Article
Ufa Mathematical Journal
Volume 4, Number 1, pp. 35-42
Iterations of the entire transcendental functions with regular behavior of the minimum of the modulus.
Gaisin A.M., Rakhmatullina Zh.G.
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In the paper the Fatou set of an entire trans\-cen\-dental 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 infinite lower order. The pair of conditions on the indexes of the series under which 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 Fej\'{e}r gap condition. The result under stronger sufficient conditions was proved earlier by Yu. Wang and Zh. Rakhmatullina.