DEMONSTRATIO MATHEMATICA Vol. XLIII No 4 2010

Janusz Matkowski

GENERALIZATIONS OF LAGRANGE AND CAUCHY -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 are presented.

1. Introduction Recall that a 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, . 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 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

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. The each x,y G I the function is a strict mean in I.p The£ mean C^M ^s symmetric if M is symmetric. For

Moreover, if M — A then p —»• JC^IM(x, y) is continuous in R.

C% = &\ PeR, where 1 is the of order p.

Thus the family of means {¿^M : p £ 1} generalizes {C^ : p G M}, a well known family of logarithmic means of order p (cf. for instance [2], p. 385).

3. A generalization of Cauchy's Mean-Value Theorem THEOREM 2. Let I,JcRbe intervals. Suppose that M : I2 -» I is a strict mean in I and K : J2 —> J is a mean. If the functions f,g:I—>M. are differentiable, g'(x) ^ 0 for all x G I, and C J, then there exists a strict mean C : I2 —> I such that, for all x,y G I, x ^ y, ff(x)-f(M(x,y)) f(M(x,y))-f(y)\ ^ f'(C(x,y)) 1 > \g(x)-g(M(x,y)yg(M(x,y))-g(y)J g>(C(x,y)y Moreover, if ^r is one-to-one, then C =: cj^'^j- is unique, (12)

CM9K^y) = S ^ \g(x)-g(M(x,y)) ' g(M(x,y))-g(y) J J i0T X T f> ^ x for x = y, and CM'9^ symmetric if M and K are symmetric. Proof. Take x,y G I, x ^ y. Without any loss of generality we can assume that x < y. Since M is strict, we have x < M(x, y) < y. By the Cauchy Mean-Value Theorem there are r,s G I, r = r(x, y), s = s(x, y),

(13) x < r < M(x, y), M(x, y) < s

Since g'(x) ^ 0 for all x € /, the Darboux property of derivative implies that g is continuous, strictly increasing in I and, consequently, the inverse function g~l : g(I) —> I is differentiate in the interval g(I). The relation

—, = U °g~1)' °g, 9 the continuity of g and the Darboux property of the derivative (f ° g~l) imply that the function ^ has the Darboux property. Since mm• Mo(/'(r -77-r) , -/'(*)}TTT I << K„(fir | —), f'(s)\I <

(16) K(mm) = m

From (13) and (15) we have x = min(i, y) < t(r(x, y), s(x, y)) < max(x, y) = y. Thus, putting

C(x,y) := t(r(x,y),s(x,y)) for x ± y, and C(x, y) := x for x = y, we get a mean C which, according to (14) and (16), satisfies (11). If ^r is one-to-one then formula (12) is a consequence of (11). The re- maining statement is obvious. •

REMARK 2. Let the functions / and g satisfy the conditions of Theorem 2 and let w € (0,1) be fixed. Denote by Aw the weighted arithmetic mean

Aw(x,y)\=wx + (\-w)y, x,yeR. Since g is continuous and strictly monotonic, the weighted quasi-arithmetic mean : I2 —> I given by

M${x,y) g~l{wg(x) + {1 -w)g(y)), x,y G /, is well defined. It easy to check that for M := and K =: A\~w formula (11) in Theorem 2 reduces to (3) that is to the classical Cauchy's theorem, and for w — ^ we get relation (4) mentioned in the Introduction. Thus we have the following

COROLLARY 3. Suppose that M : I2 —> I is a strict mean in an interval I and w € (0,1). If f,g : I R are differentiable, g'(x) ^ 0 for all x G I, 772 J. Matkowski and jp- is one to one, then

ciffi = C^l

Taking here I = (0, oo), w G (0,1), f(x) = xp, g(x) = xq where p, q 6 R, PQ(P ~ q) + 0, we get

The family of Stolarsky means :p,g£E} can be defined as follows ( [3], cf. also [1], p. 385):

fCM ifw(p-9)^0, if 0,^ = 0, 1 £[P.

£ if p = q = 0.

