PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 85(99) (2009), 107–110 DOI:10.2298/PIM0999107S

ON A NEW CONVERSE OF JENSEN’S INEQUALITY

Slavko Simić

Communicated by Stevan Pilipović

Abstract. We give another global upper bound for Jensen’s discrete inequality which is better than already existing ones. For instance, we determine a new converses for generalized A–G and G–H inequalities.

1. Introduction

Throughout this paper x̃ = {x_{i}} represents a finite sequence of real numbers belonging to a fixed closed interval I = [a, b], a < b and p̃ = {p_{i}}, ∑ p_{i} = 1 is a sequence of positive weights associated with x̃.

If f is a convex function on I, then the well-known Jensen’s inequality [1], [4] asserts that:

0 ⩽ ∑ p_{i}f(x_{i}) − f( ∑ p_{i}x_{i}) .

There are many important inequalities which are particular cases of Jensen’s inequality such as the weighted A–G–H inequality, Cauchy’s inequality, the Ky Fan and Hölder inequalities, etc.

One can see that the lower bound zero is of global nature since it does not depend on p̃ and x̃, but only on f and the interval I, whereupon f is convex.

An upper global bound (i.e., depending in f and I only) for Jensen’s inequality was given by Dragomir [3],

Theorem 1. If f is a differentiable convex mapping on I, then we have

(1)  ∑ p_{i}f(x_{i}) − f( ∑ p_{i}x_{i}) ⩽ \frac{1}{4}(b − a)(f′(b) − f′(a)) := D_{f} (a, b).

There is a number of papers where inequality (1) is utilized in applications concerning some parts of Analysis, Numerical Analysis, Information Theory etc (cf. [1], [2], [3]).

2000 Mathematics Subject Classification: Primary 26B25, 26D20.

Key words and phrases: Jensen’s discrete inequality; global bounds; generalized A-G-H inequality.

In [5] we obtained an upper global bound without differentiability restriction on f. Namely, we proved the following

Theorem 2. If p̃, x̃ are defined as above, we have that

(2)  ∑ p_{i}f(x_{i}) − f( ∑ p_{i}x_{i}) ⩽ f(a) + f(b) − 2f( \frac{a + b}{2} ) := S_{f} (a, b),

for any f that is convex over I := [a, b].

In many cases the bound S_{f} (a, b) is better than D_{f} (a, b). For instance, for f(x) = −x^{s}, 0 < s < 1; f(x) = x^{s}, s > 1; I ⊂ R^{+} , we have that S_{f} (a, b) ⩽ D_{f} (a, b), for each s ∈ (0, 1) ∪ (1, 2) ∪ (3, +∞).

In this article we establish another global bound T_{f} (a, b) for Jensen’s inequality, which is better than both of aforementioned bounds D_{f} (a, b) and S_{f} (a, b).

Finally, we determine T_{f} (a, b) in the case of a generalized A–G inequality as a combination of some well-known classical means. As a consequence, new global upper bounds for A–G and G–H inequalities are established.

2. Results

Our main result is contained in the following

Theorem 3. Let f, p̃, x̃ be defined as above and p, q > 0, p + q = 1. Then

(3) ∑ p_{i}f(x_{i}) − f( ∑ p_{i}x_{i}) ⩽ max_{p}[pf(a) + qf(b) − f(pa + qb)] := T_{f} (a, b).

For example,

T_{x^{2}} (a, b) = max_{p}(pa^{2} + qb^{2} − (pa + qb)^{2}) = max_{p}(pq(b − a)^{2}) = \frac{1}{4}(b − a)^{2},

and we obtain at once the well-known pre-Grüss inequality

∑ p_{i}x_{i}^{2} − ( ∑ p_{i}x_{i})^{2} ⩽ \frac{1}{4}(b − a)^{2},

with 1/4 as the best possible constant.

Remark 2.1. It is easy to see that g(p) := pf(a)+(1−p)f(b)−f(pa+(1−p)b) is

concave for 0 ⩽ p ⩽ 1 with g(0) = g(1) = 0. Hence, there exists the unique positive max_{p} g(p) = T_{f} (a, b).

The next theorem approves that inequality (3) is stronger than (1) or (2).

Theorem 4. Let Ĩ be the domain of a convex function f and I := [a, b] ⊂ Ĩ. Then

(i) T_{f} (a, b) ⩽ D_{f} (a, b); (ii) T_{f} (a, b) ⩽ S_{f} (a, b),

for each I ⊂ Ĩ.

The following well known A-G inequality [4], asserts that A(p̃, x̃) ⩾ G(p̃, x̃), where A(p̃, x̃) := ∑ p_{i}x_{i}, G(p̃, x̃) := ∏ x_{i}^{p_{i}} , are the generalized arithmetic and geometric means, respectively.

Applying theorems 1 and 2 with f(x) = − logx, one obtains the following converses of A–G inequality:

1 ⩽ \frac{A(p̃, x̃)}{G(p̃, x̃)} ⩽ exp( \frac{(b − a)^{2}}{4ab} ) (cf. [2]); (4)

1 ⩽ \frac{A(p̃, x̃)}{G(p̃, x̃)} ⩽ \frac{(a + b)^{2}}{4ab} . (5)

Since 1 + x ⩽ e^{x}, x ∈ R, putting x = \frac{(b−a)^{2}}{4ab} , one can see that the inequality (5) is stronger than (4) for each a, b ∈ R^{+} .

An even stronger converse of A–G inequality can be obtained applying Theorem C.

