Journal of Inequalities in Pure and
Applied Mathematics
COMPANION INEQUALITIES TO JENSEN’S INEQUALITY FOR m –
CONVEX AND (α, m)–CONVEX FUNCTIONS
volume 7, issue 5, article 194,
2006.
M. KLARIČIĆ BAKULA, J. PEČARIĆ AND M. RIBIČIĆ
Department of Mathematics
Faculty of Natural Sciences, Mathematics and Education
University of Split
Teslina 12, 21000 Split
Croatia
EMail: milica@pmfst.hr
Faculty of Textile Technology
University of Zagreb
Pierottijeva 6, 10000 Zagreb
Croatia
EMail: pecaric@hazu.hr
Department of Mathematics
University of Osijek
Trg Ljudevita Gaja 6, 31000 Osijek
Croatia
EMail: mihaela@mathos.hr
Received 21 April, 2006;
accepted 06 December, 2006.
Communicated by: S.S. Dragomir
Abstract
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Victoria University
ISSN (electronic): 1443-5756
117-06
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Abstract
General companion inequalities related to Jensen’s inequality for the classes of
m-convex and (α, m)-convex functions are presented. We show how Jensen’s
inequality for these two classes, as well as Slater’s inequality, can be obtained
from these general companion inequalities as special cases. We also present
several variants of the converse Jensen’s inequality, weighted Hermit-Hadamard’s
inequalities and inequalities of Giaccardi and Petrović for these two classes of
functions.
2000 Mathematics Subject Classification: 26A51, 26B25.
Key words: m-convex functions, (α, m)-convex functions, Jensen’s inequality,
Slater’s inequality, Hermite-Hadamard’s inequalities, Giaccardi’s inequality, Fejér’s inequalities.
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Companion Inequalities to Jensen’s Inequality for m-convex
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3
Some Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4
Inequalities for (α, m)-convex Functions . . . . . . . . . . . . . . . . . 21
References
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
1.
Introduction
Let [0, b] , b > 0, be an interval of the real line R, and let K (b) be the class of
all functions f : [0, b] → R which are continuous and nonnegative on [0, b] and
such that f (0) = 0. We define the mean function F of the function f ∈ K (b)
as
( 1 Rx
f (t) dt, x ∈ (0, b]
x 0
F (x) =
.
0,
x=0
We say that the function f is convex on [0, b] if
f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f (y)
holds for all x, y ∈ [0, b] and t ∈ [0, 1] . Let KC (b) denote the class of all
functions f ∈ K (b) convex on [0, b] , and let KF (b) be the class of all functions
f ∈ K (b) convex in mean on [0, b] , i.e., the class of all functions f ∈ K (b)
for which F ∈ KC (b) . Let KS (b) denote the class of all functions f ∈ K (b)
which are starshaped with respect to the origin on [0, b] , i.e., the class of all
functions f with the property that
f (tx) ≤ tf (x)
holds for all x ∈ [0, b] and t ∈ [0, 1] . In the paper [1], Bruckner and Ostrow,
among others, proved that
KC (b) ⊂ KF (b) ⊂ KS (b) .
In the paper [10] G. Toader defined the m-convexity: another intermediate
between the usual convexity and starshaped convexity.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Definition 1.1. The function f : [0, b] → R, b > 0, is said to be m-convex,
where m ∈ [0, 1] , if we have
f (tx + m (1 − t) y) ≤ tf (x) + m (1 − t) f (y)
for all x, y ∈ [0, b] and t ∈ [0, 1] . We say that f is m-concave if −f is m-convex.
Denote by Km (b) the class of all m-convex functions on [0, b] for which f (0) ≤
0.
Obviously, for m = 1 Definition 1.1 recaptures the concept of standard convex functions on [0, b] , and for m = 0 the concept of starshaped functions.
The following lemmas hold (see [11]).
Lemma A. If f is in the class Km (b) , then it is starshaped.
Lemma B. If f is in the class Km (b) and 0 < n < m ≤ 1, then f is in the
class Kn (b).
From Lemma A and Lemma B it follows that
K1 (b) ⊂ Km (b) ⊂ K0 (b) ,
whenever m ∈ (0, 1) . Note that in the class K1 (b) we have only the convex
functions f : [0, b] → R for which f (0) ≤ 0, i.e., K1 (b) is a proper subclass of
the class of convex functions on [0, b] .
It is interesting to point out that for any m ∈ (0, 1) there are continuous
and differentiable functions which are m-convex, but which are not convex in
the standard sense. Furthermore, in the paper [12], the following theorem was
proved.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Theorem A. For each m ∈ (0, 1) there is an m-convex polynomial f such that
f is not n-convex for any m < n ≤ 1.
