Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 1, Article 10, 2006
A VARIANT OF JESSEN’S INEQUALITY AND GENERALIZED MEANS
W.S. CHEUNG∗ , A. MATKOVIĆ, AND J. PEČARIĆ
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF H ONG KONG
P OKFULAM ROAD
H ONG KONG
wscheung@hku.hk
D EPARTMENT OF M ATHEMATICS
FACULTY OF NATURAL S CIENCES , M ATHEMATICS AND E DUCATION
U NIVERSITY OF S PLIT
T ESLINA 12, 21000 S PLIT
C ROATIA
anita@pmfst.hr
FACULTY OF T EXTILE T ECHNOLOGY
U NIVERSITY OF Z AGREB
P IEROTTIJEVA 6, 10000 Z AGREB
C ROATIA
pecaric@hazu.hr
Received 26 September, 2005; accepted 08 November, 2005
Communicated by I. Gavrea
A BSTRACT. In this paper we give a variant of Jessen’s inequality for isotonic linear functionals.
Our results generalize some recent results of Gavrea. We also give comparison theorems for
generalized means.
Key words and phrases: Isotonic linear functionals, Jessen’s inequality, Generalized means.
2000 Mathematics Subject Classification. 26D15, 39B62.
1. I NTRODUCTION
Let E be a nonempty set and L be a linear class of real valued functions f : E → R having
the properties:
L1: f, g ∈ L ⇒ (αf + βg) ∈ L for all α, β ∈ R;
L2: 1 ∈ L, i.e., if f (t) = 1 for t ∈ E, then f ∈ L.
An isotonic linear functional is a functional A : L → R having properties:
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
∗ Corresponding author. Research is supported in part by the Research Grants Council of the Hong Kong SAR (Project No. HKU7017/05P).
The authors would like to thank the referee for his invaluable comments and insightful suggestions.
290-05
2
W.S. C HEUNG , A. M ATKOVI Ć , AND J. P E ČARI Ć
A1: A (αf + βg) = αA(f ) + βA(g) for f, g ∈ L, α, β ∈ R (A is linear);
A2: f ∈ L, f (t) ≥ 0 on E ⇒ A(f ) ≥ 0 (A is isotonic).
The following result is Jessen’s generalization of the well known Jensen’s inequality for
convex functions [3] (see also [5, p. 47]):
Theorem 1.1. Let L satisfy properties L1, L2 on a nonempty set E, and let ϕ be a continuous
convex function on an interval I ⊂ R. If A is an isotonic linear functional on L with A(1) = 1,
then for all g ∈ L such that ϕ (g) ∈ L we have A(g) ∈ I and
ϕ(A (g)) ≤ A(ϕ (g)).
Similar to Jensen’s inequality, Jessen’s inequality has a converse [1] (see also [5, p. 98]):
Theorem 1.2. Let L satisfy properties L1, L2 on a nonempty set E, and let ϕ be a convex
function on an interval I = [m, M ] (−∞ < m < M < ∞). If A is an isotonic linear functional
on L with A(1) = 1, then for all g ∈ L such that ϕ (g) ∈ L (so that m ≤ g(t) ≤ M for all
t ∈ E), we have
A (g) − m
M − A (g)
A(ϕ (g)) ≤
· ϕ(m) +
· ϕ(M ).
M −m
M −m
Recently I. Gavrea [2] has obtained the following result which is in connection with Mercer’s
variant of Jensen’s inequality [4]:
Theorem 1.3. Let A be an isotonic linear functional defined on C[a, b] such that A(1) = 1.
Then for any convex function ϕ on [a, b],
ϕ(a + b − a1 ) ≤ A(ψ)
a1 − a
b − a1
− ϕ(b)
b−a
b−a
≤ ϕ(a) + ϕ(b) − A(ϕ),
≤ ϕ(a) + ϕ(b) − ϕ(a)
where ψ(t) = ϕ(a + b − t) and a1 = A(id).
Remark 1.4. Although it is not explicitly stated above, it is obvious that function ϕ needs to be
