Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 5, Article 194, 2006
COMPANION INEQUALITIES TO JENSEN’S INEQUALITY FOR m-CONVEX
AND (α, m)-CONVEX FUNCTIONS
M. KLARIČIĆ BAKULA, J. PEČARIĆ, AND M. RIBIČIĆ
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
milica@pmfst.hr
FACULTY OF T EXTILE T ECHNOLOGY
U NIVERSITY OF Z AGREB
P IEROTTIJEVA 6, 10000 Z AGREB
C ROATIA
pecaric@hazu.hr
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF O SIJEK
T RG L JUDEVITA G AJA 6, 31000 O SIJEK
C ROATIA
mihaela@mathos.hr
Received 21 April, 2006; accepted 06 December, 2006
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: m-convex functions, (α, m)-convex functions, Jensen’s inequality, Slater’s inequality, HermiteHadamard’s inequalities, Giaccardi’s inequality, Fejér’s inequalities.
2000 Mathematics Subject Classification. 26A51, 26B25.
1. I NTRODUCTION
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
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
117-06
2
M. K LARI ČI Ć BAKULA , J. P E ČARI Ć ,
AND
M. R IBI ČI Ć
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.
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.
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
f (x) =
x4 − 5x3 + 9x2 − 5x
12
16
is 17 -convex, but it is not m-convex for any m ∈ 16
,
1
(see [7]).
17
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
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O N J ENSEN ’ S INEQUALITY
FOR
m- CONVEX AND (α, m)- CONVEX FUNCTIONS
3
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) ,
for all x ∈ (a, b) , where λ ∈ R depends on x0 and is given by λ = f 0 (x0 ) when f 0 (x0 ) exists,
and λ ∈ f−0 (x0 ) , f+0 (x0 ) when f−0 (x0 ) 6= f+0 (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.
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
(b) the class of all (α, m)-convex functions on [0, b] for which f (0) ≤ 0.
Denote by Km
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 mconvex functions, as well as Slater’s inequality, 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.
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
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4
M. K LARI ČI Ć BAKULA , J. P E ČARI Ć ,
AND
M. R IBI ČI Ć
2. C OMPANION I NEQUALITIES TO J ENSEN ’ S I NEQUALITY FOR m- CONVEX
F UNCTIONS
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) ,
(2.3)
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
f (ξ) + f 0 (ξ) (mu (t) − ξ) ≤ m (f ◦ u) (t) ,
t ∈ Ω.
Integrating over Ω we obtain
0
Z
µ (Ω) f (ξ) + f (ξ) m
Z
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
Ω
from which the right hand side of (2.1) easily follows.
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.
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
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
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O N J ENSEN ’ S INEQUALITY
FOR
m- CONVEX AND (α, m)- CONVEX FUNCTIONS
5
Proof. As a special case of Lemma D for m = 0 we obtain
f (x) ≤ xf 0 (x) .
After putting
t ∈ Ω,
x = u (t) ,
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] with
P
m ∈ (0, 1] . Let p1 , ..., pn be nonnegative real numbers such that Pn = ni=1 pi 6= 0 and let
xi ∈ [0, b] be given real numbers. Then for any ξ, η ∈ [0, b] we have
!
n
X
f (ξ)
1
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
Ω = {1, 2, ..., n} ,
µ ({i}) = pi , i = 1, 2, ..., n,
u (i) = xi , i = 1, 2, ..., n.
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
m2 − 1
m
1
m2
≤
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
m
ξ=η=
µ (Ω)
Z
udµ ∈ [0, b] ,
Ω
we obtain
Z
Z
1
m
1
f
udµ ≤
(f ◦ u) dµ
m
µ (Ω) Ω
µ (Ω) Ω
Z
Z Z
m
1
m2
≤ mf
udµ +
u−
udµ (f 0 ◦ u) dµ.
µ (Ω) Ω
µ (Ω) Ω
µ (Ω) Ω
Our next result is the integral Jensen’s inequality for m-convex functions.
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
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6
M. K LARI ČI Ć BAKULA , J. P E ČARI Ć ,
AND
M. R IBI ČI Ć
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
0
ΩR
∈ [0, b] ,
(2.4)
(f ◦ u) dµ 6= 0,
m Ω (f 0 ◦ u) dµ
Ω
then we have
1
µ (Ω)
u (f 0 ◦ u) dµ
m Ω (f 0 ◦ u) dµ
R
Z
(f ◦ u) dµ ≤ mf
Ω
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
since in this case
R
Z
Z u (f 0 ◦ u) dµ
0
Ω
(u − mη) (f ◦ u) dµ =
u− R
(f 0 ◦ u) dµ = 0.