For instance, f : [0, ∞) → R defined as
1
x4 − 5x3 + 9x2 − 5x
12
16
is 16
-convex, but it is not m-convex for any m ∈ 17
, 1 (see [7]).
17
It is well known (see for example [8, p. 5]) that the function f : (a, b) → R
is convex iff there is at least one line support for f at each point x0 ∈ (a, b) ,
i.e.,
f (x0 ) ≤ f (x) + λ (x0 − x) ,
f (x) =
0
for all x ∈ (a, b) , where λ ∈ R depends on
x0 and0 is given by0 λ = f (x0 ) when
0
0
0
f (x0 ) exists, and λ ∈ f− (x0 ) , f+ (x0 ) when f− (x0 ) 6= f+ (x0 ) .
The following Lemma [3] gives an analogous result for m-convex functions.
Lemma C. If f is differentiable, then f is m-convex iff
f (x) ≤ mf (y) + f 0 (x) (x − my)
for all x, y ∈ [0, b] .
The notion of m-convexity can be further generalized via introduction of
another parameter α ∈ [0, 1] in the definition of m-convexity. The class of
(α, m)-convex functions was first introduced in [6] and it is defined as follows.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Definition 1.2. The function f : [0, b] → R, b > 0, is said to be (α, m)-convex,
where (α, m) ∈ [0, 1]2 , if we have
f (tx + m (1 − t) y) ≤ tα f (x) + m (1 − tα ) f (y)
for all x, y ∈ [0, b] and t ∈ [0, 1] .
α
Denote by Km
(b) the class of all (α, m)-convex functions on [0, b] for which
f (0) ≤ 0.
It can be easily seen that for (α, m) ∈ {(0, 0) , (α, 0) , (1, 0) , (1, m) , (1, 1) ,
(α, 1)} one obtains the following classes of functions: increasing, α-starshaped,
starshaped, m-convex, convex and α-convex functions. Note that in the class
K11 (b) are only convex functions f : [0, b] → R for which f (0) ≤ 0, i.e., K11 (b)
is a proper subclass of the class of all convex functions on [0, b] . The interested
reader can find more about partial ordering of convexity in [8, p. 8, 280].
Lemma C for m-convex functions has its analogue for the class of (α, m)convex functions, as it is stated below (see [6]).
Lemma D. If f is differentiable, then f is (α, m)-convex on [0, b] iff we have
f 0 (x) (x − my) ≥ α (f (x) − mf (y)) ,
for all x, y ∈ [0, b] .
The paper is organized as follows.
In Section 2 we first prove a general companion inequality related to Jensen’s
inequality for m-convex functions in its integral and discrete form. We show
that Jensen’s inequality for m-convex functions, as well as Slater’s inequality,
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
can be obtained from this general inequality as two special cases. In this section
we also present two converse Jensen’s inequalities for m-convex functions.
In Section 3 we use results from Section 2 to prove several more inequalities for m-convex functions: weighted Hermite-Hadamard’s inequalities and
inequalities of Giaccardi and Petrović.
In Section 4 we give a selection of the results presented in Sections 2 and 3,
but for the class of (α, m)-convex functions.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
2.
Companion Inequalities to Jensen’s Inequality
for m-convex Functions
Theorem 2.1. Let (Ω, A, µ) be a measure space with 0 < µ (Ω) < ∞ and let
f : [0, b] → R, b > 0, be a differentiable m−convex function on [0, b] with
m ∈ (0, 1] . If u : Ω → [0, b] is a measurable function such that f 0 ◦ u is in
L1 (µ) , then for any ξ, η ∈ [0, b] we have
Z
f (ξ)
1
ξ
0
(2.1)
+ f (ξ)
udµ −
m
µ (Ω) Ω
m
Z
Z
1
1
≤
(f ◦ u) dµ ≤ mf (η) +
(u − mη) (f 0 ◦ u) dµ.
µ (Ω) Ω
µ (Ω) Ω
Proof. First observe that since u is measurable and bounded we have u ∈
L∞ (µ) and since f is differentiable we also know that f ◦u ∈ L1 (µ) (moreover,
it is in L∞ (µ)). On the other hand, by the assumption we have f 0 ◦ u ∈ L1 (µ) ,
so it also follows that u · (f 0 ◦ u) ∈ L1 (µ) .
From Lemma C we know that the inequalities
(2.2)
f (x) + f 0 (x) (my − x) ≤ mf (y) ,
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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f (y) ≤ mf (x) + f 0 (y) (y − mx) ,
hold for all x, y ∈ [0, b] . If in (2.2) we let x = ξ and y = u (t) , t ∈ Ω, we get
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Page 8 of 32
f (ξ) + f 0 (ξ) (mu (t) − ξ) ≤ m (f ◦ u) (t) ,
t ∈ Ω.
