Certain inequalities for
( , )
α
m
- convex functions
Marjan Dema*
University of Prishtina, Faculty of Electrical and Computer Engineering Bregu i Diellit, p.n., 10000 Prishtinë, Kosova
E-mail: [email protected]
(Received on: 02-07-11; Accepted on: 12-10-11)
________________________________________________________________________________________________
ABSTRACT
In this paper two inequalities of Hadamard type involving two
( , )
α
m
-convex functions are proved.Key words:
( , )
α
m
-convex functions, Hadamard type inequality,( , )
α
m
−
convex function.AMS Subject Classification: 26D15, 26A51.
________________________________________________________________________________________________
INTRODUCTION:
We say that the function
f
is convex on[0, ]
b
if(
(1
) )
( )
(1
) ( )
f tx
+
−
t y
≤
tf x
+
−
t f y
holds for allx y
,
∈
[0, ]
b
andt
∈
[0,1]
.In the paper [1], G. Toader defined
m
−
convexity, an intermediate between the usual convexity and starshaped property, as the following:Definition: 1.1 The function
f
:[0, ]
b
→
is said to bem
−
convex, wherem
∈
[0,1]
, if for everyx y
,
∈
[0, ]
b
andt
∈
[0,1]
we have(
(1
) )
( )
(1
) ( ).
f tx
+
m
−
t y
≤
tf x
+
m
−
t f y
Definition: 1.2 The function
f
: [0, ]
b
→
,b
>
0
, is said to be starshaped if for everyx
∈
[0, ]
b
andt
∈
[0,1]
we have( )
( ).
f tx
≤
tf x
Obviously, for
m
=
1
, we recapture the concept of convex functions defined on[0, ]
b
and form
=
0
we get the concept of starshaped functions on[0, ]
b
.It is interesting to emphasize here that for any
m
∈
(0,1)
there are continuous and differentiable functions which arem
−
convex, but which are not convex in the standard sense. Moreover, in [3], the following fact was proved.Theorem: 1.3 For
m
∈
(0,1)
there is anm
−
convex polynomialf
such thatf
is notn
−
convex for any1.
m
<
n
≤
For example, the function
f
: [0, ]
∞ →
defined as(
4 3 2)
1
( )
5
9
5
12
f x
=
x
−
x
+
x
−
x
is
16 /17
−
convex, but it is notn
−
convex for anyn
∈
(16 / 17,1]
(see [4]).( , )
m
* + ' $ + $ ,
The notion of
m
−
convexity can be further generalized through introduction of another parameterα
∈
[0,1]
in the definition ofm
−
convexity. The class of( , )
α
m
−
convex functions was first introduced in [5] as follows.Definition: 1.4 The function
f
:[0, ]
b
→
,b
>
0
, is said to be( , )
α
m
−
convex, where( , ) [0,1]
α
m
∈
2, if(
(1
) )
( )
(1
) ( )
f tx
+
m
−
t y
≤
t f x
α+
m
−
t
αf y
for all
x y
,
∈
[0, ]
b
andt
∈
[0,1]
.It is clear that for
( , ) {(0, 0), ( , 0), (1, 0), (1, ), (1,1), ( ,1)}
α
m
∈
α
m
α
we obtain the following classes of functions: increasing,α
−
starshaped, starshaped,m
−
convex, convex, andα
−
convex functions.The reader could find some properties of convex and starshaped functions in the paper of S. S. Dragomir [2] and papers cited therein. Also, in the same paper are proved some new inequalities of Hermite-Hadamard type for
m
−
convex functions. The main goal of this paper is to establish two inequalities of Hadamard type involving two( , )
α
m
−
convex functions.Now we pass to the main results.
1. MAIN RESULTS:
Theorem: 2.1 Let
f g
,
: [0, ]
b
→
be two( , )
α
m
−
convex functions with( , ) [0,1]
α
m
∈
2 and0
≤
a
<
b
. If1
,
[ , ]
f g
∈
L a b
, then the following inequality holds
1
1
1
1
1 1 1
( )
(
)
( )
(
)
( )
[(
)
(
) ] ( )
(
)
( )
(
)
( )
(
)
( )
[(
)
(
) ] ( )
(
)
( , ; )
( , ; )
1
(
)
( ) ( )
3
6
(
)
mb
a
mb
a
mb
a
mb
a
mb
a
g a
mb
x
f x dx
mb
a
mg b
mb
a
mb
x
f x dx
mb
a
f a
mb
x
g x dx
mb
a
mf b
mb
a
mb
x
g x dx
mb
a
M a b m
N a b m
mb
x
f x g x dx
mb
a
α α
α α
α
α α
α α
α
α
α
α
+
+
+
+
−
−
−
+
−
−
−
−
+
−
−
+
−
−
−
−
≤
+
+
−
−
,
(2.1)
where
M a b m
( , ; ) :
=
f a g a
( ) ( )
+
m f b g b
2( ) ( )
,N a b m
( , ; ) :
=
m f a g b
[ ( ) ( )
+
f b g a
( ) ( )]
.Proof: From the fact that functions
f
andg
are( , )
α
m
−
convex on[ , ]
a b
, then fort
∈
[0,1]
we have[
(1
) ]
( )
(1
) ( )
f ta
+
m
−
t b
≤
t f a
α+
m
−
t
αf b
[
(1
) ]
( )
(1
) ( ).
