ON AN UPPER BOUND FOR JENSEN’S
INEQUALITY
On An Upper Bound
SLAVKO SIMIC
Mathematical Institute SANU, Kneza Mihaila 36
11000 Belgrade, Serbia
EMail: ssimic@turing.mi.sanu.ac.rs
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
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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 generalized A − G inequality.
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Contents
1
Introduction
3
2
Results
5
3
Proofs
8
On An Upper Bound
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
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1.
Introduction
Throughout this paper, x̃ = {xi } is a finite sequence of P
real numbers belonging to
a fixed closed interval I = [a, b], a < b, and p̃ = {pi },
pi = 1 is a sequence of
positive weights associated with x̃. If f is a convex function on I, then the wellknown Jensen’s inequality [1, 4] asserts that:
X
X
(1.1)
0≤
pi f (xi ) − f
pi xi .
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 on f and I only) for Jensen’s inequality
is given by Dragomir [3].
On An Upper Bound
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
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Theorem 1.1. If f is a differentiable convex mapping on I, then we have
X
1
X
(1.2)
pi f (xi ) − f
pi xi ≤ (b − a)(f 0 (b) − f 0 (a)) := Df (a, b).
4
In [5] we obtain an upper global bound without a differentiability restriction on
f . Namely, we proved the following:
Theorem 1.2. If p̃, x̃ are defined as above, we have
X
X
a+b
(1.3)
pi f (xi ) − f
:= Sf (a, b),
pi xi ≤ f (a) + f (b) − 2f
2
for any f that is convex over I := [a, b].
In many cases the bound Sf (a, b) is better than Df (a, b).
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For instance, for f (x) = −xs , 0 < s < 1; f (x) = xs , s > 1; I ⊂ R+ , we have
that
(1.4)
Sf (a, b) ≤ Df (a, b),
S
S
for each s ∈ (0, 1) (1, 2] [3, +∞).
In this article we establish another global bound Tf (a, b) for Jensen’s inequality,
which is better than both of the aforementioned bounds Df (a, b) and Sf (a, b).
Finally, we determine Tf (a, b) in the case of the generalized A − G inequality.
On An Upper Bound
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
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2.
Results
Our main result is contained in the following
Theorem 2.1. Let f, p̃, x̃ be defined as above and p, q > 0, p + q = 1. Then
X
X
(2.1)
pi f (xi ) − f
pi xi ≤ max[pf (a) + qf (b) − f (pa + qb)]
p
On An Upper Bound
:= Tf (a, b).
Slavko Simic
Remark 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 a unique positive
maxp g(p) = Tf (a, b).
vol. 10, iss. 2, art. 60, 2009
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The next theorem demonstrates that the inequality (2.1) is stronger than (1.2) or
(1.3).
˜ Then
Theorem 2.2. Let I˜ be the domain of a convex function f and I := [a, b] ⊂ I.
I. Tf (a, b) ≤ Df (a, b);
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II. Tf (a, b) ≤ Sf (a, b),
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˜
for each I ⊂ I.
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The following well known A − G inequality [4] asserts that
A(p̃, x̃) ≥ G(p̃, x̃),
(2.2)
where
(2.3)
A(p̃, x̃) :=
X
pi xi ;
G(p̃, x̃) :=
Y
xpi i ,
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are generalized arithmetic, i.e., geometric means, respectively.
Applying Theorems 2.1 (cf [2]) and 2.2 with f (x) = − log x, one obtains the
following converses of the A − G inequality:
A(p̃, x̃)
(b − a)2
(2.4)
1≤
≤ exp
G(p̃, x̃)
4ab
and
(2.5)
A(p̃, x̃)
(a + b)2
1≤
≤
.
G(p̃, x̃)
4ab
On An Upper Bound
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
2
Since 1 + x ≤ ex , x ∈ R, putting x = (b−a)
, one can see that the inequality (2.5) is
4ab
stronger than (2.4) for each a, b ∈ R+ .
An even stronger converse of the A − G inequality can be obtained by applying
Theorem 2.1.
Theorem 2.3. Let p̃, x̃, A(p̃, x̃), G(p̃, x̃) be defined as above and xi ∈ [a, b], 0 <
a < b.
Denote u := log(b/a); w := (eu − 1)/u. Then
(2.6)
w
1
A(p̃, x̃)
1≤
≤ exp := T (w).
G(p̃, x̃)
e
w
Comparing the bounds D, S and T, i.e., (2.4), (2.5) and (2.6) for arbitrary p̃ and
xi ∈ [a, 2a], a > 0, we obtain
(2.7)
A(p̃, x̃)
1≤
≤ e1/8 ≈ 1.1331,
G(p̃, x̃)
(2.8)
1≤
A(p̃, x̃)
≤ 9/8 = 1.125,
G(p̃, x̃)
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and
(2.9)
1≤
A(p̃, x̃)
≤ 2(e log 2)−1 ≈ 1.0615
G(p̃, x̃)
respectively.
