Abstract
Small functions were defined in complex analysis and next in ultrametric analysis. Order of growth and type of growth were also defined in complex analysis and have a similar definition in ultrametric analysis. Here we compare these two notions in the same way, on a complete ultrametric algebraically closed field \( \mathbb K \) of characteristic \(0\) such as \( \mathbb C _p\) . The set of small functions with respect to an entire function \(f\) is a ring. The set of entire functions with an order of growth strictly inferior to a number \(t\) is also a ring which is included in the previous one when \(f\) is regular, but not always when \(f\) is not. If an entire function \(h\) is small with respect to an entire function \(f\) , that does not imply that its order of growth is inferior to that of \(f\) . The cotype was defined in previous papers and is equal to the product of the order by the type for a clean function. But that is not always true for a function that is not clean. If \(f\) is clean and has the same order as \(h\) and if the type or the cotype of \(h\) is zero, then \(h\) is a small function with respect to \(f\) . If \(f\) is clean and the cotype of \(f\) is strictly superior to the cotype of \(h\) , while the type of \(f\) is less than the type of \(h\) , then \(h\) is a small function with respect to \(f\) ; similarly, if \(f\) is clean and the type of \(h\) is strictly superior to the type of \(f\) , while the cotype of \(h\) is less than the cotype of \(f\) , then \(h\) is a small function with respect to \(f\) . Applications are obtained to a pair of entire functions sharing 3 functions (ignoring multiplicity).
