classify the admissible solutions. This paper ends with a positive answer to a 50-year-old problem that originated with Briot, Bouquet, and Picard.