View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Math. Anal. Appl. 373 (2011) 227–234 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa A mean-value theorem and its applications ∗ Janusz Matkowski a,b, a Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Podgórna 50, PL-65246 Zielona Góra, Poland b Institute of Mathematics, Silesian University, Bankowa 14, PL-42007 Katowice, Poland article info abstract Article history: Forafunction f defined in an interval I, satisfying the conditions ensuring the existence [ ] Received 8 March 2010 and uniqueness of the Lagrange mean L f , we prove that there exists a unique two variable [ ] Available online 10 July 2010 mean M f such that Submitted by M. Laczkovich f (x) − f (y) [ ] = M f f (x), f (y) Keywords: x − y Mean-value theorem [ ] [ ] Lagrange mean-value theorem for all x, y ∈ I, x = y.ThemeanM f is closely related L f . Necessary and sufficient [ ] [ ] [ ] Cauchy mean-value theorem condition for the equality M f = M g is given. A family of means {M t : t ∈ R} relevant Mean to the logarithmic means is introduced. The invariance of geometric mean with respect to Stolarsky means mean-type mappings of this type is considered. A result on convergence of the sequences Invariant mean of iterates of some mean-type mappings and its application in solving some functional Mean-type mapping Iteration equations is given. A counterpart of the Cauchy mean-value theorem is presented. Some [t] Functional equation relations between Stolarsky means and M means are discussed. © 2010 Elsevier Inc. All rights reserved. 0. Introduction We present some mean-value theorems, closely related to the classical results of Lagrange and Cauchy. [ ] In the first section, under the conditions assuring the existence and uniqueness of the Lagrange mean L f for a real [ ] [ ] function f in an interval I, we prove the existence of a unique two variable mean M f , accompanying to L f , such that f (x) − f (y) [ ] = M f f (x), f (y) x − y [ ] for all x, y ∈ I, x = y as well as we give the formula of M f . We give necessary and sufficient condition for the equality [ ] [ ] M f = M g . It is analogous to that of Berrone and Moro [1] for the Lagrange means. Some examples of the accompanying Lagrange means, a mean-type mapping of such means for which the geometric mean is invariant, and application of the invariance in solving some functional equation, are presented. [ ] It is well known that {L p : p ∈ R}, the one-parameter family of logarithmic means, is, with respect to p, a continuous [ ] extension of the family of all L f means generated by the power functions f (x) = xp for all p ∈ R such that p(p − 1) = 0. {M[t] ∈ R} = 1 We show that the relevant family of accompanying means : t , with the parametrization t p−1 , is continuously extendable at the points t =−1 and t = 0. It turns out that the classical means, arithmetic, harmonic, geometric and [− ] logarithmic, belong to this family. Moreover, for any t ∈ R,thegeometricmeanG = M 1/2 is invariant with respect to * Address for correspondence: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Podgórna 50, PL-65246 Zielona Góra, Poland. E-mail address: [email protected]. 0022-247X/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2010.06.057 228 J. Matkowski / J. Math. Anal. Appl. 373 (2011) 227–234 [ ] [− + ] the mean-type mapping (M t , M (t 1) ), and the sequence of iterates of this mean-type mapping converges pointwise to (G, G) in (0, ∞)2. We apply this fact in solving a functional equation. In the second section, applying the main result of Section 1, we prove the relevant counterpart of the Cauchy mean- value theorem. Under the assumptions on real functions f , g defined on an interval, which guarantee the existence and [ ] uniqueness of the Cauchy mean C f ,g ,weprovetheexistenceofauniquemeanM such that f (x) − f (y) f (x) f (y) = M , g(x) − g(y) g(x) g(y) [ ◦ −1] − for all x, y ∈ I, x = y.MoreoverM = M f g is the accompanying Lagrange mean of the function f ◦ g 1. Some representa- tion formulas of the Cauchy mean with the aid of a Lagrange and its accompanying mean are proposed. A relation between [ ] Stolarsky means and the M t means is presented. 