continuous on [a, b].
In Section 2 we give the main result of this paper which is an extension of Theorem 1.3 on a
linear class L satisfying properties L1, L2. In Section 3 we use that result to prove the monotonicity property of generalized power means. We also consider in the same way generalized
means with respect to isotonic functionals.
2. M AIN R ESULT
Theorem 2.1. Let L satisfy properties L1, L2 on a nonempty set E, and let ϕ be a convex
function on an interval I = [m, M ] (−∞ < m < M < ∞). If A is an isotonic linear functional
on L with A(1) = 1, then for all g ∈ L such that ϕ (g) , ϕ (m + M − g) ∈ L (so that m ≤
g(t) ≤ M for all t ∈ E), we have the following variant of Jessen’s inequality
(2.1)
ϕ (m + M − A (g)) ≤ ϕ (m) + ϕ (M ) − A (ϕ (g)) .
In fact, to be more specific, we have the following series of inequalities
ϕ (m + M − A (g)) ≤ A (ϕ (m + M − g))
(2.2)
J. Inequal. Pure and Appl. Math., 7(1) Art. 10, 2006
M − A (g)
A (g) − m
· ϕ(M ) +
· ϕ(m)
M −m
M −m
≤ ϕ (m) + ϕ (M ) − A (ϕ (g)) .
≤
http://jipam.vu.edu.au/
A VARIANT OF J ESSEN ’ S I NEQUALITY
AND
G ENERALIZED M EANS
3
If the function ϕ is concave, inequalities (2.1) and (2.2) are reversed.
Proof. Since ϕ is continuous and convex, the same is also true for the function
ψ : [m, M ] → R
defined by
ψ(t) = ϕ(m + M − t) ,
t ∈ [m, M ] .
By Theorem 1.1,
ψ(A (g)) ≤ A(ψ (g)),
i.e.,
ϕ (m + M − A (g)) ≤ A (ϕ(m + M − g)) .
Applying Theorem 1.2 to ψ and then to ϕ, we have
A (g) − m
M − A (g)
· ψ (m) +
· ψ (M )
M −m
M −m
A (g) − m
M − A (g)
=
· ϕ (M ) +
· ϕ (m)
M −m
M −m
M − A (g)
A (g) − m
= ϕ (m) + ϕ (M ) −
· ϕ (m) +
· ϕ (M )
M −m
M −m
≤ ϕ (m) + ϕ (M ) − A (ϕ (g)) .
A (ϕ(m + M − g)) ≤
The last statement follows immediately from the facts that if ϕ is concave then −ϕ is convex,
and that A is linear on L.
Remark 2.2. In Theorem 2.1, taking L = C [a, b] and g = id (so that m = a and M = b),
we obtain the results of Theorem 1.3. On the other hand, the results of Theorem 1.3 for the
functional B defined on L by B(ϕ) = A(ϕ(g)), for which B(1) = 1 and B(id) = A(g),
become the results of Theorem 2.1. Hence, these results are equivalent.
Corollary 2.3. Let (Ω, A, µ) be a probability measure space, and let g : Ω → [m, M ]
(−∞ < m < M < ∞) be a measurable function. Then for any continuous convex function
ϕ : [m, M ] → R,
Z
Z
ϕ m + M − gdµ ≤
ϕ (m + M − g) dµ
Ω
Ω
R
R
M − Ω gdµ
gdµ − m
≤
· ϕ (M ) + Ω
· ϕ (m)
M −m
M
−
m
Z
≤ ϕ (m) + ϕ (M ) −
ϕ (g) dµ.
Ω
1
Proof. This
R is a special case of Theorem 2.1 for the functional A defined on class L (µ) as
A(g) = Ω gdµ.
3. S OME A PPLICATIONS
3.1. Generalized Power Means. Throughout this subsection we suppose that:
(i) L is a linear class having properties L1, L2 on a nonempty set E.
(ii) A is an isotonic linear functional on L such that A(1) = 1.
(iii) g ∈ L is a function of E to [m, M ] (−∞ < m < M < ∞) such that all of the following
expressions are well defined.
J. Inequal. Pure and Appl. Math., 7(1) Art. 10, 2006
http://jipam.vu.edu.au/
4
W.S. C HEUNG , A. M ATKOVI Ć , AND J. P E ČARI Ć
From (iii) it follows especially that 0 < m < M < ∞, and we define, for any r, s ∈ R,

