A VARIANT OF JESSEN’S INEQUALITY OF
MERCER’S TYPE FOR SUPERQUADRATIC
FUNCTIONS
S. ABRAMOVICH
Department of Mathematics
University of Haifa, Haifa, 31905, Israel
EMail: abramos@math.haifa.ac.il
J. BARIĆ
Fac. of Electrical Engin., Mechanical Engin. & Naval Architecture
University of Split, Rudjera Boškovića bb, 21000 Split, Croatia
EMail: jbaric@fesb.hr
J. PEČARIĆ
Faculty of Textile Technology, University of Zagreb
Pierottijeva 6, 10000 Zagreb, Croatia
EMail: pecaric@hazu.hr
Received:
14 December, 2007
Accepted:
03 April, 2008
Communicated by:
I. Gavrea
2000 AMS Sub. Class.:
26D15, 39B62.
Key words:
Isotonic linear functionals, Jessen’s inequality, Superquadratic functions, Functional quasi-arithmetic and power means of Mercer’s type.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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Abstract:
A variant of Jessen’s inequality for superquadratic functions is proved.
This is a refinement of a variant of Jessen’s inequality of Mercer’s type
for convex functions. The result is used to refine some comparison inequalities of Mercer’s type between functional power means and between
functional quasi-arithmetic means.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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Contents
1
Introduction
4
2
Main Results
8
3
Applications
15
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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1.
Introduction
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 the properties:
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).
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
The following result is Jessen’s generalization of the well known Jensen’s inequality for convex functions [10] (see also [12, p. 47]):
vol. 9, iss. 3, art. 62, 2008
Theorem A. 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
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ϕ(A (g)) ≤ A(ϕ (g)).
Similar to Jensen’s inequality, Jessen’s inequality has a converse [7] (see also [12,
p. 98]):
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Theorem B. 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(ϕ (g)) ≤
M − A (g)
A (g) − m
· ϕ(m) +
· ϕ(M ).
M −m
M −m
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Inspired by I.Gavrea’s [9] result, which is a generalization of Mercer’s variant
of Jensen’s inequality [11], recently, W.S. Cheung, A. Matković and J. Pečarić, [8]
gave the following extension on a linear class L satisfying properties L1, L2.
Theorem C. Let L satisfy properties L1, L2 on a nonempty set E, and let ϕ be a
continuous 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
(1.1)
ϕ (m + M − A (g)) ≤ ϕ (m) + ϕ (M ) − A (ϕ (g)) .
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
In fact, to be more specific we have the following series of inequalities
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(1.2)
ϕ (m + M − A (g))
≤ A (ϕ (m + M − g))
M − A (g)
A (g) − m
≤
· ϕ(M ) +
· ϕ(m)
M −m
M −m
≤ ϕ (m) + ϕ (M ) − A (ϕ (g)) .
If the function ϕ is concave, inequalities (1.1) and (1.2) are reversed.
In this paper we give an analogous result for superquadratic function (see also
different analogous results in [6]). We start with the following definition.
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Definition A ([1, Definition 2.1]). A function ϕ : [0, ∞) → R is superquadratic
provided that for all x ≥ 0 there exists a constant C(x) ∈ R such that
(1.3)
ϕ (y) − ϕ (x) − ϕ (|y − x|) ≥ C (x) (y − x)
for all y ≥ 0. We say that f is subquadratic if −f is a superquadratic function.
For example, the function ϕ(x) = xp is superquadratic for p ≥ 2 and subquadratic for p ∈ (0, 2].
Theorem D ([1, Theorem 2.3]). The inequality
Z
Z Z
gdµ ≤
f
f (g (s)) − f g (s) − gdµ
dµ (s)
holds for all probability measures µ and all non-negative µ−integrable functions g,
if and only if f is superquadratic.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
The following discrete version that follows from the above theorem is also used
in the sequel.
vol. 9, iss. 3, art. 62, 2008
Lemma
Let xr ≥ 0, 1 ≤ r ≤ n and let
P A. Suppose that f is superquadratic.
