A Reverse H¨older Type Inequality for the Logarithmic and Generalizations

John Maloney ∗ Jack Heidel † Josip Pe˘cari´c ‡ December 13, 1994

Abstract An inequality involving the logarithmic mean is established. Specifi- cally, we show that

ln(c/x) ln(x/a) L(c, x) ln(c/a) L(x, a) ln(c/a) < L(c, a) (1)

y−x where 0 < a < x < c and L(x, y) = ln y−ln x , 0 < x < y. Then several generalizations are given.

Key words Logarithmic mean, inequality

∗Department of Mathematics University of Nebraska at Omaha 68182 †Department of Mathematics University of Nebraska at Omaha 68182 ‡Faculty of Textile Technology, Zagreb Croatia

1 AMS(MOS) subjeAct classiflcation 15, 65, 68

1 Introduction

The logarithmic mean y x L(y, x) = − 0 < x < y, ln y ln x − has many applications in statistics and economics [8]. It is well known, and easily established [1,3,6,9]that

G(y, x) L(y, x) A(y, x) ≤ ≤ where G(y, x) = √xy is the geometric mean and A(y, x) = (x + y)/2 is the arithmetic mean. In fact, writing A(y, x) = M1(y, x) where

yp + xp 1/p M (y, x) = p 2 µ ¶ it is known [6] that Mp1(y, x) Mp2(y, x) for p1 p2 It is also known, [4,5,8,11,13], that ≤ ≤ L(y, x) M (y, x) ≤ 1/3 On the other hand, Hlder’s inequality states that

M (y y , x x ) M (y , x )M (y , x ) 1 1 2 1 2 ≤ p 1 1 q 2 2 if 1/p + 1/q = 1 with p, q > 0. . It is thus curious that the logarithmic mean L(y, x) satisfles the inequality

ln(c/x) ln(x/a) L(c, x) ln(c/a) L(x, a) ln(c/a) < L(c, a) (2) where 0 < a < x < c and it is noted that ln(c/x) ln(x/a) + = 1 ln(c/a) ln(c/a)

It is the reverse H˜older type inequality (1) which is the subject of this note and will be established below. (1) arises in a parameter identiflcation problem for a fractal Michaelis-Mention equation [7]. In the following, use will be made of Jensen’s inequality [10] which we now state for the reader’s convenience:

2 1.1. Jensen’s Inequality 1). if w > 0 i = 1, 2, . . . , n i ∀ 2). α0, α1, α2, . . . , αn R 3). ' : [0, ) R is ∈a strictly convex then ∞ → n n w α n w ' i=1 i i w '(α ) i n w ≤ i i Ãi=1 ! i=1 i i=1 X µP ¶ X and the inequality is strict unless αP0 = α1 = α2 = = αn. · · · 2 Main Result

nu Lemma 2.1. Let g(u) = u 1 where g(1) = 1. Then u > 0 i). g is a strictly decreasing−functin of u ∀ ii). iii). lim g(u) = , lim g(u) = 0, lim g(u) = 1 u 0+ ∞ u u 1 iv). g→(1/u) = ug(u). →∞ → Proof:

1 1 Set z(u) = 1 1/u ln u then z 0(u) = u u 1 which is positive for 0 < u < 1 and negativ− e for− u > 1. Thus z(u) increase− from to 0 at u = 1 ¡ ¢ −∞ and then decreases to as u tends to . Thus g 0(u) is negative except at u = 1. This establishes (i). The−∞ limits in (ii) can∞ be computed in the usual fashion using L’hopital’s rule. For (iii) we have ln(1/u) g(1/u) = = ug(u). 1/u 1 − ¤ Lemma 2.2. Let f(x) = x ln x, then i). f is decreasing on (0,1) and− increasing on (1, ) ii). lim f(x) = , f(1) = 1, and lim f(x) =∞ x 0+ ∞ x ∞ iii). →if α > 0, x > 0 then f(αx) =→∞f(x) for x = g(α) so that f(αg(α)) = f(g(α)). Proof:

