Power Means Generated by Some Mean-value Theorems

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 f(x)− f(y) x− y = M ( f ′(x), f ′(y) ) . 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. Introduction In a recent paper [4] the following counterpart of the Lagrange mean-value theorem has been proved. If a real function f defined on an interval I ⊂ R is differentiable, and f ′ is one-to-one, then there exists a unique mean function M : f ′ (I)× f ′ (I) → f ′ (I) , such that f(x)− f(y) x− y = M (f ′(x), f ′(y)) , x, y ∈ I, x = y. In this connection the following problem arises. Given a mean M, determine all differentiable real functions f such that (1) f(x)− f(y) x− y = M (f ′(x), f ′(y)) , x, y ∈ I, x = y. In the case when M is the geometric mean this equation has appeared in [3] and was useful in solving an open problem related to convex functions (cf. Remark 5). In the first section we consider equation (1) in the case when M = M , where M (u, v) = φ−1 ( φ(u) + φ(v) 2 ) , u, v ∈ J, and φ : f ′ (I) → R is a continuous and strictly monotonic function; so M is a quasiarithmetic mean of a generator φ. Assuming three times continuous differentiability of f , and twice continuous differentiability of φ, we give some necessary conditions for equality (1) (Theorem 1). Applying this result, in the next section we give a Received by the editors August 26, 2010. 2010 Mathematics Subject Classification. Primary 26A24, 26E60; Secondary 39B22.