J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Integrating over Ω we obtain
Z
Z
0
µ (Ω) f (ξ) + f (ξ) m udµ − ξµ (Ω) ≤ m (f ◦ u) dµ,
Ω
Ω
from which the left hand side of (2.1) immediately follows.
In order to obtain the right hand side of (2.1) we proceed in a similar way:
if in (2.3) we let x = η and y = u (t) , t ∈ Ω, we get
(f ◦ u) (t) ≤ mf (η) + (f 0 ◦ u) (t) (u (t) − mη) ,
t ∈ Ω,
so after integration over Ω we obtain
Z
Z
(f ◦ u) dµ ≤ mµ (Ω) f (η) + (u − mη) (f 0 ◦ u) dµ,
Ω
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
Ω
from which the right hand side of (2.1) easily follows.
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If m = 1, Theorem 2.1 gives an analogous result for convex functions which
was proved in [5].
The following theorem is a variant of Theorem 2.1 for the class of starshaped
functions.
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Theorem 2.2. Let (Ω, A, µ) be a measure space with 0 < µ (Ω) < ∞ and let
f : [0, b] → R, b > 0, be a differentiable starshaped function. If u : Ω → [0, b]
is a measurable function such that f 0 ◦ u is in L1 (µ) , then we have
Z
Z
1
1
(f ◦ u) dµ ≤
u (f 0 ◦ u) dµ.
µ (Ω) Ω
µ (Ω) Ω
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Proof. As a special case of Lemma D for m = 0 we obtain
f (x) ≤ xf 0 (x) .
After putting
x = u (t) ,
t ∈ Ω,
we follow the same idea as in the proof of the previous theorem.
Our next corollary is the discrete version of Theorem 2.1.
Corollary 2.3. Let f : [0, b] → R, b > 0, be a differentiable m−convex function
on [0, b]
m ∈ (0, 1] . Let p1 , ..., pn be nonnegative real numbers such that
Pwith
n
Pn = i=1 pi 6= 0 and let xi ∈ [0, b] be given real numbers. Then for any ξ,
η ∈ [0, b] we have
!
n
X
ξ
1
f (ξ)
+ f 0 (ξ)
pi xi −
m
Pn i=1
m
n
n
1 X
1 X
pi f (xi ) ≤ mf (η) +
pi (xi − mη) f 0 (xi ) .
≤
Pn i=1
Pn i=1
Proof. This is a direct consequence of Theorem 2.1: we simply choose
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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Ω = {1, 2, ..., n} ,
µ ({i}) = pi , i = 1, 2, ..., n,
u (i) = xi , i = 1, 2, ..., n.
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Now we give an estimation of the difference between the first two inequalities in (2.1) . The obtained inequality incorporates the integral version of the
Dragomir-Goh result [2] for convex functions defined on an open interval in R.
Corollary 2.4. Let all the assumptions of Theorem 2.1 be satisfied. We have
Z
Z
1
1
m
0≤
(f ◦ u) dµ − f
udµ
µ (Ω) Ω
m
µ (Ω) Ω
Z
Z Z
m
1
m2
m2 − 1
≤
f
udµ +
u−
udµ (f 0 ◦ u) dµ.
m
µ (Ω) Ω
µ (Ω) Ω
µ (Ω) Ω
Proof. If we let ξ and η in (2.1) be defined as
Z
m
udµ ∈ [0, b] ,
ξ=η=
µ (Ω) Ω
we obtain
Z
m
1
f
udµ
m
µ (Ω) Ω
Z
1
≤
(f ◦ u) dµ
µ (Ω) Ω
Z
Z Z
1
m2
m
≤ mf
udµ +
u−
udµ (f 0 ◦ u) dµ.
µ (Ω) Ω
µ (Ω) Ω
µ (Ω) Ω
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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Our next result is the integral Jensen’s inequality for m-convex functions.
J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Corollary 2.5. Let (Ω, A, µ) be a measure space with 0 < µ (Ω) < ∞ and
let f : [0, b] → R, b > 0, be a differentiable m−convex function on [0, b] with
m ∈ (0, 1] . If u : Ω → [0, b] is a measurable function, then we have
Z
Z
m
1
1
f
udµ ≤
(f ◦ u) dµ.
m
µ (Ω) Ω
µ (Ω) Ω
The following theorem gives Slater’s inequality for m-convex functions.