Theorem 5. Let p̃, x̃, A(p̃, x̃), G(p̃, x̃) be defined as above and x_{i} ∈ [a, b], 0 < a < b. We have,

(6)  1 ⩽ \frac{A(p̃, x̃)}{G(p̃, x̃)} ⩽ \frac{L(a, b)I(a, b)}{G^{2}(a, b)} := T (a, b),

where G(a, b) := \sqrt{ab}, L(a, b) := \frac{b − a}{logb − loga} , I(a, b) := \frac{1}{e}( \frac{b^{b}}{a^{a}}) ^{1/(b−a)} , are the

geometric, logarithmic and identric means, respectively.

Since, for a ≠ b, G(a, b) < L(a, b) < I(a, b) < A(a, b), one can see that the inequality (6) is stronger than (5).

In addition to the above we quote a converse for G–H inequality, where H = H(p̃, x̃) := (∑ p_{i}/x_{i})^{−1} , denotes the generalized harmonic mean.

Theorem 6. With the notation of Theorem E, we have

1 ⩽ \frac{G(p̃, x̃)}{H(p̃, x̃)} ⩽ T (a, b).

3. Proofs

Proof of Theorem 3. Since x_{i} ∈ [a, b], there is a sequence {λ_{i}}, λ_{i} ∈ [0, 1], such that x_{i} = λ_{i}a + (1 − λ_{i})b, i = 1, 2, . . .. Hence,

∑ p_{i}f(x_{i}) − f( ∑ p_{i}x_{i}) = ∑ p_{i}f(λ_{i}a + (1 − λ_{i})b) − f( ∑ p_{i}(λ_{i}a + (1 − λ_{i})b))
⩽ ∑ p_{i}(λ_{i}f(a) + (1 − λ_{i})f(b)) − f( a ∑ p_{i}λ_{i} + b ∑ p_{i}(1 − λ_{i}))
= f(a)( ∑ p_{i}λ_{i}) + f(b)( 1 − ∑ p_{i}λ_{i}) − f( a( ∑ p_{i}λ_{i}) + b( 1 − ∑ p_{i}λ_{i})) .

Denoting ∑ p_{i}λ_{i} := p, 1 − ∑ p_{i}λ_{i} := q, we have that 0 ⩽ p, q ⩽ 1, p + q = 1. Consequently,

∑ p_{i}f(x_{i}) − f( ∑ p_{i}x_{i}) ⩽ pf(a) + qf(b) − f(pa + qb)

⩽ max_{p}[pf(a) + qf(b) − f(pa + qb)] := T_{f} (a, b),

and the proof of Theorem 3 is done. □

Proof of Theorem 4. Applying Theorems 1 and 2 with #x̃ = 2, for each p ∈ [0, 1] and a, b ∈ Ĩ we get

(i) pf(a) + qf(b) − f(pa + qb) ⩽ D_{f} (a, b),
(ii) pf(a) + qf(b) − f(pa + qb) ⩽ S_{f} (a, b).

Since the right-hand sides of (i) and (ii) do not depend on p, we get at once

T_{f} (a, b)= max_{p}[pf(a) + qf(b) − f(pa + qb)] ⩽ D_{f} (a, b);
T_{f} (a, b)= max_{p}[pf(a) + qf(b) − f(pa + qb)] ⩽ S_{f} (a, b).  □

Proof of Theorem 5. By Theorem 3, applied with f(x) = − logx, we obtain

0 ⩽ log \frac{A(p̃, x̃)}{G(p̃, x̃)} ⩽ T_{− log x} (a, b) = max_{p}[log(pa + qb) − p loga − q logb].

By standard argument it is easy to find that the unique maximum is attained at

p = \frac{b}{b − a} − \frac{1}{logb − loga} .

Since 0 < a < b, we get 0 < p < 1 and, after some calculation, it follows that

0 ⩽ log \frac{A(p̃, x̃)}{G(p̃, x̃)} ⩽ log( \frac{b − a}{logb − loga} ) − log(ab) + \frac{b logb − a loga}{b − a} − 1,

Exponentiating, we obtain the assertion from Theorem 5. □

Proof of Theorem 6. By change of variable x_{i} → \frac{1}{x_{i}} , the proof easily follows from (6) since then

A(p̃, x̃) → \frac{1}{H(p̃, x̃)} ; G(p̃, x̃) → \frac{1}{G(p̃, x̃)},

and the right-hand side of (6) stays unaltered under the transformation. □

References

[1] P. R. Beesack, J. Pečarić, On Jensen’s inequality for convex functions, J. Math. Anal. Appl. 110 (1985), 536–552.

[2] I. Budimir, S. S. Dragomir, J. Pečarić, Further reverse results for Jensen’s discrete inequality and applications in Information Theory, J. Inequal. Pure Appl. Math. 2(1) (2001), Art. 5.

[3] S. S. Dragomir, A converse result for Jensen’s discrete inequality via Grüss inequality and applications in Information Theory, An. Univ. Oradea, Fasc. Mat. 7 (1999–2000), 178–189.

[4] D. S. Mitrinović, Analytic Inequalities, Springer-Verlag, New York, 1970.

[5] S. Simić, On a global upper bound for Jensen’s inequality, J. Math. Anal. Appl. 343 (2008), 414–419.

[6] G. H. Hardy, J. E. Littlewood, G. Pölya, Inequalities, Cambridge University Press, Cambridge, 1978.

Matematički institut SANU (Received 31 12 2007) Kneza Mihaila 36 Beograd, Serbia

    ssimic@mi.sanu.ac.rs