DEMONSTRATIO MATHEMATICA Vol. XLIII No 4 2010 Janusz Matkowski GENERALIZATIONS OF LAGRANGE AND CAUCHY MEAN-VALUE THEOREMS Abstract. Some generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented. 1. Introduction Recall that a function M : I2 —> I is called a mean in a nontrivial interval / C R I if it is internal, that is if min(x,y) < M(x,y) < max(i,y) for all x,y € I. The mean M is called strict if these inequalities are strict for all x,y € /, x ^ y, and symmetric if M(x,y) = M(y,x) for all x,y € I. The Lagrange Mean-Value Theorem can be formulated in the following way. If a function f : I —> R is differentiable, then there exists a strict symmetric mean L : I2 —> I such that, for all x,y € /,x ^ y, (i) Mzx-y M = meii)). If f is one-to-one then, obviously, L^ := L is uniquely determined and is called a Lagrange mean generated by /. Note that formula (1) can be written in the form x x iim-H -¥), n -¥)-m\ fl(T( „ cT , 21 + £±g-t, )=/(L(^))' x,y€l,x^y, \ x 2 2 y ' 2000 Mathematics Subject Classification: Primary 26A24. Key words and phrases: mean, mean-value theorem, Lagrange theorem, Cauchy the- orem, generalization, Lagrange mean, Cauchy mean, logrithmic mean, Stolarsky mean. 766 J. Matkowski or, setting A(x,y) := in the form f(x)-f(A(x,y)) f(A(x,y)) - f(y)\ (2) A = f'(L(x,y)), x — A(x, y) ' A(x, y) — y ) x,y€ I, x^y, which shows a relationship of the mean-value theorem and the arithmetic mean. This equation has the following geometrical interpretation. The arithmetic mean of the slope of chord of the graph of / passing through the points (x, f(x)) and / and the slope of chord of the graph passing through the points / and (y, f(y)) is equal to the slope of tangent to the graph at a point (L(x,y), f(L(x,y))) (cf. Figure 1). /(f) Fig. 1. Our idea of generalization of the Lagrange Mean-Value Theorem is based on formula (2). Let I, J C R be intervals. Assume that M : I2 —> I is a strict mean in I and K : J2 J is a mean in J. In section 2 we show that if / : I —» K is a differentiate function and /'(/) C J, then there exists a strict mean L : I2 —> I such that, for all x,y G I, x ¿y, 'f(x)-f(M(x,y)) f(M(x,y))-f(yY K = f'(L(x,y)). x — M{x,y) ' M(x,y) — y Moreover, L := L^ K is unique if /' is one-to-one, and symmetric if M and K are symmetric. Putting Aw(x,y) := wx + (1 - w)y, x,yeR, Mean-value theorems 767 for arbitrarily fixed w G (0,1), we observe that for M Aw and K = A\-w the above result reduces to the original Lagrange theorem. In particular, in this case, L does not depend on w. Since = L^, the mean L^jj K gener- alizes the Lagrange mean u. As an application we obtain a generalization of the family of logarithmic means. The Cauchy Mean-Value Theorem can be formulated as follows. If f,g: I —> R are differentiable and g'(x) ^ 0 for all x & I then there exists a strict symmetric mean C : I2 I such that for all x,y G I,x ^ y, m f(x)~f(y) = f'(C(x,y)) lJ g(x)-g(y) g'(C(x,y))- Moreover, if ¿7 is one-to-one then C is unique and it is called a Cauchy mean generated by / and g. The assumptions of Cauchy's Mean-Value Theorem imply that g is con- tinuous, strictly monotonic. It follows that the quasi-arithmetic mean Afbl : is well defined, and (3), the original Cauchy formula, can be written in the form (f{x)-f(MW(x,y)) ttMW(x,y))-f(y)\ = f'(C(x,y)) { 1 \g(x)-g(M\9](x,y))'g{M\9\(x,y))-g{y)J g'(C(x,y)) for all x,y G I, x ^ y. In Section 3 we generalize the Cauchy Mean-Value Theorem showing that if f,g : I —• R are differentiable, g'(x) ^ 0 for all x G I, and {¡>(1) C J, then formula (4) remains true on replacing the quasi-arithmetic mean by an arbitrary strict mean M and the arithmetic mean A by an arbitrary mean K. Moreover, C[,:= C is unique if ^ is one-to-one, and symmetric if M and K are symmetric. In the case when M = where M$(x,y) :— l g~ (wg(x) + (1 — w)g(y)) and K = A\-w for some w G (0,1), this gener- alization reduces the original Cauchy theorem. Applying among other the Stolarsky means ([4], [5]) we obtain some new classes of means. 