0 ◦ u) dµ
(f
Ω
Ω
Ω
RIf m 0= 1 Theorem 2.6 recaptures Slater’s result from [9]: if f is convex and increasing and
if Ω (f ◦ u) dµ 6= 0 we have
R
Z
u (f 0 ◦ u) dµ
1
Ω
R
=
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
≤ min
f (a) + m
f
,m
f
+
f (b) ,
b−a
b−a
m
b−a
m
b−a
where
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
1
u=
µ (Ω)
Z
udµ.
Ω
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O N J ENSEN ’ S INEQUALITY
FOR
m- CONVEX AND (α, m)- CONVEX FUNCTIONS
7
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
b
(f ◦ u) (t) ≤
f (a) + m
f
,
t ∈ Ω,
b−a
b−a
m
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
a
f (a) + mf mb mf m
+ f (b)
1
(f ◦ u) dµ ≤ min
,
.
µ (Ω) Ω
2
2
Proof. Since u is symmetric about
a+b
2
we have
u (t) = a + b − u (t) =
u (t) − a
b − u (t) b
a+m
b−a
b−a m
from which we get
(2.7)
u−a
b−u
(f ◦ u) dµ ≤ µ (Ω)
f (a) + m
f
b−a
b−a
Ω
Z
b
m
.
If we add (2.6) to (2.7) and then divide the sum by 2µ (Ω) we obtain
Z
1
1 b−u
u−a
b
(f ◦ u) dµ ≤
f (a) + m
f
µ (Ω) Ω
2 b−a
b−a
m
u−a
b−u
b
+
f (a) + m
f
b−a
b−a
m
b
f (a) + mf m
=
.
2
Analogously we obtain
Z
a
mf m
+ f (b)
1
(f ◦ u) dµ ≤
.
µ (Ω) Ω
2
Corollary 2.9. Let f : [0, ∞) → R be an m−convex function
P on [0, ∞) with m ∈ (0, 1] .
Let p1 , ..., pn be nonnegative real numbers such that Pn = ni=1 pi 6= 0 and let xi ∈ [a, b] ,
0 ≤ a < b < ∞, be given real numbers. Than we have
n
1 X
b−x
x−a
b
b−x a x−a
pi f (xi ) ≤ min
f (a) + m
f
,m
f
+
f (b) ,
Pn i=1
b−a
b−a
m
b−a
m
b−a
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
http://jipam.vu.edu.au/
8
M. K LARI ČI Ć BAKULA , J. P E ČARI Ć ,
AND
M. R IBI ČI Ć
where
n
1 X
x=
pi xi .
Pn i=1
Proof. The proof is analogous to the proof of Corollary 2.3.
3. S OME F URTHER R ESULTS
In this section we first show that the Fejér inequalities [4] (i.e., weighted Hermite-Hadamard’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
, where 0 ≤ a < b < ∞. If f is
2
0
1
differentiable and f is in L ([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
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
Rb
Z
xg (x) dx
1
a+b
udµ = Ra b
=
.
µ (Ω) Ω
2
g
(x)
dx
a
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
, where 0 ≤ a < b < ∞. If f is in
2
1
L ([a, b]), then
(
)Z
Z b
a
b
f (a) + mf mb mf m
+ f (b)
(3.1)
f (x) g (x) dx ≤ min
,
g (x) dx.
2
2
a
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
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.
In two following theorems we prove inequalities of Giaccardi and Petrović for m-convex
functions.
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
(3.3)
(xi − x0 ) (e
x − xi ) ≥ 0
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
(i = 1, 2, . . . , n),
x
e 6= x0 ,
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O N J ENSEN ’ S INEQUALITY
where x
e=
Pn
k=1
n
X
(3.4)
FOR
m- CONVEX AND (α, m)- CONVEX FUNCTIONS
9
pk xk , then
pk f (xk )
k=1
x x
e
0
+ (Pn − 1) Bf (x0 ) , Af (e
x) + m (Pn − 1) Bf
,
≤ min mAf
m
m
where
Pn
k=1
A=
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
1 X
pk f (xk )
Pn k=1
x
e−x
x − x0
x
e
x
e − x x0 x − x0
f (x0 ) + m
f
,m
f
+
f (e
x) .
≤ min
x
e − x0
x
e − x0
m
x
e − x0
m
x
e − x0
Since
Pn
x
e
m
Pn
x
e
x
e
k=1 pk (xk − x0 )
= (Pn − 1)
f (x0 ) + m
f
,
x
e − x0
x
e − x0
m
x
e−x
x − x0
f (x0 ) + m
f
x
e − x0
x
e − x0
and
x
e − x x0 x − x0
f
+
f (e
x)
Pn m
x
e − x0
x
e − x0
m
x x
e
0
= m (Pn − 1)
f
+
x
e − x0
m
Pn
k=1
pk (xk − x0 )
f (e
x) ,
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
pk xk ≥ xi
(i = 1, ..., n),
k=1
then
n
X
pk f (xk ) ≤ min mf
k=1
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.