P
x̄ = nr=1 λr xr where λr ≥ 0 and nr=1 λr = 1. Then
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n
X
r=1
λr f (xr ) ≥ f (x̄) +
n
X
Contents
λr f (|xr − x̄|).
r=1
In [3] and [4] the following converse of Jensen’s inequality for superquadratic
functions was proved.
Theorem E. Let(Ω, A, µ) be a measurable space with 0 < µ(r) < ∞ and let f :
[0, ∞) → R be a superquadratic function. If g : Ω → [m, M ] ≤ [0, ∞) is such that
g, f ◦ g ∈ L1 (µ), then we have
Z
1
M − ḡ
ḡ − m
f (g)dµ ≤
f (m) +
f (M )
µ(Ω) Ω
M −m
M −m
Z
1
1
−
((M − g) f (g − m) + (g − m) f (M − g)) dµ,
µ(Ω) M − m Ω
R
1
for ḡ = µ(Ω)
gdµ.
Ω
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The discrete version of this theorem is:
Theorem F. Let f : [0, ∞) → R be a superquadratic function. Let (x1 , . . . , xn )
n
be an n−tuple in [m, M ]P
(0 ≤ m < M < ∞), and (p
1 , . . . , pn ) be a non-negative
P
n−tuple such that Pn = ni=1 pi > 0. Denote x̄ = P1n ni=1 pi xi , then
n
M − x̄
x̄ − m
1 X
pi f (xi ) ≤
f (m) +
f (M )
Pn i=1
M −m
M −m
n
X
1
−
pi [(M − xi ) f (xi − m) + (xi − m) f (M − xi )] .
Pn (M − m) i=1
In Section 2 we give the main result of our paper which is an analogue of Theorem C for superquadratic functions. In Section 3 we use that result to derive some
refinements of the inequalities obtained in [8] which involve functional power means
of Mercer’s type and functional quasi-arithmetic means of Mercer’s type.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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2.
Main Results
Theorem 2.1. Let L satisfy properties L1, L2, on a nonempty set E, ϕ : [0, ∞) → R
be a continuous superquadratic function, and 0 ≤ m < M < ∞. Assume that
A is an isotonic linear functional on L with A(1) = 1. If g ∈ L is such that
m ≤ g(t) ≤ M , for all t ∈ E, and such that ϕ(g), ϕ(m+M −g), (M −g)ϕ(g −m),
(g − m)ϕ(M − g) ∈ L, then we have
ϕ(m + M − A(g))
M − A(g)
A(g) − m
ϕ(m) +
ϕ(M )
≤
M −m
M −m
1
−
[(A(g) − m)ϕ(M − A(g)) + (M − A(g))ϕ(A(g) − m)]
M −m
(2.1) ≤ ϕ(m) + ϕ(M ) − A (ϕ(g))
1
−
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m))
M −m
1
−
[(A(g) − m)ϕ(M − A(g)) + (M − A(g))ϕ(A(g) − m)] .
M −m
If the function ϕ is subquadratic, then all the inequalities above are reversed.
Proof. From Lemma A for n = 2, as well as from Theorem F, we get that for
0 ≤ m ≤ t ≤ M,
(2.2) ϕ(t) ≤
M −t
t−m
ϕ(m) +
ϕ(M )
M −m
M −m
t−m
M −t
−
ϕ(t − m) −
ϕ(M − t).
M −m
M −m
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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Replacing t with M + m − t in (2.2) it follows that
t−m
M −t
ϕ(m) +
ϕ(M )
M −m
M −m
t−m
M −t
ϕ(M − t) −
ϕ(t − m)
−
M − m
M −m
M −t
t−m
ϕ(M ) +
ϕ(m)
= ϕ(m) + ϕ(M ) −
M −m
M −m
t−m
M −t
ϕ(M − t) −
ϕ(t − m).
−
M −m
M −m
ϕ(M + m − t) ≤
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
Since m ≤ g(t) ≤ M for all t ∈ E, it follows that m ≤ A(g) ≤ M and we have
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(2.3) ϕ(m + M − A(g))
Contents
M − A(g)
A(g) − m
ϕ(M ) +
ϕ(m)
≤ ϕ(m) + ϕ(M ) −
M −m
M −m
A(g) − m
M − A(g)
−
ϕ(M − A(g)) −
ϕ(A(g) − m).