2. A generalization of Lagrange's Mean-Value Theorem THEOREM 1. Let I, J C R be intervals. Suppose that M : I2 —> I is a strict mean and K : J2 —> J is a mean. If a function f : I —* R is differentiable 768 J. Matkowski and f'(I) C J, then there exists a strict mean L : I2 —> / such that m „(ft?) - f(M(x, y)) f(M(x, y))-f(y)\ x,y e I, x^y. Moreover, if f is one-to-one then L =: ^M K ^ unique, (6) 1 f{x) f{Mix y)) /(M(x y)) fiy) i(f')- (K( ~ - ' ~ )) for x + u Llf] (xv)=)V> x-M(x,y) ' M(x,y)~y )) J r Vi for x = y, and K is symmetric if M and K are symmetric. Proof. Take x,y 6 I, x ^ y. We can assume, without any loss of generality, that x < y. Since M is strict, we have x < M(x, y) < y. By the Lagrange Mean-Value Theorem there are r, s G I, r = r(x, y), s = s(x, y), (7) x<r < M(x, y), M(x, y) < s <y, such that f(x)-f(M(x,y)) _ , f(M(x,y))-f(y) _ () x-M(x,y) ~J[rh M(x,y) — y ~1 [S)' Since min (f'(r)j'(s)) < K (/' (r), f (a)) < max(/'(r),/'(*)), the Darboux property of the derivative (cf. [3], p. 108, Theorem 5.12) implies that there exists t = t(r,s) E I, (9) r < t < s, such that (10) K(f'(r)J'(s)) = f'(t). From (7) and (9) we have x = min(x, y) < t(r(x,y),s(x,y)) < max(x, y) = y. Putting L(x,y) := t(r{x,y),s(x,y)) for x ± y, and L(x, y) := x for x = y, we get a mean L which, according to (8) and (10), satisfies (5). If /' is one-to-one, formula (6) is a consequence of (5). The remaining statement is obvious. • REMARK 1. Let a function / satisfy the conditions of Theorem 1 and let w G (0,1) be fixed. Denote by Aw the weighted arithmetic mean Aw(x,y) := wx + (1 - w)y, x,yeR. Mean-value theorems 769 It easy to verify that formula (5) of Theorem 1 with M := Aw and K := A\~w reduces to the classical Lagrange theorem. For w = ^ we get relation (2). Note also that, for each w 6 (0,1), A = Aw <8> A\~w, that A is a Gauss composition of the means Aw and Ai-W (cf. for instance [1]). In particular we have 2 COROLLARY 1. Suppose that M : I —> / is a strict mean in an interval I and w G (0,1). ///:/—> K is differentiate and f is one to one, then L[{] a =L[f]- »Hui i^H 1 — xi} p+l Taking I C (0, oo), K = A and f(x) = x for some p e M, -1 ^ p ± 0, in Theorem 1, we obtain 1 fxP+1 - [M(x, y)]f+1 [M(x, y)]p+1 - yP+1 \ \ 1/p 2(P+1)\ x-M{x,y) M(x,y)-y for all x,y G I, x ^ y. To complete this definition it is enough to put Simple calculations show that, for all x,y € /, x ^ y, [pj N _ (I flogx-logM(x,y) | log M (x, y) - logy W'1 p—1 M>A y2 \ x-M(x,y) + M(x,y)-y )) = H(£(x,M(x,y)),£(M(x,y),y)), and / y/<-*<-» fM(x,yr^ y/(M(^ lunLMA(x,y)-e [M(xy)M{x,y)) ^ yV J = g{J(x, M(X, y)), J{M(x, y),y)), where 7i, C, 0, J denote, respectively, the harmonic, logarithmic, geometric and identric means defined by H(x,y) = -^-, C(x,y)= X V x + y' ' log x — log y' \ f \ x — y 0(x, y) = y/xy, J(x, y) = ~ ( J for all x, y > 0, x ^ y. Hence, denoting by V\ and V2 the projective means, we get the following 2 COROLLARY 2. Let I C (0,00) be an interval, M : I —> I a strict mean and p € R. Then the function A : I2 —» / defined by 770 J. Matkowski -{no (Co(VuM),£ O (M, p2)) if p= -1, G°(JO(VUM),JO{M,P2)) ifp = 0, [pi I.