New properties on the growth of ultrametric entire functions and applications

Abstract

Let IK be a complete ultrametric algebraically closed field and let f be an entire function in IK whose order of growth is finite. We show that the type of growth is finite if and only if so is the cotype. We give bounds for the cotype of growth and also for the lower cotype of growth. We show that the type of growth of f is equal to its lower type if and only if its cotype is equal to its lower cotype and when these are realised, then the cotype is the product of the type by the order of growth and the order of growth (if > 0), is then equal to the lower order of growth. If an entire function h has an order of growth strictly inferior to the lower order of an entire function f, then h is a small function with respect to f. A similar comparison is made with the type of growth. Conversely, if h is a small function with respect to f, then f +h and f have same order, same type and same cotype of growth. Links are showed with the Nevanlinna Theory. Suppose that IK is of characteristic 0. Given a meromorphic function f = g h , if f admits primitives and if the type or the cotype of h is finite, then f assumes all values infinitely many times. A counter-example is constructed where the lower order of growth is equal to the order of growth but the lower type of growth is not equal to the type of growth and where the the cotype is not equal to the product of the type by the order of growth. In complex analysis, a claim was made for complex meromorphic functions stating that if the lower order of growth equals the order, then the lower type equals the type but we contest the proof.