M −m
M −m
On the other hand, since m ≤ g(t) ≤ M for all t ∈ E it follows that
ϕ(g(t)) ≤
M − g(t)
g(t) − m
ϕ(m) +
ϕ(M )
M −m
M −m
M − g(t)
g(t) − m
ϕ(g(t) − m) −
ϕ(M − g(t)).
−
M −m
M −m
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Using functional calculus we have
M − A(g)
ϕ(m)
M −m
A(g) − m
1
+
ϕ(M ) −
A ((M − g(t)ϕ(g(t) − m))
M −m
M −m
1
A ((g(t) − m)ϕ(M − g(t))) .
−
M −m
(2.4) A(ϕ(g)) ≤
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
Using inequalities (2.3) and (2.4), we obtain the desired inequality (2.1).
The last statement follows immediately from the fact that if ϕ is subquadratic
then −ϕ is a superquadratic function.
Remark 1. If a function ϕ is superquadratic and nonnegative, then it is convex [1,
Lema 2.2]. Hence, in this case inequality (2.1) is a refinement of inequality (1.1).
On the other hand, we can get one more inequality in (2.1) if we use a result of
S. Banić and S. Varosănec [5] on Jessen’s inequality for superquadratic functions:
Theorem 2.2 ([5, Theorem 8, Remark 1]). Let L satisfy properties L1, L2, on a
nonempty set E, and let ϕ : [0, ∞) → R be a continuous superquadratic function.
Assume that A is an isotonic linear functional on L with A(1) = 1. If f ∈ L is
nonnegative and such that ϕ(f ), ϕ(|f − A(f )|) ∈ L, then we have
(2.5)
ϕ(A(f )) ≤ A(ϕ(f )) − A(ϕ(|f − A(f )|)).
If the function ϕ is subquadratic, then the inequality above is reversed.
Using Theorem 2.2 and some basic properties of superquadratic functions we
prove the next theorem.
vol. 9, iss. 3, art. 62, 2008
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Theorem 2.3. Let L satisfy properties L1, L2, on a nonempty set E, and let ϕ :
[0, ∞) → R be a continuous superquadratic function, and let 0 ≤ m < M < ∞.
Assume that A is an isotonic linear functional on L with A(1) = 1. If g ∈ L is
such that m ≤ g(t) ≤ M , for all t ∈ E, and such that ϕ(g), ϕ(m + M − g),
(M − g)ϕ(g − m), (g − m)ϕ(M − g), ϕ (|g − A (g)|) ∈ L, then we have
(2.6)
(2.7)
(2.8)
ϕ(m + M − A(g))
≤ A(ϕ(m + M − g)) − A(ϕ(|g − A(g)|))
M − A(g)
A(g) − m
≤
ϕ(m) +
ϕ(M )
M −m
M −m
1
−
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m))
M −m
− A(ϕ(|g − A(g)|))
≤ ϕ(m) + ϕ(M ) − A (ϕ(g))
2
−
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m))
M −m
− A(ϕ(|g − A(g)|)).
If the function ϕ is subquadratic, then all the inequalities above are reversed.
Proof. Notice that (m + M − g) ∈ L. Since m ≤ g(t) ≤ M for all t ∈ E, it
follows that m ≤ m + M − g(t) ≤ M for all t ∈ E. Applying (2.5) to the function
f = m + M − g we get
ϕ(A(m + M − g))
= ϕ(m + M − A(g))
≤ A(ϕ(m + M − g)) − A(ϕ(|m + M − g − A(m + M − g)|))
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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= A(ϕ(m + M − g)) − A(ϕ(|m + M − g − m − M + A(g)|))
= A(ϕ(m + M − g)) − A(ϕ(|g − A(g)|)),
which is the inequality (2.6).
From the discrete Jensen’s inequality for superquadratic functions we get for all
m ≤ x ≤ M,
x−m
M −x
ϕ(m) +
ϕ(M )
M −m
M −m
M −x
x−m
−
ϕ(x − m) −
ϕ(M − x).
