The \(p\)-adic theorem of finite increments. (Le théorème des accroissements finis \(p\)-adique.) (French) Zbl 0831.26015

Let \(K\) be a complete valued extension of the \(p\)-adic number field. It is well-known that there is no mean value theorem for functions with domain and range in \(K\), as there exist \(C^1\)-injections with everywhere vanishing derivative. But even polynomials behave badly: let \(f: x\mapsto x^p\), then \(f(1)- f(0)= 1\neq f'(c)\) for all \(c\) in the unit disk.
Yet, in this elegant paper the author shows that if \(f: x\mapsto \sum_{n\geq 0} a_n x^n\) is analytic on \(B_k:= \{\lambda\in K: |\lambda|\leq 1\}\) (where the \(a_n\) are taken from a \(K\)-Banach space and \(|a_n|\to -\)) then \(|f(t+ h)- f(t)|\leq |h||f'|\) (where \(|f'|= \max|na_n|\)) for all \(t, h\in B_k\), \(|h|\leq r_p= p^{1/(1- p)}\).
It is shown that this result no longer holes if we admit \(|h|> r_p\). Three applications are given. First yet another proof of the Kummer and von Staudt congruences about Bernoulli numbers, also some binomial congruences and a fixed point theorem.
Also the following second-order result is proved. With \(f\), \(t\) as above we have \[ |f(t+ h)- f(t)- f'(t) h|\leq |{1\over 2} t^2||f''| \] provided \(|h|\leq p^{1/2p}\) if \(p\geq 3\), \(|h|\leq {1\over \sqrt 2}\) if \(p= 2\). As an application one deduces the fact that the numerator of \(1+ {1\over 2}+\cdots+ {1\over p} n\) is divisible by \(p^2\) (\(p\) odd). Finally, a \(p\)-adic Rolle theorem is discussed.


26E30 Non-Archimedean analysis
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