Theorem 2.6. Let all the assumptions of Theorem 2.1 be satisfied. If
R
Z
u (f 0 ◦ u) dµ
ΩR
(2.4)
(f 0 ◦ u) dµ 6= 0,
∈ [0, b] ,
m Ω (f 0 ◦ u) dµ
Ω
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
then we have
1
µ (Ω)
(f ◦ u) dµ ≤ mf
Ω
u (f 0 ◦ u) dµ
m Ω (f 0 ◦ u) dµ
R
Z
ΩR
.
Proof. If the conditions (2.4) are satisfied, then in Theorem 2.1 we may choose
R
u (f 0 ◦ u) dµ
η = ΩR
,
m Ω (f 0 ◦ u) dµ
so from the right hand side of the inequality (2.1) we obtain
R
Z
u (f 0 ◦ u) dµ
1
ΩR
(f ◦ u) dµ ≤ mf
,
µ (Ω) Ω
m Ω (f 0 ◦ u) dµ
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
since in this case
R
Z
Z u (f 0 ◦ u) dµ
0
Ω
R
(u − mη) (f ◦ u) dµ =
u−
(f 0 ◦ u) dµ = 0.
0 ◦ u) dµ
(f
Ω
Ω
Ω
If m = 1 Theorem
2.6 recaptures Slater’s result from [9]: if f is convex and
R
increasing and if Ω (f 0 ◦ u) dµ 6= 0 we have
R
Z
u (f 0 ◦ u) dµ
1
RΩ
=
udν ∈ [0, b] ,
ν (Ω) Ω
(f 0 ◦ u) dµ
Ω
where the positive measure ν is defined as dν = (f 0 ◦ u) dµ.
In the next two theorems we give converses of the integral Jensen’s inequality
for m-convex functions.
Theorem 2.7. Let (Ω, A, µ) be a measure space with 0 < µ (Ω) < ∞ and
let f : [0, ∞) → R be an m−convex function on [0, ∞) with m ∈ (0, 1] . If
u : Ω → [a, b] , 0 ≤ a < b < ∞, is a measurable function such that f ◦ u is in
L1 (µ) , then we have
Z
1
(2.5)
(f ◦ u) dµ
µ (Ω) Ω
b−u
u−a
b
b−u a u−a
f (a) + m
f
,m
f
+
f (b) ,
≤ min
b−a
b−a
m
b−a
m
b−a
where
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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1
u=
µ (Ω)
Z
udµ.
Page 13 of 32
Ω
J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Proof. We may write
(f ◦ u) (t) = f
b − u (t)
u (t) − a b
a+m
b−a
b−a m
t ∈ Ω.
,
Since f is m-convex on [0, ∞) we have
b − u (t)
u (t) − a
(f ◦ u) (t) ≤
f (a) + m
f
b−a
b−a
b
m
,
t ∈ Ω,
and after integration over Ω we get
Z
b
u−a
b−u
f
.
(2.6)
(f ◦ u) dµ ≤ µ (Ω)
f (a) + m
b−a
m
b−a
Ω
In the similar way we obtain
Z
b−u a u−a
(f ◦ u) dµ ≤ µ (Ω) m
f
+
f (b) ,
b−a
m
b−a
Ω
so (2.5) immediately follows.
Theorem 2.8. Let all the assumptions of Theorem 2.7 be satisfied and suppose
also that the function u is symmetric about a+b
. Then we have
2
(
)
Z
b
a
f
(a)
+
mf
mf
+
f
(b)
1
m
m
(f ◦ u) dµ ≤ min
,
.
µ (Ω) Ω
2
2
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
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M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Proof. Since u is symmetric about
a+b
2
u (t) = a + b − u (t) =
we have
u (t) − a
b − u (t) b
a+m
b−a
b−a m
from which we get
Z
b
b−u
u−a
f
.
(2.7)
(f ◦ u) dµ ≤ µ (Ω)
f (a) + m
b−a
m
b−a
Ω
If we add (2.6) to (2.7) and then divide the sum by 2µ (Ω) we obtain
Z
b
u−a
1
1 b−u
f
f (a) + m
(f ◦ u) dµ ≤
b−a
m
µ (Ω) Ω
2 b−a
b
b−u
u−a
f
f (a) + m
+
b−a
m
b−a
b
f (a) + mf m
=
.