(i) and (ii) can be established in the usual way. For (iii) we have

f(α x) = f(x) α x ln(α x) = x ln x (α 1)x = ln α x = g(α). ⇒ − − ⇒ − ⇒ ¤ Let y(x) denote the left hand side of (1) and set α = ln c ln a. Note that y(x) > 0 a < x < c. Then − ∀ α ln y = [ln c ln x] [ln (c x) ln(ln c ln x)]+[ln x ln a] [ln(x a) ln(ln x ln a)] − − − − − − − −

3 and so

α y 0 1 1 1/x = [ln(c x) ln(ln c ln x)] + [ln c ln x] − − + y −x − − − − c x − ln c ln x · − − ¸ 1 1 1/x [ln(x a) ln(ln x ln a)] + [ln x ln a] x − − − − x a − ln x ln a · − − ¸ 1 x a ln x ln a 1 1 ln c ln x 1 c x = ln − + − + − ln − x ln x ln a x a − x x − c x − x ln c ln x · · − ¸¸ · − ¸ − · − ¸ 1 a/x 1 1 ln(a/x) 1 ln(c/x) 1 c/x 1 = ln x − + ln x − x ln(a/x) x a/x 1 − x c/x 1 − x ln(c/x) · ¸ · ¸ − − (3) or α y 1 1 0 = [f(g(a/x)) f(g(c/x))] = h(x) (4) y x − x Now f(g(a/x)) is an increasing function of x while f(g(c/x)) is a decreasing function of x so that h(x) is an increasing function of x. Clearly αy 0/y is zero at exactly one point which implies that y 0 is zero at exactly one point.

Lemma 2.3. y 0 is zero at the point x = √ac. Proof: a Now f(g(c/x)) = f(g(a/x)) = f x g(a/x)) , from lemma 2.3 (iii), so that g(c/x) = (a/x)g(a/x) = g(x/a) by lemma 2.2 (iii). Thus c/x = x/a giving ¡ ¢ x = √ac. ¤ Theorem 2.1. For all values of 0 < a < x < c

ln c ln x ln x ln a ln c ln a c x − x a − c a − − − < − (5) ln c ln x ln x ln a ln c ln a µ − ¶ µ − ¶ µ − ¶ Proof: The results hold ifi c x x a c a (ln c ln x) − +(ln x ln a) − < (ln c ln a) − − ln c ln x − ln x ln a − ln c ln a µ − ¶ µ − ¶ µ − ¶

xi xi 1 Set x = a, x = x, x = c and let w = ln x ln x , α = − − 0 1 2 i i i 1 i ln xi ln xi 1 − − − and let ' (x) = ln x, the result follows from the Jensen’s inequalit− y with rather than− <. But ≤ y α y 0 = [f(g(a/x)) f(g(c/x))] x − so that y 0 is negative on [a, √ac] and positive on [√ac, c]. Strict inequal- ity in Theorem 2.4 now follows from the previous results since the is strictly negative on [a, √ac] and positive on the interval [√ac, c]. Thus equality holds only at a and c. ¤

4 3 Convexity

Theorem 3.1. The function

ln c ln x ln x ln a ln c−ln a ln c−ln a c x − x a − y(x) = − − (6) ln c ln x ln x ln a µ − ¶ µ − ¶ is log-convex, and hence convex, on the interval √ac. Proof: Let w = α ln y, then w 0 = αy 0/y and hence from (5) xw 0 = f(g(a/x)) − f(g(c/x)) is an increasing function so that w 0 + xw 00 0. Thus xw 00 ≥ ≥ w 0. Now on [a, √ac], w 0 0, and so w 00 0 so that w is convex (and − ≤ ≥ hence log convex) on [a, √ac]. ¤ Lemma 3.1. The curve ln c ln x ln x ln a ln c−ln a ln c−ln a c x − x a − y(x) = − − (6) ln c ln x ln x ln a µ − ¶ µ − ¶ is invariant under the transformation x ac/x. ← Proof:

ln c ln ac ln ac ln a − ( x ) ( x )− ac ln c ln a ac ln c ln a c − a − z(x) = − x x − ln c ln ac ln ac ln a à − x ! à x − ! ln c ln(ac)+ln x ln(ac) ln x ln a c¡(x ¢a) −ln c¡ ln a¢ a(c x) ln−c ln −a − − − − = x x ln c ln a ln c + ln x ln(ac) ln x ln a à − − ! à − − ! ln x ln a ln c ln x c(x a) ln c−ln a a(c x) ln c−ln a − − − − = x x ln x ln a ln c ln x à − ! à − ! ln x ln a ln c ln x c ln c−ln a a ln c−ln a = − − y(x). x x ³ ´ ³ ´ (7) Now ln x ln a ln c ln x − + − = 1 ln c ln a ln c ln a − − Thus from (7)

