# zbMATH — the first resource for mathematics

Lagrangian means. (English) Zbl 0903.39006
The authors call $$M(x,y)$$ a Lagrangian mean on $$I^{2}$$ if there exists a continuous strictly monotonic function $$f$$ such that $$f(M(x,y))=\int_{x}^{y}f(t) dt/(y-x)$$ if $$x\neq y$$, while $$M(x,x)=x;$$ throughout $$x,y\in I$$. R. Bojanić gave necessary and sufficient conditions for $$M$$ to be a Lagrangian mean under the supposition that $$M$$ and $$f$$ are twice differentiable. The authors offer a necessary and sufficient condition without such differentiability conditions, and a connection between these means and quasiarithmetic means $$(2g(Q(x,y)) = g(x)+g(y))$$.

##### MSC:
 39B22 Functional equations for real functions 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
##### Keywords:
quasiarithmetic means; Lagrange mean
Full Text: