## 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.

### MSC:

 2.6e+31 Non-Archimedean analysis
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### References:

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