ln x ln a ln c ln x ln x ln a ln x ln a c ln c−ln a a ln c−ln a c ln c−ln a a 1 ln c−ln a − − = − − − x x x x ³ ´ ³ ´ ³ ´ln x ln a ³ ´ c ln c−ln a x − a = ln x ln a a ln c−ln a x ¡ x ¢ − a c ln(x/a)/ ln(c/a) = ¡ ¢ x a ³a x´ ³ ´ = = 1 since bx = ex ln b. x a

5 Thus z(x) = y(x) and the lemma is proved. ¤

4 Generalizations and Applications

The following theorems follow directly from Jensen’s inequality and are generalizations of Theorem 2.1. Theorem 4.1. if: i). ' : [0, ) R is a function ii). f, g : [0∞. →) R are increasing functions ∞ → 3). A0, A1, . . . , An then 1). If ' is convex

n f(An) f(A0) f(Ai) f(Ai 1) (g(An) g(A0)) ' − (g(Ai) g(Ai 1)) ' − − − g(A ) g(A ) ≤ − − g(A ) g(A ) n 0 i=1 i i 1 µ − ¶ X µ − − ¶ 2). If ' is concave then

n f(An) f(A0) f(Ai) f(Ai 1) (g(An) g(A0)) ' − (g(Ai) g(Ai 1)) ' − − − g(A ) g(A ) ≥ − − g(A ) g(A ) n 0 i=1 i i 1 µ − ¶ X µ − − ¶ 3). If ' is log convex then

(g(An) g(A0)) n (g(Ai) g(Ai 1)) f(An) f(A0) − f(Ai) f(Ai 1) − − ' − ' − − g(A ) g(A ) ≤ g(A ) g(A ) n 0 i=1 i i 1 µ − ¶ Y µ − − ¶ 4). If ' log concave then

(g(An) g(A0)) n (g(Ai) g(Ai 1)) f(An) f(A0) − f(Ai) f(Ai 1) − − ' − ' − − g(A ) g(A ) ≥ g(A ) g(A ) n 0 i=1 i i 1 µ − ¶ Y µ − − ¶ Proof: f(Ai) f(Ai 1) In Jensen’s inequality set w = g(A ) g(A ) and α = − − i i i 1 i g(Ai) g(Ai 1) − − − − and the result follows. ¤ As a flrst application let M, N : R R N strictly monotone. Given any two numbers a and b, there is a n→umber c, according to the , such that

M(b) M(a) M 0(c) − = N(b) N(a) N (c) − 0 for some c, a < c < b. If c is uniquely determined then it is called the (M.N) mean-value mean of a and b [2]. In this case let H be the inverse of M 0/N 0 and write M(b) M(a) c = H − N(b) N(a) µ − ¶

6 If M and N are both increasing and H is either log-convex or log- concave, we can apply one of the inequalities in Theorem 4.1 to write

N(A ) N(A ) i − i 1 n N(A ) N(A− ) M(An) M(A0) M(Ai) M(Ai 1) n − 0 H − H − − N(A ) N(A ) ≤ N(A ) N(A ) n 0 i=1 i i 1 µ − ¶ Y µ − − ¶ or

N(A ) N(A ) i − i 1 n N(A ) N(A− ) M(An) M(A0) M(Ai) M(Ai 1) n − 0 H − H − − N(A ) N(A ) ≥ N(A ) N(A ) n 0 i=1 i i 1 µ − ¶ Y µ − − ¶ where we have made the associations that ' = H, f = M, g = N, An = b, A0 = a Now specializing to the case of ' (x) (log-concave ' ) in theorem 4.1 we obtain

g(A ) g(A ) i − i 1 n g(A ) g(A− ) f(An) f(A0) f(Ai) f(Ai 1) n − 0 − − − g(A ) g(A ) ≥ g(A ) g(A ) n 0 i=1 i i 1 − Y µ − − ¶ and interchanging f and g we can write

f(A ) f(A ) i − i 1 n f(A ) f(A− ) f(An) f(A0) f(Ai) f(Ai 1) n − 0 − − − g(A ) g(A ) ≤ g(A ) g(A ) n 0 i=1 i i 1 − Y µ − − ¶ From these expressions we can obtain inequalities for Stolarsky’s ([2], [12]) extended mean value

