Power means generated by some mean-value theorems

Abstract:

According to a new mean-value theorem, under the conditions of a function $f$ ensuring the existence and uniqueness of Lagrange’s mean, there exists a unique mean $M$ such that\[ \frac {f(x)-f(y)}{x-y}=M\left ( f^{\prime }(x),f^{\prime }(y)\right ). \] The main result says that, in this equality, $M$ is a power mean if, and only if, $M$ is either geometric, arithmetic or harmonic. A Cauchy relevant type result is also presented.