Characterization of the equality of Cauchy means to quasiarithmetic means

Abstract

The main result of this paper provides six necessary and sufficient conditions under various regularity assumptions for a so-called Cauchy mean to be identical to a two-variable quasiarithmetic mean. One of these conditions says that a Cauchy mean is quasiarithmetic if and only if the range of its generating functions is covered by a nondegenerate conic section.

Characterizations of the equality of two-variable generalized quasiarithmetic means

“…Assume that (v) holds for some real constants α, β, γ, δ, ε, ζ, η. Then implication (ii)⇒(iii) of [25,Theorem 10] and implication (iv)⇒(vi) of [19,Theorem 7] ϕ, from which we get that A ϕ = A ψ is valid on I 2 . Consequently, (vii) holds.…”

“…Therefore, all the assertions from (i) to (viii) are equivalent. Finally, the equivalence of (viii) and (ix) is a direct consequence of [25,Corollary 9] and [19], respectively.…”

“…The second equality in (5. 19) is a consequence of W 1,0 h,k = k 2 (h/k) ′ and the second identity in (5.21). Furthermore, using (5.21), we have that…”

“…is constant. Applying implication (vii)⇒(iv) of [19,Theorem 7], then there exist real constants δ, ε, ζ such that the second equality in (5.18) is valid. Thus, we have proved that assertion (v) holds.…”

On the equality of two-variable general functional means

“…The following theorem characterizes the equality of generalized quasiarithmetic means under 3 times differentiability assumptions in the case when the third central moment is different from zero. x;f,g;μ (0) = m (2) x;F,G;μ (0) and m (4) x;f,g;μ (0) = m (4) x;F,G;μ (0) (21) hold, then the equality Φ f,g = Φ F,G =: Φ holds on I and there exists a constant γ ∈ R such that…”