M −m
M −m
Replacing x in (2.9) with m + M − g(t) ∈ [m, M ] for all t ∈ E, we have
(2.9)
ϕ(x) ≤
g(t) − m
M − g(t)
ϕ(m + M − g(t)) ≤
ϕ(m) +
ϕ(M )
M −m
M −m
g(t) − m
M − g(t)
−
ϕ(M − g(t)) −
ϕ(g(t) − m).
M −m
M −m
Since A is linear, isotonic and satisfies A (1) = 1, from the above inequality it
follows that
A(g) − m
M − A(g)
ϕ(m) +
ϕ(M )
M −m
M −m
1
−
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m)) .
M −m
Adding −A(ϕ(|g − A(g)|)) on both sides of (2.10) we get
(2.10) A(ϕ(m + M − g)) ≤
(2.11) A(ϕ(m + M − g)) − A(ϕ(|g − A(g)|))
M − A(g)
A(g) − m
≤
ϕ(m) +
ϕ(M )
M −m
M −m
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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−
1
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m)) − A(ϕ(|g − A(g)|)),
M −m
which is the inequality (2.7).
The right hand side of (2.11) can be written as follows
(2.12) ϕ(m) + ϕ(M ) −
−
M − A(g)
A(g) − m
ϕ(m) −
ϕ(M )
M −m
M −m
1
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m)) − A(ϕ(|g − A(g)|)).
M −m
On the other hand, replacing x, in (2.9), with g(t) ∈ [m, M ], for all t ∈ E, we get
(2.13) ϕ(g(t)) ≤
g(t) − m
M − g(t)
ϕ(m) +
ϕ(M )
M −m
M −m
g(t) − m
M − g(t)
−
ϕ(g(t) − m) −
ϕ(M − g(t)).
M −m
M −m
Applying the functional A on (2.13) we have
(2.14) A(ϕ(g)) ≤
A(g) − m
M − A(g)
ϕ(m) +
ϕ(M )
M −m
M −m
1
−
A ((M − g)ϕ(g − m) + (g − m)ϕ(M − g)) ,
M −m
The inequality (2.14) can be written as follows
−
M − A(g)
A(g) − m
ϕ(m) −
ϕ(M )
M −m
M −m
1
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m)) .
≤ −A(ϕ(g)) −
M −m
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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Using (2.12) we get
A(g) − m
M − A(g)
ϕ(m) +
ϕ(M )
M −m
M −m
1
−
A ((g − m)ϕ(M − g) + (M
M −m
≤ ϕ(m) + ϕ(M ) − A(ϕ(g))
1
−
A ((g − m)ϕ(M − g) + (M
M −m
1
−
A ((g − m)ϕ(M − g) + (M
M −m
= ϕ(m) + ϕ(M ) − A(ϕ(g))
2
−
A ((g − m)ϕ(M − g) + (M
M −m
− g)ϕ(g − m)) − A(ϕ(|g − A(g)|))
− g)ϕ(g − m))
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
− g)ϕ(g − m)) − A(ϕ(|g − A(g)|))
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− g)ϕ(g − m)) − A(ϕ(|g − A(g)|)).
Now, it follows that
A(g) − m
M − A(g)
ϕ(m) +
ϕ(M )
M −m
M −m
1
−
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m)) − A(ϕ(|g − A(g)|))
M −m
≤ ϕ(m) + ϕ(M ) − A(ϕ(g))
2
−
A ((g − m)ϕ(M − g) + (M − g)ϕ(g − m)) − A(ϕ(|g − A(g)|)),
M −m
which is the inequality (2.8).
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
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3.
Applications
Throughout this section 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 ] (0 < m < M < ∞) such that all of the
following expressions are well defined.
Let ψ be a continuous and strictly monotonic function on an interval I = [m, M ],
(0 < m < M < ∞).