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
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M. K LARI ČI Ć BAKULA , J. P E ČARI Ć , AND M. R IBI ČI Ć
4. I NEQUALITIES FOR (α, m)- CONVEX F UNCTIONS
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
Proof. From Lemma D we know that the inequalities
(4.2)
αf (x) + f 0 (x) (my − x) ≤ αmf (y) ,
(4.3)
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
αf (ξ) + f 0 (ξ) (mu (t) − ξ) ≤ αm (f ◦ u) (t) ,
t ∈ Ω.
Integrating over Ω we obtain
0
Z
µ (Ω) αf (ξ) + f (ξ) m
Z
udµ − ξµ (Ω) ≤ αm (f ◦ u) dµ,
Ω
Ω
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
from which the right hand side of (4.1) easily follows.
Ω
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.
Our next corollary gives the integral version of the Dragomir-Goh result [2] for the class of
(α, m)-functions.
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O N J ENSEN ’ S INEQUALITY FOR m- CONVEX
AND
(α, m)- CONVEX FUNCTIONS
11
Corollary 4.3. Let all the assumptions of Theorem 4.1 be satisfied. We have
Z
Z
1
m
1
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
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
1
m
1
f
udµ ≤
(f ◦ u) dµ
m
µ (Ω) Ω
µ (Ω) Ω
Z
m
udµ
≤ mf
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.
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
Z
1
(f ◦ u) dµ
µ (Ω) Ω
Z α
b
1
b
b − u (t)
≤ min mf
+
f (a) − mf
dµ,
m
µ (Ω)
m
b−a
Ω
a
a i Z u (t) − a α 1 h
mf
+
f (b) − mf
dµ .
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
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
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M. K LARI ČI Ć BAKULA , J. P E ČARI Ć , AND M. R IBI ČI Ć
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
b − u (t)
f (a) + m 1 −
f
(f ◦ u) (t) ≤
b−a
b−a
m
α
b
b
b − u (t)
= mf
+ f (a) − mf
, t ∈ Ω,
m
m
b−a
so after integration over Ω we obtain
Z α
Z
1
b
1
b
b − u (t)
(4.4)
(f ◦ u) dµ ≤ mf
+
f (a) − mf
dµ.
µ (Ω) Ω
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
t ∈ Ω,
we obtain
a i Z u (t) − a α
1 h
(4.5)
(f ◦ u) dµ ≤ mf
+
f (b) − mf
dµ.
m
µ (Ω)
m
b−a
Ω
Ω
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
1
b − u (t)
1
b − u (t)
b−u
dµ ≤
dµ
=
.
µ (Ω) Ω
b−a
µ (Ω) Ω b − a
b−a
1
µ (Ω)
Z
a
Using that, from (4.4) we obtain
α
Z
1
b
b
b−u
(f ◦ u) dµ ≤ mf
+ f (a) − mf
.
µ (Ω) Ω
m
m
b−a
On the other hand, from the generalized Bernoulli’s inequality we have
α
b−u
b−u
u−a
≤1−α 1−
=1−α
,
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
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O N J ENSEN ’ S INEQUALITY FOR m- CONVEX
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(α, m)- CONVEX FUNCTIONS
13
and
mf
a
m
h
+ f (b) − mf
a i u − a α
b−a
a i a h
b−u
.
≤ mf
+ f (b) − mf
1−α
b−a
m
m
m
This completes the proof.
Remark 4.5. 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
b − u (t)
b−u
1
,
dµ =
b−a
µ (Ω) Ω b − a
Z
1
u (t) − a
u−a
dµ =
.
µ (Ω) Ω b − a
b−a
a
This means that in this case the conditions f (a) − mf mb ≥ 0 and f (b) − 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 4.6. 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 a+b
, where 0 ≤ a < b < ∞. If f
2
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.
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 g : [a, b] → [0, ∞) be integrable and symmetric about a+b
, where
2
1
0 ≤ a < b < ∞. If f is in L ([a, b]) , then
Z b
f (x) g (x) dx
a
Z b
Z b α
b
b
b−x
g (x) dx + f (a) − mf
g (x) dx,
≤ min mf
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
J. Inequal. Pure and Appl. Math., 7(5) Art. 194, 2006
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14
M. K LARI ČI Ć BAKULA , J. P E ČARI Ć , AND M. R IBI ČI Ć
If f is differentiable, then we also have
Z b
Z b
a+b
1
f m
g (x) dx ≤
f (x) g (x) dx.
(4.6)
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 4.8. 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
1
b
1
b
f (x) dx ≤ min mf
+
f (a) − mf
,
b−a a
m
α+1
m
a
a i
1 h
mf
+
f (b) − 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
Proof. Directly from Corollary 4.7. 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.
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