Complex and p-adic branched functions and growth of entire functions

International audienceFollowing a previous paper by Jacqueline Ojeda and the first author, here we examine the number of possible branched values and branched functions for certain p-adic and complex meromorphic functions where numerator and denominator have different kind of growth, either when the denominator is small comparatively to the numerator, or vice-versa, or (for p-adic functions) when the order or the type of growth of the numerator is different from this of the denominator: this implies that one is a small function comparatively to the other. Finally, if a complex meromorphic function f g admits four perfectly branched small functions, then T (r, f) and T (r, g) are close. If a p-adic meromorphic function f g admits four branched values, then f and g have close growth. We also show that, given a p-adic meromorphic function f , there exists at most one small function w such that f − w admits finitely many zeros and an entire function admits no such a small function