1. An accompanying of the Lagrange theorem We begin this section with the following: Theorem 1. Let I ⊆ R be an interval, f : I → R be a differentiable function. Then J := f (I) is an interval and, if f is one-to-one, then there exists a unique strict mean M : J 2 → J suchthatforanyx, y ∈ I, x= y, f (x) − f (y) = M f (x), f (y) . x − y Moreover, − − f [( f ) 1(u)]− f [( f ) 1(v)] M(u, v) = , u, v ∈ J, u = v. (1) ( f )−1(u) − ( f )−1(v) Proof. The Darboux property of derivative implies that J is an interval and f is strictly monotonic and continuous. Take any u, v ∈ J , u = v. There are unique x, y ∈ I, x = y,suchthatu = f (x), v = f (y). By the Lagrange mean-value theorem and the strict monotonicity of f , there is a unique ξ ∈ (min(x, y), max(x, y)) such that f (x) − f (y) = f ξ(x, y) , x − y that is −1 −1 f [( f ) (u)]− f [( f ) (v)] −1 −1 = f ξ f (u), f (v) . (2) ( f )−1(u) − ( f )−1(v) Setting − − f [( f ) 1(u)]− f [( f ) 1(v)] M(u, v) := , (3) ( f )−1(u) − ( f )−1(v) and taking into account that min f (x), f (y) < f (ξ) < min f (x), f (y) , we have min(u, v)<M(u, v)<max(u, v). Putting M(u, u) = u for all u ∈ J we see that M is a strict mean. For arbitrary x, y ∈ I, x = y, setting u := f (x), v := f (y) in (3) we obtain f (x) − f (y) M f (x), f (y) = . x − y − − The uniqueness of M follows from this formula (it is enough to put here x = ( f ) 1(u) and y = ( f ) 1(v) for u = v). 2 [ ] [ ] Denote by M f the mean M in formula (1). Thus, under the assumptions of Theorem 1, the mean M f : J 2 → J is defined by − − f [( f ) 1(u)]− f [( f ) 1(v)] [ f ] for u = v, M (u, v) := ( f )−1(u)−( f )−1(v) u, v ∈ J := f (I). (4) u for u = v, J. Matkowski / J. Math. Anal. Appl. 373 (2011) 227–234 229 [ ] Note that M f is continuous and symmetric. It is well known that, under the assumptions of Theorem 1, the function f generates the Lagrange mean defined by − f (x)− f (y) [ ] ( f ) 1( ) for x = y, L f (x, y) := x−y x, y ∈ I. x for x = y, [ ] As ξ in (2) coincides with L f , we have the following obvious Proposition 1. Let I ⊆ R be an interval, f : I → R be a differentiable function such that f is one-to-one. Then [ ] [ ] −1 −1 M f (u, v) = f L f f (u), f (v) , u, v ∈ J := f (I), and [ ] −1 [ ] L f (x, y) = f M f f (x), f (y) , x, y ∈ I. [ ] [ ] [ ] Thus the means L f and M f are closely related. The mean M f could be referred to as accompanying of the Lagrange [ ] mean L f . Theorem 2. Let I, I1 ⊆ R be intervals. Suppose that f : I → R,g: I1 → R, are differentiable and such that f ,g are one-to-one and [ f ] [g] f (I) = g (I1).ThenM = M if, and only if, there are a, b ∈ R,a= 0,suchthat −1 −1 g (u) = a f (u) + b, u ∈ f (I). [ ] [ ] Proof. Suppose that M f = M g . Putting J := f (I),wehave − − − − f [( f ) 1(u)]− f [( f ) 1(v)] g[(g ) 1(u)]−g[(g ) 1(v)] = , u, v ∈ J, u = v. ( f )−1(u) − ( f )−1(v) (g)−1(u) − (g)−1(v) Hence, setting −1 h := g ◦ f , we get f (x) − f (y) g[h(x)]−g[h(y)] = , x, y ∈ I, x = y. (5) x − y h(x) − h(y) Without any loss of generality we may assume that 0 ∈ I and 0 ∈ I1. (If it is not the case we can translate these intervals and redefine in an obvious way the functions f and g.) Since for arbitrary c, c1 ∈ R, the functions f + c and g + c1 also satisfy this equation, i.e. [ f (x) + c]−[f (y) + c] {g[h(x)]+c1}−{g[h(y)]+c1} = , x, y ∈ I, x = y, x − y h(x) − h(y) we can assume that f (0) = g(0) = 0. Hence, setting y = 0 in (5), we obtain f (x) g[h(x)] = , x ∈ I, x = 0, x h(x) − h(0) whence f (x) g h(x) = h(x) − h(0) , x ∈ I, x = 0. x Applying this formula in (5) we obtain f (x) f (y) f (x) − f (y) [h(x) − h(0)]− [h(y) − h(0)] = x x , x, y ∈ I\{0}, x = y, x − y [h(x) − h(0)]−[h(y) − h(0)] which, after simple calculations, reduces to the equality yf(x) − xf (y) x h(y) − h(0) − y h(x) − h(0) = 0(6) for all x, y ∈ I\{0}, x = y.