For any r ∈ R, a power mean of Mercer’s type functional

1
r
r
r r

 [m + M − A(g )] , r 6= 0
Q(r, g) :=
mM


,
r = 0,
exp (A(log g))
and a quasi-arithmetic mean functional of Mercer’s type
fψ (g, A) = ψ −1 (ψ(m) + ψ(M ) − A(ψ(g)))
M
are defined in [8] and the following theorems are proved.
Theorem G. If r, s ∈ R and r ≤ s, then
Q(r, g) ≤ Q(s, g).
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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Theorem H.
(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
(3.1)
fψ (g, A) ≤ M
fχ (g, A) .
M
(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 inequality (3.1) is reversed.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
Applying the inequality (2.1) to the adequate superquadratic functions we shall
give some refinements of the inequalities in Theorems G and H. To do this, we will
define following functions.
vol. 9, iss. 3, art. 62, 2008
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1
r
r
r
r rs
♦(m, M, r, s, g, A) = r
A
(M
−
g
)(g
−
m
)
M − mr
1
r
r
r
r rs
+ r
A
(g
−
m
)(M
−
g
)
M − mr
s
1
+ r
(A(g r ) − mr ) (M r − A(g r )) r
r
M −m
s
1
+ r
(M r − A(g r )) (A(g r ) − mr ) r .
r
M −m
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and
♦(m, M, ψ, χ, g, A)
1
=
A (ψ(M ) − ψ(g))χ ψ −1 (ψ(g) − ψ(m)))
ψ(M ) − ψ(m)
1
+
A (ψ(g) − ψ(m))χ ψ −1 (ψ(M ) − ψ(g))
ψ(M ) − ψ(m)
Close
1
(A(ψ(g)) − ψ(m)) χ ψ −1 (ψ(M ) − A(ψ(g)))
ψ(M ) − ψ(m)
1
+
(ψ(M ) − A(ψ(g))) χ ψ −1 (A(ψ(g)) − ψ(m)) .
ψ(M ) − ψ(m)
+
Now, the following theorems are valid.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
Theorem 3.1. Let r, s ∈ R.
(i) If 0 < 2r ≤ s, then
and J. Pečarić
1
(3.2)
Q(r, g) ≤ [(Q(s, g))s − ♦(m, M, r, s, g, A)] s .
(ii) If 2r ≤ s < 0, then for (Q (s, g))s − ♦ (M, m, r, s, g, A) > 0
vol. 9, iss. 3, art. 62, 2008
Title Page
1
(3.3)
Q(r, g) ≤ [(Q(s, g))s − ♦(M, m, r, s, g, A)] s ,
where we used ♦(M, m, r, s, g, A) to denote the new function derived from the
function ♦(m, M, r, s, g, A) by changing the places of m and M .
s
(iii) If 0 < s ≤ 2r, then for (Q (s, g)) − ♦ (M, m, r, s, g, A) > 0 the reverse
inequality (3.2) holds.
(iv) If s ≤ 2r < 0, then the reversed inequality (3.3) holds.
Proof.
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(i) It is given that
0 < m ≤ g ≤ M < ∞.
Since 0 < 2r ≤ s, it follows that
0 < mr ≤ g r ≤ M r < ∞.
Applying Theorem 2.1, or more precisely inequality (2.1) to the superquadratic
s
function ϕ(t) = t r (note that rs ≥ 2 here) and replacing g, m and M with g r ,
mr and M r , respectively, we have
s
s
1
r
r
r
r r
(A(g
)
−
m
)
(M
−
A(g
))
M r − mr
s
1
r
r
r
r r
+ r
(M
−
A(g
))
(A(g
)
−
m
)
M − mr
s
≤ m + M s − A(g s )
1
r
r
r
r rs
−
g
)(g
−
m
)
− r
A
(M
M − mr
1
r
r
r
r rs
− r
A
(g
−
m
)(M
−
g
)
.
M − mr
[mr + M r − A(g r )] r +
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
Title Page
Contents
i.e.
(3.4)
[Q(r, g)]s ≤ [Q(s, g)]s − ♦(m, M, r, s, g, A).
Raising both sides of (3.4) to the power
1
s
> 0, we get desired inequality (3.2).