Remark 2. One can see that T (w) = S(t), where Specht’s ratio S(t) is defined as
(2.10)
t1/(t−1)
S(t) :=
e log t1/(t−1)
with t = b/a.
It is known [6, 7] that S(t) represents the best possible upper bound for the A − G
inequality with uniform weights, i.e.
1
1
x1 + x2 + · · · + xn n
n
≥ (x1 x2 · · · xn ) .
(2.11)
S(t)(x1 x2 · · · xn ) ≥
n
Therefore Theorem 2.3 shows that Specht’s ratio is the best upper bound for the
generalized A − G inequality also.
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Slavko Simic
vol. 10, iss. 2, art. 60, 2009
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3.
Proofs
Proof of Theorem 2.1. Since xi ∈ [a, b], there is a sequence {λi }, λi ∈ [0, 1], such
that xi = λi a + (1 − λi )b.
Hence
X
X
pi f (xi ) − f
pi xi
X
X
=
pi f (λi a + (1 − λi )b) − f
pi (λi a + (1 − λi )b)
X
X
X
≤
pi (λi f (a) + (1 − λi )f (b)) − f (a
p i λi + b
pi (1 − λi )
X
h X
i
X
X
= f (a)
pi λi + f (b) 1 −
p i λi − f a
p i λi + b 1 −
p i λi .
P
P
Denoting pi λi := p; 1 − pi λi := q, we have that 0 ≤ p, q ≤ 1, p + q = 1.
Consequently,
X
X
pi f (xi ) − f
pi xi ≤ pf (a) + qf (b) − f (pa + qb)
≤ max[pf (a) + qf (b) − f (pa + qb)]
On An Upper Bound
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
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p
:= Tf (a, b),
and the proof of Theorem 2.1 is complete.
Proof of Theorem 2.2.
Part I.
Since f is convex (and differentiable, in this case), we have
(3.1)
∀x, t ∈ I : f (x) ≥ f (t) + (x − t)f 0 (t).
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In particular,
(3.2) f (pa + qb) ≥ f (a) + q(b − a)f 0 (a); f (pa + qb) ≥ f (b) + p(a − b)f 0 (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)f 0 (a)) + q(p(b − a)f 0 (b))
= pq(b − a)(f 0 (b) − f 0 (a)).
On An Upper Bound
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
Hence
Tf (a, b) := max[pf (a) + qf (b) − f (pa + qb)]
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p
≤ max[pq(b − a)(f 0 (b) − f 0 (a))]
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p
1
= (b − a)(f 0 (b) − f 0 (a))
4
:= Df (a, b).
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Part II.
We shall prove that, for each 0 ≤ p, q, p + q = 1,
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(3.3)
pf (a) + qf (b) − f (pa + qb) ≤ f (a) + f (b) − 2f
a+b
2
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.
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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1
1
≤ f (a) + f (b) − 2f
(qa + pb) + (pa + qb)
2
2
a+b
= f (a) + f (b) − 2f
.
2
Since the right-hand side of the above inequality does not depend on p, we immediately get
On An Upper Bound
Tf (a, b) ≤ Sf (a, b).
(3.4)
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
Proof of Theorem 2.3. By Theorem 2.1, applied with f (x) = − log x, we obtain
A(p̃, x̃)
0 ≤ log
G(p̃, x̃)
≤ T− log x (a, b)
= max[log(pa + qb) − p log a − q log b].
p
Using standard arguments it is easy to find that the unique maximum is attained at
the point p:
b
1
(3.5)
p=
−
.
b − a log b − log a
Since 0 < a < b, we get 0 < p < 1 and after some calculations, it follows that
b−a
a log b − b log a
A(p̃, x̃)
(3.6)
0 ≤ log
≤ log
+
− 1.
G(p̃, x̃)
log b − log a
b−a
Putting log(b/a) := u, (eu − 1)/u := w and taking the exponent, one obtains the
result of Theorem 2.3.
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References
[1] P.R. BEESACK AND J. PEČARIĆ, On Jensen’s inequality for convex functions,
J. Math. Anal. Appl., 110 (1985), 536–552.
[2] I. BUDIMIR, S.S. DRAGOMIR AND 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. [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.
On An Upper Bound
Slavko Simic
vol. 10, iss. 2, art. 60, 2009
Title Page
[4] D.S. MITRINOVIĆ, Analytic Inequalities, Springer, New York, 1970.
[5] S. SIMIĆ, 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.
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