ON AN UPPER BOUND FOR JENSEN’S INEQUALITY

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On An Upper Bound Slavko Simic vol. 10, iss. 2, art. 60, 2009

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ON AN UPPER BOUND FOR JENSEN’S INEQUALITY

SLAVKO SIMIC

Mathematical Institute SANU, Kneza Mihaila 36 11000 Belgrade, Serbia

EMail:[email protected]

Received: 25 May, 2007

Accepted: 16 November, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26B25.

Key words: Jensen’s discrete inequality, global bounds, generalized A-G inequality.

Abstract: In this paper we shall give another global upper bound for Jensen’s discrete inequality which is better than existing ones. For instance, we determine a new converse for the generalizedA−Ginequality.

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1. Introduction

Throughout this paper,x˜ = {x_i} is a finite sequence of real numbers belonging to a fixed closed intervalI = [a, b], a < b, andp˜= {p_i},P

p_i = 1is a sequence of positive weights associated withx. If˜ f is a convex function on I, then the well- known Jensen’s inequality [1,4] asserts that:

(1.1) 0≤X

pif(xi)−fX pixi

.

One can see that the lower bound zero is of global nature since it does not depend on

˜

pandx˜but only onf and the intervalI, whereuponf is convex.

An upper global bound (i.e. depending on f andI only) for Jensen’s inequality is given by Dragomir [3].

Theorem 1.1. Iff is a differentiable convex mapping onI, then we have

(1.2) X

p_if(x_i)−fX p_ix_i

≤ 1

4(b−a)(f⁰(b)−f⁰(a)) :=D_f(a, b).

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

Theorem 1.2. Ifp,˜ x˜are defined as above, we have

(1.3) X

p_if(x_i)−fX p_ix_i

≤f(a) +f(b)−2f

a+b 2

:=S_f(a, b), for anyf that is convex overI := [a, b].

In many cases the boundS_f(a, b)is better thanD_f(a, b).

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For instance, forf(x) = −x^s,0 < s < 1; f(x) = x^s, s > 1;I ⊂ R⁺, we have that

(1.4) S_f(a, b)≤D_f(a, b),

for eachs∈(0,1)S

(1,2]S

[3,+∞).

In this article we establish another global boundT_f(a, b)for Jensen’s inequality, which is better than both of the aforementioned boundsD_f(a, b)andS_f(a, b).

Finally, we determineT_f(a, b)in the case of the generalizedA−Ginequality.

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2. Results

Our main result is contained in the following

Theorem 2.1. Letf, p,˜ x˜be defined as above andp, q >0, p+q = 1. Then Xp_if(x_i)−fX

p_ix_i

≤max

p [pf(a) +qf(b)−f(pa+qb)]

(2.1)

:=T_f(a, b).

Remark 1. It is easy to see thatg(p) :=pf(a) + (1−p)f(b)−f(pa+ (1−p)b)is concave for0 ≤p≤ 1withg(0) =g(1) = 0. Hence, there exists a unique positive max_pg(p) = T_f(a, b).

The next theorem demonstrates that the inequality(2.1)is stronger than(1.2)or (1.3).

Theorem 2.2. LetI˜be the domain of a convex functionf andI := [a, b]⊂I. Then˜ I. T_f(a, b)≤D_f(a, b);

II. T_f(a, b)≤S_f(a, b), for eachI ⊂I˜.

The following well knownA−Ginequality [4] asserts that

(2.2) A(˜p,x)˜ ≥G(˜p,x),˜

where

(2.3) A(˜p,x) :=˜ X

p_ix_i; G(˜p,x) :=˜ Y x^p_iⁱ,

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are generalized arithmetic, i.e., geometric means, respectively.

Applying Theorems 2.1 (cf [2]) and 2.2 with f(x) = −logx, one obtains the following converses of theA−Ginequality:

(2.4) 1≤ A(˜p,x)˜

G(˜p,x)˜ ≤exp

(b−a)² 4ab

and

(2.5) 1≤ A(˜p,x)˜

G(˜p,x)˜ ≤ (a+b)² 4ab .

Since1 +x≤e^x, x∈R, puttingx= ^(b−a)_4ab², one can see that the inequality (2.5) is stronger than (2.4) for eacha, b∈R⁺.

An even stronger converse of theA−Ginequality can be obtained by applying Theorem2.1.

Theorem 2.3. Letp,˜ x, A(˜˜ p,x), G(˜˜ p,x)˜ be defined as above and x_i ∈ [a, b], 0 <

a < b.

Denoteu:= log(b/a); w:= (e^u−1)/u. Then

(2.6) 1≤ A(˜p,x)˜

G(˜p,x)˜ ≤ w e exp 1

w :=T(w).

Comparing the boundsD, S andT,i.e.,(2.4),(2.5)and(2.6)for arbitraryp˜and x_i ∈[a,2a], a >0, we obtain

(2.7) 1≤ A(˜p,x)˜

G(˜p,x)˜ ≤e^1/8 ≈1.1331,

(2.8) 1≤ A(˜p,x)˜

G(˜p,x)˜ ≤9/8 = 1.125,

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and

(2.9) 1≤ A(˜p,x)˜

G(˜p,x)˜ ≤2(elog 2)⁻¹ ≈1.0615 respectively.