2
Analogously we obtain
1
µ (Ω)
Z
(f ◦ u) dµ ≤
Ω
mf
a
m
2
+ f (b)
.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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Corollary 2.9. Let f : [0, ∞) → R be an m−convex function on [0, ∞)
with m ∈ (0, 1] . Let p1 , ..., pn be nonnegative real numbers such that Pn =
Page 15 of 32
J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Pn
pi 6= 0 and let xi ∈ [a, b] , 0 ≤ a < b < ∞, be given real numbers. Than
we have
i=1
n
1 X
pi f (xi )
Pn i=1
b
x−a
b−x a x−a
b−x
f (b) ,
f
+
f
,m
≤ min
f (a) + m
b−a
b−a
m
b−a
m
b−a
where
n
1 X
x=
pi xi .
Pn i=1
Proof. The proof is analogous to the proof of Corollary 2.3.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
3.
Some Further Results
In this section we first show that the Fejér inequalities [4] (i.e., weighted HermiteHadamard’s inequalities) for m-convex functions presented in [13, Th7, Th8]
can be obtained as special cases of Theorem 2.1 and Theorem 2.7.
Corollary 3.1. Let f : [0, b] → R be an m−convex function on [0, b] with
m ∈ (0, 1] and let g : [a, b] → [0, ∞) be integrable and symmetric about a+b
,
2
where 0 ≤ a < b < ∞. If f is differentiable and f 0 is in L1 ([a, b]), then
Z b
f (mb) b − a 0
−
f (mb)
g (x) dx
m
2
a
Z b
Z b
≤
f (x) g (x) dx ≤
[(x − ma) f 0 (x) + mf (a)] g (x) dx.
a
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
a
Proof. This is a simple consequence of Theorem 2.1. We just choose µ to be
the Lebesgue measure defined as dµ = g (x) dx, Ω = [a, b], u (x) = x for all
x ∈ [a, b] , ξ = mb and η = a. Note that in this case we have
1
µ (Ω)
Rb
xg (x) dx
a+b
udµ = Ra b
=
.
2
g
(x)
dx
Ω
a
Z
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Corollary 3.2. Let f : [0, ∞) → R be an m−convex function on [0, ∞) with
m ∈ (0, 1] and let g : [a, b] → [0, ∞) be integrable and symmetric about a+b
,
2
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
where 0 ≤ a < b < ∞. If f is in L1 ([a, b]), then
Z b
(3.1)
f (x) g (x) dx
a
(
f (a) + mf mb mf
,
≤ min
2
a
m
+ f (b)
)Z
2
b
g (x) dx.
a
If f is also differentiable, then
Z b
Z b
1
a+b
(3.2)
f m
g (x) dx ≤
f (x) g (x) dx.
m
2
a
a
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
Proof. If we choose µ to be the Lebesgue measure defined as dµ = g (x) dx,
Ω = [a, b], u (x) = x for all x ∈ [a, b] , then (3.1) is obtain directly from
Theorem 2.7. Similarly, the inequality (3.2) is obtained from Corollary 2.5.
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
In two following theorems we prove inequalities of Giaccardi and Petrović
for m-convex functions.
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Theorem 3.3. Let f : [0, ∞) → R be an m−convex function on [0, ∞) with
m ∈ (0, 1] . Let x0 , xi and pi (i = 1, ..., n) be nonnegative real numbers. If
(xi − x0 ) (e
x − xi ) ≥ 0 (i = 1, 2, . . . , n), x
e 6= x0 ,
Pn
where x
e = k=1 pk xk , then
n
X
x
e
(3.4)
pk f (xk ) ≤ min mAf
+ (Pn − 1) Bf (x0 ) ,
m
k=1
(3.3)
Af (e
x) + m (Pn − 1) Bf
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x o
0
m
,
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
where
Pn
A=
k=1
pk (xk − x0 )
,
x
e − x0
B=
x
e
.
x
e − x0
Proof. From the condition (3.3) we may deduce that
x0 ≤ xi ≤ x
e,
(i = 1, ..., n) ,
x
e ≤ xi ≤ x0 ,
(i = 1, ..., n) .
or
Suppose that the first conclusion holds true. We may apply Corollary 2.9 to
obtain
n
x
e
x − x0
x
e−x
1 X
f
,
f (x0 ) + m
pk f (xk ) ≤ min
x
e − x0
m
x
e − x0
Pn k=1
x
e − x x0 x − x0
f
+
f (e
x) .
m
x
e − x0
m
x
e − x0
Since
x
e−x
x − x0
x
e
Pn
f (x0 ) + m
f
x
e − x0
x
e − x0
m
Pn
x
e
x
e
k=1 pk (xk − x0 )
= (Pn − 1)
f (x0 ) + m
f
,
x
e − x0
x
e − x0
m
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
and
x
e − x x0 x − x0
f
+
f (e
x)
Pn m
x
e − x0
m
x
e − x0
x Pn p (x − x )
x
e
k
k
0
0
= m (Pn − 1)
f
+ k=1
f (e
x) ,
x
e − x0
m
x
e − x0
the inequality (3.4) is proved.
