Iterations of entire transcendental functions with a regular behavior of the modulus minimum

In the paper the Fatou set of an entire transcendental function is considered, i.e. the largest open set of the complex plane, where the family of iterations of the given function forms a normal family according to Montel. The entire function is assumed to be of an infinite lower order. The pair of conditions on the indices of the series providing that every component of the Fatou set is bounded is found. This pair of conditions is optimal in a certain sense and is stronger than the Fejér gap condition. The result under stronger sufficient conditions was proved earlier by Yu. Wang and Zh. Rakhmatullina.