(Here £ is the logarithmic mean, J is the identric mean, V\ and V2 the projective means, and Q is the geometric mean.) Note that

£M = ¿M for an p>qeR.

EXAMPLE 1. Taking I c (0,00), K A, f(x) = xp, g(x) = xq where pq{p — 9) / 0 in Theorem 2, for x y, we obtain r<[/.9], 1 /(P-9)

\2p\xi- [M(x,y)]i [M(x,y)}q -yq

Since, for all x,y £ I, x ^ y,

I/P limCr!(i,j/)-A = Y^V h—^ xP-M(x,y)P + logM(x,y)P logM(x,y)P-logyP

= (A(C(xp, [M(x, y)]p), £([M(x, y)]P, yP)))1^,

9 -l/9 1 / log x - log M(x, y)q log M(x, y)q - log yq y + p—>limCK(x,y0 M,A ) = ( 2£ V xq - M(x, y)q M(x, y)q - yq

= (H (C(xq, [M{x, y)}q),C([M(x, Mean-value theorems 773

YunCt'i(x,y) p->'i \/(x"-M{x,yY) /M(X; y^M(x,y)" \ 1 / {M {x ,y)"-y") 1/2 x = e"i/9 M(x,y)M(x>y)q <,yq = M(x, y)),£q'q(M(x, y), y)], and

= (xyM(x, y)2)1/4 = yj\/EyM(x,y) = Qo{Q, M)(x, y), we obtain the following COROLLARY 4. Let I C (0, OO) be an interval, M : I2 —> / a strict mean and p, q G R. T/ien i/ie function : /2 —> / defined by M C M,A if PQ(P -9)7*0,

r\p

o (£9.9 o (Pi, o (M,V2)) ifp = q*0, [Go(g,M) ifp = q = 0, is a strict mean in I and C^'f^ is symmetric if M is symmetric. Moreover, for all p, q G M, p ^ q, [<7>P] -A,M iC A,M" Only the last statement needs an argument. To show it suppose, on the contrary, that C^'fj = C^'^ for a strict mean M and some p ^ q. First con- sider the case pq ± 0. From the definition of cJ'J, after obvious calculations, we hence get, for all x,y G I, x ^ y,

XP-MP Mp-yp xp - Mp MP-yP = n where M = M(x,y). xi - Mi'Mi - yi) \xi - Mi' Mi - yit Since the values of the arithmetic and harmonic means are equal only on the diagonal, we hence get the equation

XP -MP MP -yP tt- = , x, y ^ i, % t1 y, which implies that (xp - Mp)(Mq - yq) - (xq - Mq)(Mp - yp) = 0, x, y G I. Let us fix x, y G I. Without any loss of generality we can assume that x < y. 774 J. Matkowski

Define a function $ : [x, y] —> R by the formula $(m) := (xp - mp)(mq - yq) - (xq - mq)(mp - yp), m € [x,y]. Note that = = 0. Since $'(m) = 0 iff

m = £^q\x,y), there is no M € (x, y) satisfying the above equation. In the case when pq = 0 we can argue similarly.

References

[1] J. M. Borwein, P. B. Borwein, Pi and the AGM, a Study in Analytic Number Theory and Computational Complexity, John Wiley & Sons Inc., New York, 1987. [2] P. S. Bullen, Handbook of Means and their Inequalities, Mathematics and Its Appli- cations, Vol. 560, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003. [3] W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, Inc., New York, 1976. [4] K. B. Stolarsky, Generalizations of the logarithmic mean, Math. Mag. 48 (1975), 87-92. [5] K. B. Stolarsky, The power and generalized logarithmic mean, Amer. Math. Monthly 87 (1980), 545-548.

FACULTY OF MATHEMATICS COMPUTER SCIENCE AND ECONOMETRY UNIVERSITY OF ZIELONA G6RA Podg6rna 50 65-246 ZIELONA GORA, POLAND and INSTITUTE OF MATHEMATICS SILESIAN UNIVERSITY Bankowa 14 42-007 KATOWICE, POLAND E-mail: [email protected]

Received July 18, 2009.