The other case is similar.
Corollary 3.4. Let f : [0, ∞) → R be an m−convex function on [0, ∞) with
m ∈ (0, 1] . Let xi and pi (i = 1, ..., n) be nonnegative real numbers. If
0 6= x
e=
n
X
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
pk xk ≥ xi
(i = 1, ..., n),
k=1
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then
n
X
Companion Inequalities to
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-convex and (α, m)-convex
Functions
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pk f (xk )
k=1
≤ min mf
x
e
m
+ (Pn − 1) f (0) , f (e
x) + m (Pn − 1) f (0) .
Proof. This is a special case of Theorem 3.3 for x0 = 0.
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
4.
Inequalities for (α, m)-convex Functions
In this section we first give a general companion inequality to Jensen’s inequality for (α, m) −convex functions.
Theorem 4.1. Let (Ω, A, µ) be a measure space with 0 < µ (Ω) < ∞ and let
f : [0, b] → R, b > 0, be a differentiable (α, m) −convex function on [0, b] with
(α, m) ∈ (0, 1]2 . If u : Ω → [0, b] is a measurable function such that f 0 ◦ u is
in L1 (µ) , then for any ξ, η ∈ [0, b] we have
Z
f (ξ) f 0 (ξ)
1
ξ
(4.1)
+
udµ −
m
α
µ (Ω) Ω
m
Z
Z
1
1
≤
(f ◦ u) dµ ≤ mf (η) +
(u − mη) (f 0 ◦ u) dµ.
µ (Ω) Ω
αµ (Ω) Ω
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
Proof. From Lemma D we know that the inequalities
(4.2)
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αf (x) + f 0 (x) (my − x) ≤ αmf (y) ,
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(4.3)
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αf (y) ≤ αmf (x) + f 0 (y) (y − mx) ,
hold for all x, y ∈ [0, b] . If in (4.2) we let x = ξ and y = u (t) , t ∈ Ω, we get
0
αf (ξ) + f (ξ) (mu (t) − ξ) ≤ αm (f ◦ u) (t) ,
t ∈ Ω.
Integrating over Ω we obtain
Z
Z
0
µ (Ω) αf (ξ) + f (ξ) m udµ − ξµ (Ω) ≤ αm (f ◦ u) dµ,
Ω
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
from which the left hand side of (4.1) immediately follows.
In order to obtain the right hand side of (4.1) we proceed in a similar way:
if in (4.3) we let x = η and y = u (t) , t ∈ Ω, we get
α (f ◦ u) (t) ≤ αmf (η) + (f 0 ◦ u) (t) (u (t) − mη) ,
t ∈ Ω,
so after integration over Ω we obtain
Z
Z
α (f ◦ u) dµ ≤ αmµ (Ω) f (η) + (u − mη) (f 0 ◦ u) dµ,
Ω
Ω
from which the right hand side of (4.1) easily follows.
The following theorem is a variant of Theorem 4.1 for the class of α-starshaped
functions (i.e. (α, 0)-convex functions).
Theorem 4.2. Let (Ω, A, µ) be a measure space with 0 < µ (Ω) < ∞ and let
f : [0, b] → R, b > 0, be a differentiable α-starshaped function. If u : Ω →
[0, b] is a measurable function such that f 0 ◦ u is in L1 (µ) , then we have
Z
Z
1
1
(f ◦ u) dµ ≤
u (f 0 ◦ u) dµ.
µ (Ω) Ω
αµ (Ω) Ω
Proof. Similarly to the proof of Theorem 2.2.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
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Our next corollary gives the integral version of the Dragomir-Goh result [2]
for the class of (α, m)-functions.
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Corollary 4.3. Let all the assumptions of Theorem 4.1 be satisfied. We have
Z
Z
m
1
1
0≤
(f ◦ u) dµ − f
udµ
µ (Ω) Ω
m
µ (Ω) Ω
Z
Z Z
m
m2 − 1
1
m2
f
udµ +
u−
udµ (f 0 ◦ u) dµ.
≤
m
µ (Ω) Ω
αµ (Ω) Ω
µ (Ω) Ω
Proof. If we let ξ and η in (4.1) be defined as
Z
m
ξ=η=
udµ ∈ [0, b] ,
µ (Ω) Ω
we obtain
Z
Z
m
1
1
f
udµ ≤
(f ◦ u) dµ
m
µ (Ω) Ω
µ (Ω) Ω
Z
m
≤ mf
udµ
µ (Ω) Ω
Z Z
m2
1
+
u−
udµ (f 0 ◦ u) dµ.
