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A mean-value theorem and its applications. (English) Zbl 1206.26032
Let $$I\subset\mathbb R$$ be an interval and $$f:I\rightarrow\mathbb R$$ a differentiable function. Following the Lagrange mean-value theorem, if $$f'$$ is one-to-one, the function $$L^{[f]}:I^2\rightarrow\mathbb R$$ defined by
$L^{[f]}(x,y):=(f')^{-1}\left( \frac{f(x)-f(y)}{x-y} \right), \quad x\neq y,$ is a mean. The author proves that, under the same assumptions, $$J:=f'(I)$$ is also an interval and there exists a unique strict mean $$M^{[f]}:J^2\rightarrow J$$ such that
$\frac{f(x)-f(y)}{x-y}=M^{[f]}(f'(x),f'(y)), \quad x,y\in I,\;x\neq y.$ Moreover,
$M^{[f]}(u,v)=f'(L^{[f]}((f')^{-1}(u),(f')^{-1}(v))), \quad u,v\in J.$ A counterpart of the Cauchy mean-value theorem is also presented. The family of means $$\{M^{[e_p]}$$; $$p\in \mathbb R\}$$, where $$e_p(x)=x^p$$, $$p(p-1)\neq 0$$, is studied.

##### MSC:
 2.6e+61 Means
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##### References:
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