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volume 4, issue 1, article 11, 2003.

Received 23 July 2002;

accepted 22 November 2002.

Communicated by:P.S. Bullen

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Journal of Inequalities in Pure and Applied Mathematics

EXTENSIONS OF POPOVICIU’S INEQUALITY USING A GENERAL METHOD

This note is dedicated to my wife Mari – a very brave lady.

A. McD. MERCER

Department of Mathematics and Statistics University of Guelph

Guelph, Ontario, N1G 1J4, Canada;

Box 12, RR 7. Belleville, Ontario, K8N 4Z7, Canada.

EMail:amercer@reach.net

c

2000Victoria University ISSN (electronic): 1443-5756 132-02

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Extensions of Popoviciu’s Inequality Using a General

Method A. McD. Mercer

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J. Ineq. Pure and Appl. Math. 4(1) Art. 11, 2003

http://jipam.vu.edu.au

Abstract

A lemma of considerable generality is proved from which one can obtain inequalities of Popoviciu’s type involving norms in a Banach space and Gram determinants.

2000 Mathematics Subject Classification:26D15.

Key words: Aczel’s inequality, Popoviciu’s inequality, Inequalities for Grammians.

1. Introduction

Let x_k and y_k (1 ≤ k ≤ n) be non-negative real numbers. Let ¹_p + ¹_q = 1, p, q >1, and suppose thatP

x^p_k ≤1andP

y_k^q ≤1. Then an inequality due to Popoviciu reads:

(1.1)

1−X x_ky_k

≥

1−X

x^p_k_p¹

1−X y^q_k¹_q

.

When we make the substitutions (1.2) x^p_k →w_ka_k

a p

and y_k^q →w_k b_k

b q

in (1.1) and then multiply throughout byabwe get the more usual, but no more general, form of the inequality; (see [1, p.118], or [2, p.58], for example). The casep=q = 2is called Aczèl’s Inequality [1, p.117] or [2, p.57].

Our purpose here is to present a general inequality, (see lemma below), whose proof is short but which yields many generalizations of (1.1).

We shall present all our results in a ‘reduced form’ like (1.1) but it would be a simple matter to rescale them in the spirit of (1.2) to obtain inequalities of more apparent generality.

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2. The Basic Result

Lemma 2.1. Let f be a real-valued function defined and continuous on[0,1]

that is positive, strictly decreasing and strictly log-concave on the open interval (0,1).Letx, y, z ∈[0,1]and z ≤xythere. Then withpandqdefined as above we have:

(2.1) f(z)≥f(xy)≥[f(x^p)]¹^p[f(y^q)]¹^q.

Proof. Write L(x) = logf(x). The properties of f imply that L is a strictly decreasing and strictly concave function on(0,1).Hence if x, y, z ∈(0,1)and z ≤xywe have

(2.2) L(z)≥L(xy)≥L 1

px^p+1 qy^q

≥ 1

pL(x^p) + 1 qL(y^q).

The second step here uses the arithmetic mean-geometric mean inequality and the third uses the strict concavity ofL; these inequalities are strict ifx6=y.

Taking exponentials we get

(2.3) f(z)≥f(xy)≥[f(x^p)]¹^p[f(y^q)]¹^q.

Appealing to the continuity of f we now extend this to the case in which x, y, z ∈[0,1]andz ≤xythere. This completes the proof of the lemma.

Note: The conditions in Lemma2.1are satisfied if f is twice differentiable on (0,1)and on that open intervalf > 0, f⁰ < 0, f f⁰⁰−(f⁰)² < 0. Our reason for working in(0,1)and then proceeding to[0,1]via continuity is because our main specialization below will bef(x) = 1−x.

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3. Specializations

We now state some inequalities which result by specializing (2.1).

(1) Suppose that B is a Banach space whose dual is B^∗. Let g ∈ B and F ∈ B^∗. Recalling that |F(g)| ≤ kFk kgkwe read this as z ≤ xy and then (2.1) reads:

(3.1) f(|F(g)|)≥f(kFk kgk)≥[f(kFk^p)]¹^p[f(kgk^q)]¹^q, provided thatkFk,kgk ≤1.

(2) Takingf(t) = (1−t)specializes (3.1) further to:

(3.2) (1− |F(g)|)≥(1− kFk kgk)≥(1− kFk^p)¹^p(1− kgk^q)¹^q, provided thatkFk,kgk ≤1.

