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

Abstract

Following 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 $\displaystyle{f\over g}$ admits four perfectly branched small functions, then $T(r,f)$ and $T(r,g)$ are close. If a p-adic meromorphic function $\displaystyle{f\over g}$ 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.