1
r
r
r r

[m
+
M
−
A(g
)]
, r 6= 0


Q(r, g) :=
mM
,
r=0,
exp (A(log g))
h i 1s
s

r
r
r r

,
r
A
[m
+
M
−
g
]





1

r
r
r r


exp
A
log
[m
+
M
−
A(g
)]
, r

R(r, s, g) := h s i 1s

mM

A
,
r

g





mM


exp A log
,
r
g



6= 0, s 6= 0
6= 0, s = 0
= 0, s 6= 0
=s=0,
and
h
i1
r
r)

A(g r )−mr
s
s s
 M −A(g
·
M
+
·
m
,

M r −mr
M r −mr



r

r)
r )−mr


exp MM−A(g
· log M + A(g
·
log
m
,
r −mr
r
r
M −m
S(r, s, g) := h
i1


log M −A(log g)
A(log g)−log m
s
s s

·
M
+
·
m
,

log M −log m
log M −log m





exp log M −A(log g) · log M + A(log g)−log m · log m ,
log M −log m
log M −log m
r 6= 0, s 6= 0
r 6= 0, s = 0
r = 0, s 6= 0
r = s = 0.
In [2] Gavrea proved the following result:
“If r, s ∈ R such that r ≤ s, then for every monotone positive function g ∈ C [a, b] ,
e g) ≤ Q(s,
e g),
Q(r,
where
e g) =
Q(r,