(3) Examples of (3.1) and (3.2) are afforded by taking B to be the sequence spaceB =l⁽ⁿ⁾q in which caseB^∗is the spacelp⁽ⁿ⁾.Then the outer inequalities of (3.1) and (3.2) yield:

(3.3) f

Xx_ky_k

≥h

fX

|x^p_k|i_p¹ h

fX

|y_k^q|i¹_q

and

(3.4)

1−

Xx_ky_k

≥

1−X

|x^p_k|¹_p

1−X

|y_k^q|¹_q , provided that, in each case,(P

|x^p_k|)

1p

≤1and(P

|y_k^q|)¹^q ≤1.

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The inequality (3.4) is a slightly stronger form of (1.1).

Taking other interpretations ofB andB^∗ we give two more examples of the outer inequalities of (3.2) as follows:

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1−

Z

E

uv

≥

1− Z

E

|u|^p _p¹

1− Z

E

|v|^q ¹_q

, provided that R

E|u|^p_p¹

≤1and R

E|v|^q¹_q

≤1.The integrals are Lebesgue integrals andE is a bounded measurable subset of the real numbers.

(5) When we takeB ≡ C[0,1]andB^∗ ≡BV[0,1]in (3.2) we get the some- what exotic result:

1−

Z 1

0

h(t)dα(t)

≥[1−(Max|h|)^p]¹^p[1−(Var(α))^q]¹^q, where the maximum and total variation are taken over[0,1] and each is less than or equal to1.

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4. Inequalities for Grammians

LetΓ(x,y),Γ(x)andΓ(y)denote the determinants of sizenwhose(i, j)th el- ements are respectively the inner products(xi,yj),(xi,xj)and(yi,yj)where thex’s andy’s are vectors in a Hilbert space. Then it is known, see [1, p.599], that

(4.1) [Γ(x,y)]² ≤Γ(x)Γ(y).

It is also well-known that the two factors on the right side of (3.3) are each non-negative so that we have

(4.2) |Γ(x,y)|^2α ≤[Γ(x)]^α[Γ(y)]^α if α >0.

If we read this asz ≤xyand we takef(t) = 1−tagain then (2.1) gives (1− |Γ(x,y)|^2α)≥[1−(Γ(x))^pα]¹^p[1−(Γ(y))^qα]¹^q, provided thatΓ(x)≤1andΓ(y)≤1.

Whenp= q = 2andα = ¹₂ this is a result due to J. Peˇcari´c, [4], and when p = q = 2and α = ¹₄ we get a result due to S.S. Dragomir and B. Mond, [1, Theorem 2].

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5. Some Final Remarks

In giving examples of the use of (2.1) we have used the functionf(t) = 1−t since that is the source of Popoviciu’s result. But interesting inequalities arise also from other suitable choices off. For example, taking

f(t) = (α−t)

(β−t) in [0,1] (1< α < β) we are led to the result

α− |P x_ky_k| β− |P

x_ky_k|

≥

α−P

|x^p_k| β−P

|x^p_k| ¹_p

α−P

|y^q_k| β−P

|y_k^q| ¹_q

provided that (P|x^p_k|)

1p

≤ 1, (P|y_k^q|)¹^q ≤ 1. This reduces to Popoviciu’s inequality (3.2) if we multiply throughout byβ,letβ→ ∞andα→1.

Next letf possess the same properties as in Lemma2.1above but now take x, y, z, w ∈[0,1]withw≤xyz. Then one finds that

f(w)≥f(xyz)≥[f(x^p)]¹^p[f(y^q)]¹^q[f(z^r)]¹^r, where

1 p+ 1

q + 1

r = 1 (p, q, r <1).

Specializing this by again taking f(t) = 1−tand reading the extended Hölder inequality

Xx_ky_kz_k

≤hX

|x_k|^pi¹_phX

|y_k|^qi¹_q hX

|z_k|^ri¹_r

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asw≤xyzwe get the Popoviciu-type inequality:

1−

Xx_ky_kz_k

≥

1−X

|x^p_k|¹_p

1−X

|y^q_k|¹_q

1−X

|z_k^r|¹_r , provided(P|x^p_k|)¹^p ≤1,(P|y_k^q|)¹^q ≤1,(P|z_k^r|)¹^r ≤1.

One can also construct inequalities which involve products of four or more factors, in the same way.

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References

[1] S.S. DRAGOMIR AND B. MOND, Some inequalities of Aczèl type for Grammians in inner product spaces, RGMIA. Res. Rep. Coll., 2(2) (1999), Article 10. [ONLINE:http://rgmia.vu.edu.au/v2n2.html] [2] D.S. MITRINOVI ´C, Analytic Inequalities,(1970), Springer-Verlag.

[3] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C ANDA.M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.

[4] J.E. PE ˇCARI ´C, On some classical inequalities in unitary spaces, Mat. Bil- ten. (Skopje), 16 (1992), 63–72.