1 s s s r r (a b ) − E (a, b) = − r,s s (ar br) µ − ¶ s r by making the associations f(x) = x /s, g(x) = x /r, An = b, A0 = a and then raising both sides to the power 1/(s-r). For rs > 0

br ur ur ar bs us us as s s br−ar s s br −ar s s s s bs−as s s bs −as b u − u a − b a b u − u a − − − − − − br ur ur ar ≤ br ar ≤ br ur ur ar µ − ¶ µ − ¶ − µ − ¶ µ − ¶ where a < u < b. If rs < 0, f(x) = xs/s and g(x) = xr/r are still both increasing functions and we have a similar inequality

br ur ur ar bs us us as s s br−ar s s br −ar s s s s bs−as s s bs −as r (b u ) − r (u a ) − r (b a ) r (b u ) − r (u a ) − − − − − − s (ur ar) s (ur ar) ≤ s (br ar) ≤ s (br ur) s (ur ar) µ − ¶ µ − ¶ − µ − ¶ µ − ¶ where it is now necessary to include r/s or else reverse the inequality.

7 A further application is obtained by setting f(x) = x and g(x) = ln x above to obtain

ln(An) ln(A0) n ln(Ai) ln(Ai 1) An A0 − Ai Ai 1 − − − − − ln(A ) ln(A ) ≥ ln(A ) ln(A ) n 0 i=1 i i 1 µ − ¶ Y µ − − ¶ and

An A0 n Ai Ai 1 An A0 − Ai Ai 1 − − − − − ln(A ) ln(A ) ≤ ln(A ) ln(A ) n 0 i=1 i i 1 µ − ¶ Y µ − − ¶ These two inequalities provide a direct generalization and converse to the main inequality (3) discussed in this paper

ln c ln x ln x ln a ln c ln a c x − x a − c a − − − < − . ln c ln x ln x ln a ln c ln a µ − ¶ µ − ¶ µ − ¶

8 References

[1] E.F. Beckenbach and R. Bellman. Inequalities. , Springer Verlag, Berlin, , second edition edition, 1965.

[2] P.S. Bullen, D.S. Mitronovi¶c, and P.M. Vasi¶c. and Their Inequalities. D. Reidel Publishing Co, Dordrecht, 1988.

[3] Frank Burk. By all means. Amer. Math. Monthly, 92:50, 1985.

[4] B.C. Carlson. The logarithmic mean. Amer. Math. Monthly, 79:615{618, 1972. [5] W.E. Diewart. Superlative index numbers and consistency in aggregation. Economterika, 46:883{900, 1978.

[6] G.H. Hardy, J.E. Littlewood, and G. Polya. Inequalities. Cam- bridge University Press, Great Britain, 1967.

[7] J. Heidel and J. Maloney. An analysis of a fractal michaelis- menten curve. J. Australian Math. Soc.,Series B, 41:410{422, 2000.

[8] Great Britain. Log-ratios and the logarithmic mean. Statistical Papers, 30:61{75, 1989.

[9] D.S. Mitronovi¶c. Analytic Inequalities. , Springer-Verlag, , Berlin, 1970.

[10] D.S. Mitronovi¶c, J. E. Pe‚cari¶c, and A. M. Fink. Classical and New Inequalities in Analysis. Kluner Academic Publishers, Dor- drecht, 1993.

[11] Dieter Ruthing. Eine allgemeine logarithmische ungleichung. , El. Math, 41:14{16, 1996.

[12] Kenneth B. Stolarsky. The power and generalized logarithmic means. American Mathematical Monthly, 87:545{548, 1980.

[13] H.Van Haeringen. Unimodal functions, and inequalities for the logarithmic mean. Delft Progress Report, Convex Fuzzy Sets, 8:173{179, 1983.

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