Remark 2. One can see thatT(w) = S(t), where Specht’s ratioS(t)is defined as

(2.10) S(t) := t^1/(t−1)

elogt^1/(t−1) witht=b/a.

It is known [6,7] thatS(t)represents the best possible upper bound for theA−G inequality with uniform weights, i.e.

(2.11) S(t)(x₁x₂· · ·x_n)ⁿ¹ ≥ x1+x2+· · ·+xn

n

≥(x₁x₂· · ·x_n)¹ⁿ . Therefore Theorem 2.3 shows that Specht’s ratio is the best upper bound for the generalizedA−Ginequality also.

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3. Proofs

Proof of Theorem2.1. Sincex_i ∈ [a, b], there is a sequence{λ_i}, λ_i ∈ [0,1], such thatx_i =λ_ia+ (1−λ_i)b.

Hence

Xpif(xi)−fX pixi

=X

p_if(λ_ia+ (1−λ_i)b)−fX

p_i(λ_ia+ (1−λ_i)b)

≤X

p_i(λ_if(a) + (1−λ_i)f(b))−f(aX

p_iλ_i+bX

p_i(1−λ_i)

=f(a)X p_iλ_i

+f(b)

1−X p_iλ_i

−fh

aX

p_iλ_i +b

1−X

p_iλ_ii . DenotingP

p_iλ_i :=p; 1−P

p_iλ_i :=q, we have that0≤p, q ≤1, p+q= 1.

Consequently,

Xp_if(x_i)−fX p_ix_i

≤pf(a) +qf(b)−f(pa+qb)

≤max

:=T_f(a, b), and the proof of Theorem2.1is complete.

Proof of Theorem2.2.

Part I.

Sincef is convex (and differentiable, in this case), we have (3.1) ∀x, t∈I : f(x)≥f(t) + (x−t)f⁰(t).

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In particular,

(3.2) f(pa+qb)≥f(a) +q(b−a)f⁰(a); f(pa+qb)≥f(b) +p(a−b)f⁰(b).

Therefore

pf(a) +qf(b)−f(pa+qb) = p(f(a)−f(pa+qb)) +q(f(b)−f(pa+qb))

≤p(q(a−b)f⁰(a)) +q(p(b−a)f⁰(b))

=pq(b−a)(f⁰(b)−f⁰(a)).

Hence

Tf(a, b) := max

≤max

p [pq(b−a)(f⁰(b)−f⁰(a))]

= 1

4(b−a)(f⁰(b)−f⁰(a)) :=D_f(a, b).

Part II.

We shall prove that, for each0≤p, q, p+q= 1,

(3.3) pf(a) +qf(b)−f(pa+qb)≤f(a) +f(b)−2f

a+b 2

.

Indeed,

pf(a) +qf(b)−f(pa+qb) = f(a) +f(b)−(qf(a) +pf(b))−f(pa+qb)

≤f(a) +f(b)−(f(qa+pb) +f(pa+qb))

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≤f(a) +f(b)−2f 1

2(qa+pb) + 1

2(pa+qb)

=f(a) +f(b)−2f

a+b 2

.

Since the right-hand side of the above inequality does not depend onp, we immedi- ately get

(3.4) T_f(a, b)≤S_f(a, b).

Proof of Theorem2.3. By Theorem2.1, applied withf(x) =−logx, we obtain 0≤logA(˜p,x)˜

G(˜p,x)˜

≤T−logx(a, b)

= max

p [log(pa+qb)−ploga−qlogb].

Using standard arguments it is easy to find that the unique maximum is attained at the pointp:

(3.5) p= b

b−a − 1

logb−loga.

Since0< a < b, we get0< p <1and after some calculations, it follows that (3.6) 0≤log A(˜p,x)˜

G(˜p,x)˜ ≤log

b−a logb−loga

+ alogb−bloga b−a −1.

Puttinglog(b/a) := u, (e^u −1)/u := w and taking the exponent, one obtains the result of Theorem2.3.

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References

[1] P.R. BEESACKANDJ. PE ˇCARI ´C, On Jensen’s inequality for convex functions, J. Math. Anal. Appl., 110 (1985), 536–552.

[2] I. BUDIMIR, S.S. DRAGOMIR AND J. PE ˇCARI ´C, Further reverse results for Jensen’s discrete inequality and applications in information theory, J. Inequal.

Pure Appl. Math., 2(1) (2001), Art. 5. [ONLINE:http://jipam.vu.edu.

au/article.php?sid=121]

[3] S.S. DRAGOMIR, A converse result for Jensen’s discrete inequality via Gruss inequality and applications in information theory, Analele Univ. Oradea. Fasc.

Math., 7 (1999-2000), 178–189.

[4] D.S. MITRINOVI ´C, Analytic Inequalities, Springer, New York, 1970.

[5] S. SIMI ´C, Jensen’s inequality and new entropy bounds, submitted to Appl. Math.

Letters.

[6] W. SPECHT, Zur Theorie der elementaren Mittel, Math. Z., 74 (1960), 91–98.

[7] M. TOMINAGA, Specht’s ratio in the Young inequality, Sci. Math. Japon., 55 (2002), 583–588.