1
 [g r (a) + g r (b) − M r (r, g)] r r 6= 0

g(a)g(b)
exp(A(log g))
,
r=0
and M (r, g) is power mean of order r.”
The following is an extension to Gavrea’s result.
Theorem 3.1. If r, s ∈ R and r ≤ s, then
Q(r, g) ≤ Q(s, g).
Furthermore,
(3.1)
Q(r, g) ≤ R (r, s, g) ≤ S (r, s, g) ≤ Q(s, g).
Proof. From above, we know that
0<m≤g≤M <∞.
S TEP 1: Assume 0 < r ≤ s.
In this case, we have
0 < mr ≤ g r ≤ M r < ∞.
Applying Theorem 2.1 or more precisely inequality (2.2) to the continuous convex function
ϕ : (0, ∞) → R
s
ϕ(x) = x r , x ∈ (0, ∞) ,
J. Inequal. Pure and Appl. Math., 7(1) Art. 10, 2006
http://jipam.vu.edu.au/
A VARIANT OF J ESSEN ’ S I NEQUALITY
AND
G ENERALIZED M EANS
5
we have
s
s
[mr + M r − A (g r )] r ≤ A (mr + M r − g r ) r
M r − A (g r )
A (g r ) − mr
s
·
M
+
· ms
M r − mr
M r − mr
≤ ms + M s − A (g s ) .
≤
Since s ≥ r > 0, this gives
h i 1s
1
s
[mr + M r − A(g r )] r ≤ A (mr + M r − g r ) r
r
1s
r
r
M − A (g r )
A
(g
)
−
m
≤
· Ms +
· ms
M r − mr
M r − mr
1
≤ [ms + M s − A (g s )] s ,
or
Q(r, g) ≤ R (r, s, g) ≤ S (r, s, g) ≤ Q(s, g).
S TEP 2: Assume r ≤ s < 0.
In this case we have
0 < M r ≤ g r ≤ mr < ∞.
Applying Theorem 2.1 or more precisely inequality (2.2) to the continuous concave function
(note that 0 < rs ≤ 1 here)
ϕ : (0, ∞) → R
s
ϕ(x) = x r , x ∈ (0, ∞) ,
we have
s
s
[M r + mr − A (g r )] r ≥ A (M r + mr − g r ) r
mr − A (g r )
A (g r ) − M r
s
·
m
+
· Ms
mr − M r
mr − M r
≥ M s + ms − A (g s ) .
≥
Since r ≤ s < 0, this gives
h i 1s
1
s
[mr + M r − A(g r )] r ≤ A (mr + M r − g r ) r
r
1s
M − A (g r )
A (g r ) − mr
s
s
≤
·M +
·m
M r − mr
M r − mr
1
≤ [ms + M s − A (g s )] s ,
or
Q(r, g) ≤ R (r, s, g) ≤ S (r, s, g) ≤ Q(s, g).
S TEP 3: Assume r < 0 < s.
In this case we have
0 < M r ≤ g r ≤ mr < ∞.
Applying Theorem 2.1 or more precisely inequality (2.2) to the continuous convex function
(note that rs < 0 here)
ϕ : (0, ∞) → R
s
ϕ(x) = x r , x ∈ (0, ∞) ,
J. Inequal. Pure and Appl. Math., 7(1) Art. 10, 2006
http://jipam.vu.edu.au/
6
W.S. C HEUNG , A. M ATKOVI Ć , AND J. P E ČARI Ć
we have
s
s
[M r + mr − A (g r )] r ≤ A (M r + mr − g r ) r
mr − A (g r )
A (g r ) − M r
s
·
m
+
· Ms
mr − M r
mr − M r
≤ M s + ms − A (g s ) .
≤
Since r < 0 < s, this gives
h i 1s
s
1
[mr + M r − A(g r )] r ≤ A (mr + M r − g r ) r
r
1s
M − A (g r )
A (g r ) − mr
s
s
≤
·M +
·m
M r − mr
M r − mr
1
≤ [ms + M s − A (g s )] s ,
or
Q(r, g) ≤ R (r, s, g) ≤ S (r, s, g) ≤ Q(s, g).
S TEP 4: Assume r < 0, s = 0.
In this case we have
0 < M r ≤ g r ≤ mr < ∞.
Applying Theorem 2.1 or more precisely inequality (2.2) to the continuous convex function
(0, ∞) → R
ϕ:
ϕ(x) = 1r log x , x ∈ (0, ∞) ,
we have
1
1
r
r
r
r
r
r
log (M + m − A (g )) ≤ A
log (M + m − g )
r
r
mr − A (g r ) 1
A (g r ) − M r 1
r
≤
·
log
m
+
· log M r
r − Mr
mr − M r
r
m
r
1
1
1
≤ log M r + log mr − A
log g r ,
r
r
r
or
log Q(r, g) ≤ log R (r, 0, g) ≤ log S (r, 0, g) ≤ log Q(0, g).
Hence
Q(r, g) ≤ R (r, 0, g) ≤ S (r, 0, g) ≤ Q(0, g).
S TEP 5: Assume r = 0, s > 0.
In this case we have
−∞ < log m ≤ log g ≤ log M < ∞.
Applying Theorem 2.1 or more precisely inequality (2.2) to the continuous convex function
ϕ:
R → (0, ∞)
ϕ(x) = exp (sx) , x ∈ R ,
J. Inequal. Pure and Appl. Math., 7(1) Art. 10, 2006
http://jipam.vu.edu.au/
A VARIANT OF J ESSEN ’ S I NEQUALITY
AND
G ENERALIZED M EANS
7
we have
exp (s (log m + log M − A (log g)))
≤ A (exp (s (log m + log M − log g)))
log M − A (log g)
A (log g) − log m
· exp (s log M ) +
· exp (s log m)
log M − log m
log M − log m
≤ exp (s log m) + exp s (log M ) − A (exp (s log g)) ,
≤
or
Q(0, g)s ≤ R (0, s, g)s ≤ S (0, s, g)s ≤ Q(s, g)s .
Since s > 0, we have
Q(0, g) ≤ R (0, s, g) ≤ S (0, s, g) ≤ Q(s, g).
This completes the proof of the theorem, since when r = s = 0 we have
Q(0, g) = R (0, 0, g) = S (0, 0, g) .
Corollary 3.2. Let (Ω, A, µ) be a probability measure space, and let g :RΩ → [m, M ]