αµ (Ω) Ω
µ (Ω) Ω
It may be interesting to note here that the variants of Jensen’s inequality and
Slater’s inequality for the class of (α, m)-convex functions are the same as the
variants for the class of m-convex functions (α 6= 0) , i.e., they do not depend
on α.
In the next theorem we give a converse of the integral Jensen’s inequality for
(α, m)-convex functions.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Theorem 4.4. Let (Ω, A, µ) be a measure space with 0 < µ (Ω) < ∞ and let
f : [0, ∞) → R be an (α, m)-convex function on [0, ∞) , with α ∈ (0, 1) and
m ∈ (0, 1] . If u : Ω → [a, b] , 0 ≤ a < b < ∞, is a measurable function such
that f ◦ u is in L1 (µ) , then we have
1
µ (Ω)
Z
(f ◦ u) dµ
Z α
b
b
b − u (t)
1
≤ min mf
+
f (a) − mf
dµ,
m
µ (Ω)
m
b−a
Ω
a i Z u (t) − a α a
1 h
dµ .
mf
+
f (b) − mf
m
µ (Ω)
m
b−a
Ω
If additionally we have f (a) − mf mb ≥ 0, then
α
Z
1
b
b
b−u
(f ◦ u) dµ ≤ mf
+ f (a) − mf
µ (Ω) Ω
m
m
b−a
b
b
u−a
≤ mf
+ f (a) − mf
1−α
,
m
m
b−a
a
or symmetrically, if we have f (b) − mf m
≥ 0, then
Z
a h
a i u − a α
1
(f ◦ u) dµ ≤ mf
+ f (b) − mf
µ (Ω) Ω
m
m
b−a
a h
a i b−u
≤ mf
+ f (b) − mf
1−α
.
m
m
b−a
Ω
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Proof. We can write
(f ◦ u) (t) = f
b − u (t)
b − u (t) b
a+m 1−
,
b−a
b−a
m
t ∈ Ω.
Since f is (α, m)-convex we have
α
α b − u (t)
b − u (t)
b
f (a) + m 1 −
(f ◦ u) (t) ≤
f
b−a
b−a
m
α
b
b − u (t)
b
, t ∈ Ω,
= mf
+ f (a) − mf
m
m
b−a
so after integration over Ω we obtain
Z
1
(4.4)
(f ◦ u) dµ
µ (Ω) Ω
Z α
b
b
b − u (t)
1
≤ mf
dµ.
+
f (a) − mf
m
µ (Ω)
m
b−a
Ω
Analogously, from
(f ◦ u) (t) = f
u (t) − a
u (t) − a a
b+m 1−
,
b−a
b−a
m
1
µ (Ω)
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t ∈ Ω,
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we obtain
(4.5)
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Functions
Z
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(f ◦ u) dµ
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Ω
a
a i
1 h
≤ mf
+
f (b) − mf
m
µ (Ω)
m
Z Ω
u (t) − a
b−a
α
dµ.
Page 25 of 32
J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Suppose now that f (a) − mf mb ≥ 0. We know that the function ϕ :
[0, ∞) → R defined as ϕ (x) = xα , where α ∈ (0, 1] is fixed, is concave on
[0, ∞) , so from the integral Jensen’s inequality we have
α
α
α Z Z
b − u (t)
1
b − u (t)
b−u
1
.
dµ ≤
dµ
=
b−a
µ (Ω) Ω
b−a
µ (Ω) Ω b − a
Using that, from (4.4) we obtain
α
Z
1
b
b
b−u
.
(f ◦ u) dµ ≤ mf
+ f (a) − mf
b−a
µ (Ω) Ω
m
m
On the other hand, from the generalized Bernoulli’s inequality we have
α
u−a
b−u
b−u
,
=1−α
≤1−α 1−
b−a
b−a
b−a
so from (4.4) we may deduce
α
b
b
b−u
mf
+ f (a) − mf
m
m
b−a
b
b
u−a
≤ mf
+ f (a) − mf
1−α
.
m
m
b−a
a
Analogously, if f (b) − mf m
≥ 0, from (4.5) we obtain
Z
a h
a i u − a α
1
(f ◦ u) dµ ≤ mf
+ f (b) − mf
,
µ (Ω) Ω
m
m
b−a
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
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and
mf
a
m
h
+ f (b) − mf
a i u − a α
b−a
a h
a i b−u
≤ mf
.
+ f (b) − mf
1−α
b−a
m
m
m
This completes the proof.