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Page 18 of 26
(ii) In this case we have
0 < M r ≤ g r ≤ mr < ∞.
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Applying Theorem 2.1 or, more precisely, the reversed inequality (2.1) to the
s
subquadratic function ϕ(t) = t r (note that now we have 0 < rs ≤ 2) and
replacing g, m and M with g r , mr and M r , respectively, we get
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s
[M r + mr − A(g r )] r +
s
1
(A(g r ) − M r ) (mr − A(g r )) r
r
−M
s
1
(mr − A(g r )) (A(g r ) − M r ) r
+ r
r
m −M
mr
Close
s
1
A (mr − g r )(g r − M r ) r
r
−M
r
r
r
r rs
.
A
(g
−
M
)(m
−
g
)
r
≥ M s + ms − A(g s ) −
−
1
mr − M
mr
Since 2r ≤ s < 0, raising both sides to the power 1s , it follows that
1
1
[M r + mr − A(g r )] r ≤ [M s + ms − A(g s ) − ♦(M, m, r, s, g, A)] s ,
or
and J. Pečarić
1
Q(r, g) ≤ [(Q(s, g))s − ♦(M, m, r, s, g, A)] s .
s
r
r
r
vol. 9, iss. 3, art. 62, 2008
r
(iii) In this case we have 0 <
≤ 2. Since 0 < m ≤ g ≤ M < ∞, we
can apply Theorem 2.1, or more precisely, the reversed inequality (2.1) to the
s
subquadratic function ϕ(t) = t r . Replacing g, m and M with g r , mr and M r ,
respectively, it follows that
s
[mr + M r − A(g r )] r
s
1
+ r
(A(g r ) − mr ) (M r − A(g r )) r
r
M −m
s
1
+ r
(M r − A(g r )) (A(g r ) − mr ) r
r
M −m
s
≥ m + M s − A(g s )
1
r
r
r
r rs
− r
A
(M
−
g
)(g
−
m
)
M − mr
s
1
− r
A (g r − mr )(M r − g r ) r ,
r
M −m
i.e.
(3.5)
[Q(r, g)]s ≥ [Q(s, g)]s − ♦(m, M, r, s, g, A).
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
Title Page
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Raising both sides of (3.5) to the power
1
s
> 0 we get
1
Q(r, g) ≥ [(Q(s, g))s − ♦(m, M, r, s, g, A)] s .
(iv) Since r < 0, from 0 < m ≤ g ≤ M < ∞ it follows that 0 < M r ≤ g r ≤
mr < ∞. Now, we are applying Theorem 2.1 to the superquadratic function
s
ϕ(t) = t r , because rs ≥ 2 here, and analogous to the previous theorem we get
[Q(r, g)]s ≤ [Q(s, g)]s − ♦(M, m, r, s, g, A).
Raising both sides to the power
1
s
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
< 0 it follows that
1
Q(r, g) ≥ [(Q(s, g))s − ♦(M, m, r, s, g, A)] s .
Title Page
Contents
Theorem 3.2. Let r, s ∈ R.
(i) If 0 < 2s ≤ r, then
(3.6)
1
Q(r, g) ≥ [(Q(s, g))r + ♦(m, M, s, r, g, A)] r ,
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Page 20 of 26
where we used ♦(m, M, s, r, g, A) to denote the new function derived from the
function ♦(m, M, r, s, g, A) by changing the places of r and s.
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(ii) If 2s ≤ r < 0, then
(3.7)
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r
1
r
Q(r, g) ≤ [(Q(s, g)) + ♦(M, m, s, r, g, A)] .
(iii) If 0 < r ≤ 2s, then the reversed inequality (3.6) holds.
(iv) If r ≤ 2s < 0, then the reversed inequality (3.7) holds.