(0 < m < M < ∞) be a measurable function. Let A be defined as A (g) = Ω gdµ. Then
for any continuous convex function ϕ : [m, M ] → R, and any r, s ∈ R with r ≤ s, (3.1) holds.
3.2. Generalized Means. Let L satisfy properties L1, L2 on a nonempty set E, and let A be an
isotonic linear functional on L with A(1) = 1. Let ψ, χ be continuous and strictly monotonic
functions on an interval I = [m, M ] (−∞ < m < M < ∞). Then for any g ∈ L such that
ψ (g) , χ (g) , χ (ψ −1 (ψ (m) + ψ (M ) − ψ (g))) ∈ L (so that m ≤ g(t) ≤ M for all t ∈ E), we
define the generalized mean of g with respect to the functional A and the function ψ by (see for
example [5, p. 107])
Mψ (g, A) = ψ −1 (A (ψ (g))) .
Observe that if ψ (m) ≤ ψ (g) ≤ ψ (M ) for t ∈ E, then by the isotonic character of A, we have
ψ (m) ≤ A (ψ (g)) ≤ ψ (M ), so that Mψ is well defined. We further define
fψ (g, A) = ψ −1 (ψ (m) + ψ (M ) − A (ψ (g))) .
M
From the above observation we know that
ψ (m) ≤ ψ (m) + ψ (M ) − A (ψ (g)) ≤ ψ (M )
fψ is also well defined.
so that M
Theorem 3.3. Under the above hypotheses, we have
(i) if either χ ◦ ψ −1 is convex and χ is strictly increasing, or χ ◦ ψ −1 is concave and χ is
strictly decreasing, then
fψ (g, A) ≤ M
fχ (g, A) .
(3.2)
M
(3.3)
In fact, to be more specific we have the following series of inequalities
fψ (g, A)
M
≤ χ−1 A χ ψ −1 (ψ (m) + ψ (M ) − ψ (g))
ψ (M ) − A (ψ (g))
A (ψ (g)) − ψ (m)
−1
· χ (M ) +
· χ (m)
≤χ
ψ (M ) − ψ (m)
ψ (M ) − ψ (m)
fχ (g, A) ;
≤M
J. Inequal. Pure and Appl. Math., 7(1) Art. 10, 2006
http://jipam.vu.edu.au/
8
W.S. C HEUNG , A. M ATKOVI Ć , AND J. P E ČARI Ć
(ii) if either χ ◦ ψ −1 is concave and χ is strictly increasing, or χ ◦ ψ −1 is convex and χ is
strictly decreasing, then the reverse inequalities hold.
Proof. Since ψ is strictly monotonic and −∞ < m ≤ g(t) ≤ M < ∞, we have −∞ <
ψ (m) ≤ ψ(g) ≤ ψ (M ) < ∞, or −∞ < ψ (M ) ≤ ψ(g) ≤ ψ (m) < ∞.
Suppose that χ ◦ ψ −1 is convex. Letting ϕ = χ ◦ ψ −1 in Theorem 2.1 we obtain
χ ◦ ψ −1 (ψ(m) + ψ(M ) − A (ψ(g)))
≤ A χ ◦ ψ −1 (ψ (m) + ψ (M ) − ψ (g))
ψ (M ) − A (ψ (g))
A (ψ (g)) − ψ (m)
≤
· χ ◦ ψ −1 (ψ (M )) +
· χ ◦ ψ −1 (ψ (m))
ψ (M ) − ψ (m)
ψ (M ) − ψ (m)
−1
−1
≤ χ◦ψ
(ψ (m)) + χ ◦ ψ
(ψ (M )) − A χ ◦ ψ −1 (ψ (g)) ,
or
χ ψ −1 (ψ(m) + ψ(M ) − A (ψ(g)))
(3.4)
≤ A χ ψ −1 (ψ (m) + ψ (M ) − ψ (g))
ψ (M ) − A (ψ (g))
A (ψ (g)) − ψ (m)
· χ (M ) +
· χ (m)
ψ (M ) − ψ (m)
ψ (M ) − ψ (m)
≤ χ (m) + χ (M ) − A (χ (g)) .
≤
If χ ◦ ψ −1 is concave we have the reverse of inequalities (3.4).
If χ is strictly increasing, then the inverse function χ−1 is also strictly increasing, so that (3.4)
implies (3.3). If χ is strictly decreasing, then the inverse function χ−1 is also strictly decreasing,
so in that case the reverse of (3.4) implies (3.3). Analogously, we get the reverse of (3.3) in the
cases when χ ◦ ψ −1 is convex and χ is strictly decreasing, or χ ◦ ψ −1 is concave and χ is strictly
increasing.
Remark 3.4. If we let
(
gr ,
(
r 6= 0
ψ (g) =
log g, r = 0
and χ (g) =
gs,
r 6= 0
,
log g, r = 0
then Theorem 3.3 reduces to Theorem 3.1.
R EFERENCES
[1] P.R. BEESACK AND J.E. PEČARIĆ, On the Jessen’s inequality for convex functions, J. Math. Anal.,
110 (1985), 536–552.
[2] I. GAVREA, Some considerations on the monotonicity property of power mean, J. Inequal. Pure
and Appl. Math., 5(4) (2004), Art. 93. [ONLINE: http://jipam.vu.edu.au/article.
php?sid=448]
[3] B. JESSEN, Bemaerkinger om konvekse Funktioner og Uligheder imellem Middelvaerdier I.,
Mat.Tidsskrift, B, 17–28. (1931).
[4] A. McD. MERCER, A variant of Jensen’s inequality, J. Inequal. Pure and Appl. Math., 4(4) (2003),
Art. 73. [ONLINE: http://jipam.vu.edu.au/article.php?sid=314]
[5] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, Inc. (1992).
J. Inequal. Pure and Appl. Math., 7(1) Art. 10, 2006
http://jipam.vu.edu.au/