Remark 1. It can be easily seen that the assertion of Theorem 4.4 remains valid
for α = 1, since in this case we directly have
Z
1
b − u (t)
b−u
,
dµ =
b−a
µ (Ω) Ω b − a
Z
u (t) − a
u−a
1
.
dµ =
b−a
µ (Ω) Ω b − a
b
This means
that
in
this
case
the
conditions
f
(a)
−
mf
≥ 0 and f (b) −
m
a
mf m
≥ 0 can be omitted, which implies that for α = 1 Theorem 4.4 gives
the previously obtained result for m-convex functions given in Theorem 2.7.
At the end of this section we give two variants of the weighted Hadamard
inequality for (α, m)-convex functions and also a variant of Hadamard’s inequality.
Corollary 4.5. Let f : [0, b] → R be an (α, m)-convex function on [0, b] with
(α, m) ∈ (0, 1]2 and let g : [a, b] → [0, ∞) be integrable and symmetric about
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
where 0 ≤ a < b < ∞. If f is differentiable and f 0 is in L1 ([a, b]), then
Z b
f (mb) b − a 0
−
f (mb)
g (x) dx
m
2α
a
Z b
Z b
x − ma 0
≤
f (x) g (x) dx ≤
f (x) + mf (a) g (x) dx.
α
a
a
Proof. The proof is similar to that of Corollary 3.1.
a+b
,
2
Corollary 4.6. Let f : [0, ∞) → R be an (α, m)-convex function on [0, ∞) ,
with α ∈ (0, 1) and m ∈ (0, 1] , and let g : [a, b] → [0, ∞) be integrable and
symmetric about a+b
, where 0 ≤ a < b < ∞. If f is in L1 ([a, b]) , then
2
Z b
f (x) g (x) dx
a
Z b α
Z b
b
b−x
b
≤ min mf
g (x) dx + f (a) − mf
g (x) dx,
m
m
b−a
a
a
a Z b
h
a i Z b x − a α
mf
g (x) dx + f (b) − mf
g (x) dx .
m a
m
b−a
a
If additionally we have f (a) − mf mb ≥ 0, then
Z b
Z b
b
1
b
f (x) g (x) dx ≤ mf
+ α f (a) − mf
g (x) dx,
m
2
m
a
a
a
or symmetrically, if we have f (b) − mf m
≥ 0, then
Z b
a
a Z b
1 f (x) g (x) dx ≤ mf
+ α f (b) − mf
g (x) dx.
m
2
m
a
a
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
If f is differentiable, then we also have
Z b
Z b
1
a+b
(4.6)
f m
g (x) dx ≤
f (x) g (x) dx.
m
2
a
a
Proof. If we choose µ to be the Lebesgue measure defined as dµ = g (x) dx,
Ω = [a, b], u (x) = x for all x ∈ [a, b] , then (3.1) is obtained directly from
Theorem 4.4. Note that in this case we have
α
α 1
b−u
u−a
= α.
=
b−a
2
b−a
The inequality (4.6) is a simple consequence of Corollary 4.3.
Corollary 4.7. Let f : [0, ∞) → R be an (α, m)-convex function on [0, ∞) ,
with α ∈ (0, 1) and m ∈ (0, 1] , and let 0 ≤ a < b < ∞. If f is in L1 ([a, b]) ,
then
Z b
b
1
b
1
f (x) dx ≤ min mf
+
f (a) − mf
,
b−a a
m
α+1
m
a
a i
1 h
+
f (b) − mf
.
mf
m
α+1
m
If f is differentiable, then we also have
Z b
1
a+b
1
f m
≤
f (x) dx.
m
2
b−a a
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006
Proof. Directly from Corollary 4.6. We simply choose the function g to be the
constant function 1, and in that case we have
α
α
Z b
Z b
x−a
b−a
b−x
g (x) dx =
g (x) dx =
.
b−a
b−a
α+1
a
a
Variants of other inequalities, which were proved for the class of m-convex
functions in Sections 2 and 3, can be also stated for this class of mappings, but
we omit the details.
Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
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References
[1] A.M. BRUCKNER AND E. OSTROW, Some function classes related to
the class of convex functions, Pacific J. Math., 12 (1962), 1203–1215.
[2] S.S. DRAGOMIR AND C.J. GOH, A counterpart of Jensen’s discrete inequality for differentiable convex mappings and applications in information theory, Math. Comput. Modelling, 24(2) (1996), 1–11.
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Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
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[8] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, Inc. (1992).
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Companion Inequalities to
Jensen’s Inequality for m
-convex and (α, m)-convex
Functions
M. Klaričić Bakula, J. Pečarić and
M. Ribičić
Title Page
Contents
JJ
J
II
I
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J. Ineq. Pure and Appl. Math. 7(5) Art. 194, 2006