Close
Proof.
r
(i) Applying inequality (2.1) to the superquadratic function ϕ(t) = t s (note that
r
≥ 2 here) and replacing g, m and M with g s , ms and M s , (0 < ms ≤ g s ≤
s
M s < ∞) respectively, we have
r
[ms + M s − A(g s )] s
r
1
s
s
s
s s
+ s
(A(g
)
−
m
)
(M
−
A(g
))
M − ms
r
1
+ s
(M s − A(g s )) (A(g s ) − ms ) s
s
M −m
r
≥ m + M r − A(g r )
1
s
s
s
s rs
− s
A
(M
−
g
)(g
−
m
)
M − ms
r
1
− s
A (g s − ms )(M s − g s ) s ,
s
M −m
i.e.
[Q(s, g)]r ≤ [Q(r, g)]r − ♦(m, M, s, r, g, A).
Raising both sides to the power
1
r
> 0, the inequality (3.6) follows.
(ii) Since s < 0, we have 0 < M s ≤ g s ≤ ms < ∞ so the function ♦ will be of
the form ♦(M, m, s, r, g, A). Since 0 < rs ≤ 2, we will apply Theorem 2.1 to
r
the subquadratic function ϕ(t) = t s and, as in previous case, it follows that
[Q(s, g)]r + ♦(M, m, s, r, g, A) ≥ [Q(r, g)]r .
Raising both sides to the power
1
r
< 0, the inequality (3.7) follows.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
Title Page
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(iii) Since 0 < rs ≤ 2, we will apply Theorem 2.1 to the subquadratic function
r
ϕ(t) = t s and then it follows that
[Q(s, g)]r + ♦(m, M, s, r, g, A) ≥ [Q(r, g)]r .
Raising both sides to the power
1
r
> 0, we get
1
Q(r, g) ≤ [(Q(s, g))r + ♦(m, M, s, r, g, A)] r .
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
(iv) Since rs ≥ 2, we will apply Theorem 2.1 to the superquadratic function ϕ(t) =
r
t s and use the function ♦(M, m, s, r, g, A) instead of ♦(m, M, s, r, g, A). Then
we get
[Q(s, g)]r + ♦(M, m, s, r, g, A) ≤ [Q(r, g)]r .
Raising both sides to the power
1
r
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
Title Page
< 0, it follows that
Contents
r
1
r
Q(r, g) ≥ [(Q(s, g)) + ♦(M, m, s, r, g, A)] .
Remark 2. Notice that some cases in the last theorems have common parts. In some
of them we can establish double inequalities. For example, if 0 < r ≤ 2s and
0 < s ≤ 2r, then for (Q (s, g))s − ♦ (M, m, r, s, g, A) > 0
1
(i) If either χ ◦ ψ −1 is superquadratic and χ is strictly increasing, or χ ◦ ψ −1 is
subquadratic and χ is strictly decreasing, then
fψ (g, A) ≤ χ−1 χ M
fχ (g, A) − ♦(m, M, ψ, χ, g, A) ,
(3.8)
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1
[(Q (s, g))r + ♦ (m, M, s, r, g, A)] r ≥ Q (r, g) ≥ [(Q (s, g))s − ♦ (m, M, r, s, g, A)] s .
Theorem 3.3. Let ψ ∈ C([m, M ]) be strictly increasing and let χ ∈ C([m, M ]) be
strictly monotonic functions.
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(ii) If either χ ◦ ψ −1 is subquadratic and χ is strictly increasing or χ ◦ ψ −1 is superquadratic and χ is strictly decreasing, then the inequality (3.8) is reversed.
Proof. Suppose that χ ◦ ψ −1 is superquadratic. Letting ϕ = χ ◦ ψ −1 in Theorem 2.1
and replacing g, m and M with ψ(g), ψ(m) and ψ(M ) respectively, we have
χ ψ −1 (ψ(m) + ψ(M ) − A(ψ(g)))
1
+
(A(ψ(g)) − ψ(m)) χ ψ −1 (ψ(M ) − A(ψ(g)))
ψ(M ) − ψ(m)
1
+
(ψ(M ) − A(ψ(g))) χ ψ −1 (A(ψ(g)) − ψ(m))
ψ(M ) − ψ(m)
≤ χ(m) + χ(M ) − A(χ(g))
1
−
A (ψ(M ) − ψ(m)) χ ψ −1 (ψ(g) − ψ(m))
ψ(M ) − ψ(m)
1
−
A (ψ(g) − ψ(m)) χ ψ −1 (ψ(M ) − ψ(g)) ,
ψ(M ) − ψ(m)
i.e.,
(3.9)
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
Title Page
Contents
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fψ (g, A)
χ M
≤ χ(m) + χ(M ) − A(χ(g)) − ♦(m, M, ψ, χ, g, A)
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−1
≤ χ ◦ χ (χ(m) + χ(M ) − A(χ(g))) − ♦(m, M, ψ, χ, g, A)
f
≤ χ Mχ (g, A) − ♦(m, M, ψ, χ, g, A).
