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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.

26E60 Means
Full Text: DOI
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