Resumen de Means and the mean value theorem

Jorma K. Merikoski, Markku Halmetoja, Timo Tossavainen

• Let I be a real interval. We call a continuous function µ : I × I ? ? a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I ? ? be a differentiable and strictly convex or strictly concave function. If a, b ? I with a ? b, then there exists a unique number ? between a and b such that f(b) - f(a) = f '(?)(b - a). We study under what conditions ? is a proper mean of a and b, and what kind of means are obtained by applying certain f 's. We also study the converse problem: Given a proper mean µ(a, b), does there exist f such that f(b) - f(a) = f '(µ(a, b))(b - a) for all a, b ? I with a ? b?