On homogeneous Lagrange means
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On homogeneous Lagrange means Janusz Matkowski
Published online: 20 May 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com
Abstract Let f be a real differentiable function in an open interval I with one-to-one derivative. We observe that if the Lagrange mean L [ f ] of a generator f is conditionally positively homogeneous, then f must be of the class C ∞ and the function g(x) := x f (x) − f (x) , L[ f ]
L [g]
x ∈ I,
L [ f ] . We show that this fact and a result on equality
is also a generator of i.e. that = of two Lagrange means allow easily to determine all positively homogeneous Lagrange means. Keywords Mean · Lagrange mean · Homogeneous mean · Equality of means · Differential equation Mathematics Subject Classification
Primary: 26A24 · 26E60 · 39B22
1 Introduction Let f be a real differentiable function defined in an open interval I ⊂ R. By the Lagrange mean-value theorem, there exists a two-place function L [ f ] : I 2 → I , called a Lagrange mean of a generator f , such that, for all x, y ∈ I, f (x) − f (y) = f L [ f ] (x, y) (x − y) . If the derivative f is one-to-one, then the mean L [ f ] is uniquely determined. It is well known that if the Lagrange mean L [ f ] is positively homogeneous, then it is a generalized logarithmic mean L[ p] ([4,5] cf. also [3], p. 404, Theorem 6). The main purpose J. Matkowski (B) Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul Prof. Z. Szafrana 5a, 65-516 Zielona Góra, Poland e-mail: [email protected]
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of this note is to present a new proof of this fact. The idea is based on the observation that any generator f of a conditionally positively homogeneous Lagrange mean L [ f ] : I 2 → I must be of the class C ∞ and that the function g(x) := x f (x) − f (x) ,
x ∈ I,
is also a generator of L [ f ] , that is L [g] = L [ f ] . These facts and a result on equality of two Lagrange means (cf. [1], [6]) imply that x f (x) − f (x) = a f (x) + bx + c,
x ∈ I,
for some a, b, c ∈ R. Solving this differential equation we obtain all the homogeneous Lagrange means. In section 2 we recall the definitions of a mean, the Lagrange mean based on Lagrange’s mean-value theorem (that is used in this paper), its integral counterpart and some of its properties. In section 3 we prove Theorem 3.2, a slight modification of the results on the equality of two Lagrange means (cf. [1] and [6]). In the first part of Theorem 3.2 we assume that at least one of two equal means is uniquely determined. In section 4 we show that any generator f of the conditionally homogeneous Lagrange mean L [ f ] in an open interval I, that is such that, for all x, y ∈ I and t > 0, t x, t y ∈ I ⇒ L [ f ] (t x, t y) = t L [ f ] (x, y) , is infinitely times differentiable (Theorem 4.1). Applying these results, in section 5 we determine all the conditionally homogeneous Lagrange means (Theorem 5.1). Applicability of the method to determine the form of all conditionally homogeneous Cauchy means is dis
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