If χ is strictly increasing, then the inverse function χ−1 is also strictly increasing
and inequality (3.9) implies the inequality (3.8). If χ is strictly decreasing, then the
inverse function χ−1 is also strictly decreasing and in that case the reverse of (3.9)
Close
implies (3.8). Analogously, we get the reverse of (3.8) in the cases when χ ◦ ψ −1
is superquadratic and χ is strictly decreasing, or χ ◦ ψ −1 is subquadratic and χ is
strictly increasing.
Remark 3. If the function ψ in Theorem 3.3 is strictly decreasing, then the inequality
(3.8) and its reversal also hold under the same assumptions, but with m and M
interchanged.
Remark 4. Obviously, Theorem 3.1 and Theorem 3.2 follow from Theorem 3.3 and
Remark 3 by choosing ψ(t) = tr and χ(t) = ts , or vice versa.
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
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References
[1] S. ABRAMOVICH, G. JAMESON AND G. SINNAMON, Refining Jensen’s
iInequality, Bull. Math. Soc. Math. Roum., 47 (2004), 3–14.
[2] S. ABRAMOVICH, G. JAMESON AND G. SINNAMON, Inequalities for averages of convex and superquadratic functions, J. Inequal. in Pure & Appl. Math.,
5(4) (2004), Art. 91. [ONLINE: http://jipam.vu.edu.au/article.
php?sid=444].
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
[3] S. BANIĆ, Superquadratic Functions, PhD. Thesis, (2007), Zagreb, (in Croatian).
vol. 9, iss. 3, art. 62, 2008
[4] S. BANIĆ, J. PEČARIĆ AND S. VAROŠANEC, Superquadratic functions and
refinements of some classical inequalities, J. Korean Math. Soc., to appear
Title Page
[5] S. BANIĆ AND S. VAROŠANEC, Functional inequalities for superquadratic
functions, submitted for publication.
[6] J. BARIĆ, A. MATKOVIĆ AND J.E. PEČARIĆ, A variant of Mercer’s operator
inequality for superquadratic functions, Scientiae Mathematicae Japonicae.
[7] P.R. BEESACK AND J.E. PEČARIĆ, On the Jessen’s inequality for convex
functions, J. Math. Anal., 110 (1985), 536–552.
[8] W.S. CHEUNG, A. MATKOVIĆ AND J.E. PEČARIĆ, A variant of Jessen’s inequality and generalized means, J. Inequal. in Pure & Appl. Math., 7(1) (2006),
Art. 10. [ONLINE: http://jipam.vu.edu.au/article.php?sid=
623].
Contents
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[9] I. GAVREA, Some considerations on the monotonicity property of power
mean, J. Inequal. in Pure & Appl. Math., 5(4) (2004), Art. 93. [ONLINE:
http://jipam.vu.edu.au/article.php?sid=448].
[10] B. JESSEN, Bemaerkinger om konvekse Funktioner og Uligheder imellem
Middelvaerdier I., Mat.Tidsskrift, B, (1931), 17-28.
[11] A.McD. MERCER, A variant of Jensen’s inequality, J. Inequal. in Pure &
Appl. Math., 4(4) (2003), Art. 73. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=314].
Jessen’s Inequality of Mercer’s
Type and Superquadracity
S. Abramovich, J. Barić
and J. Pečarić
vol. 9, iss. 3, art. 62, 2008
[12] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial
Orderings, and Statistical Applications, Academic Press, Inc. (1992).
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