g ta
+
m
−
t b
≤
t g a
α+
m
−
t
αg b
Using the inequality
ER
+
FP
≤
EP
+
FR
, which holds forE
≤
F
andP
≤
R
, we obtain[
(1
) ][
( )
(1
) ( )]
[
(1
) ][
( )
(1
) ( )] [ ( )
(1
) ( )][ ( )
(1
) ( )]
[
(1
) ] [
(1
) ]
f ta
m
t b t g a
m
t
g b
g ta
m
t b t f a
m
t
f b
tf a
m
t f b
tg a
m
t g b
f t a
m
t
b g t a
m
t
b
α α
α α
α α α α
+
−
+
−
+
+
−
+
−
≤
+
−
+
−
+
+
−
+
−
or
2 2 2
( )
[
(1
) ]
( )(1
) [
(1
) ]
( )
[
(1
) ]
( )(1
) [
(1
) ]
( ) ( )
(1
)
( ) ( )
(1
) ( ) ( )
(1
) ( ) ( )
[
(1
) ] [
(1
) ].
g a t f ta
m
t b
mg b
t
f ta
m
t b
f a t g ta
m
t b
mf b
t
g ta
m
t b
t f a g a
m
t
f b g b
m
t tf b g a
m
t tf a g b
f t a
m
t
b g t a
m
t
b
α α
α α
α α α α
+
−
+
−
+
−
+
+
−
+
−
+
−
≤
+
−
+
−
* + ' $ + $ ,
Now we integrate both sides of the latter inequality over
[0,1]
, so that we shall obtain
1 1
0 0
1 1
0 0
1 2 2 1 2 1
0 0 0
1 1
0 0
( )
[
(1
) ]
( ) (1
) [
(1
) ]
( )
[
(1
) ]
( ) (1
) [
(1
) ]
( ) ( )
( ) ( ) (1
)
( ) ( ) (1
)
( ) ( ) (1
)
[
g a
t f ta
m
t b dt
mg b
t
f ta
m
t b dt
f a
t g ta
m
t b dt
mf b
t
g ta
m
t b dt
f a g a
t dt
m f b g b
t dt
mf b g a
t tdt
mf a g b
t tdt
f t a
m
α α
α α
α
+
−
+
−
+
−
+
+
−
+
−
+
−
≤
+
−
+
−
+
−
+
+
(1
−
t
α) ] [
b g t a
α+
m
(1
−
t
α) ] .
b dt
(2.2)
Substituting
ta
+
m
(1
−
t b
)
=
x
,
t a
α+
m
(1
−
t
α)
b
=
x
we easy find1
1 0
1
1 0
1
1 0
1
1 0
1
[
(1
) ]
(
)
( )
(
)
1
(1
) [
(1
) ]
[(
)
(
) ] ( )
(
)
1
[
(1
) ]
(
)
( )
(
)
1
(1
) [
(1
) ]
[(
)
(
)
mb
a
mb
a
mb
a
t f ta
m
t b dt
mb
x
f x dx
mb a
t
f ta
m
t b dt
mb a
mb
x
f x dx
mb
a
t g ta
m
t b dt
mb
x
g x dx
mb
a
t
g ta
m
t b dt
mb
a
mb a
α α
α
α α α
α
α α
α
α
α
+
+
+
+
+
−
=
−
−
−
+
−
=
−
−
−
−
+
−
=
−
−
−
+
−
=
−
−
(
) ] ( )
mb
a
mb
x
g x dx
α α
−
−
1 1 1
1 0
1
[
(1
) ] [
(1
) ]
(
)
( ) ( )
(
)
mb
a
f t a
m
t
b g t a
m
t
b dt
mb
x
f x g x dx
mb
a
α α α α α
α
α
−
+
−
+
−
=
−
−
and1 1 1
2 2
0 0 0
1
1
(1
)
,
(1
)
.
3
6
t dt
=
−
t dt
=
t
−
t dt
=
Now inserting above equalities to the inequality (2.2), we immediately obtain (2.1).
Theorem: 2.2 Let
f g
,
:[0, ]
b
→
be twom
−
convex functions withm
∈
[0,1]
and0
≤
a
<
b
. If1
,
[ , ]
f g
∈
L a b
. Then the following inequality holds2
2
( )
( )
1
( ) ( )
2(
)
( , ; )
( , ; )
,
12
6
2
2
mb mb
a a
mb
a
a
mb
a
mb
f
g
g x dx
f x dx
mb
a
mb
a
f x g x dx
mb
a
M a b m
N a b m
a
mb
a
mb
f
g
+
+
+
−
−
≤
−
+
+
+
+
+
(2.3)
where
M a b m
( , ; )
, andN a b m
( , ; )
are same as those in theorem 2.1.Proof: From the fact that
f g
,
are( , )
α
m
−
convex functions on[ , ]
a b
, then fort
∈
[0,1]
we have(1
)
(1
)
2
2
2
[
(1
) ]
[(1
)
]
2
a
mb
ta
m
t b
t a
mtb
f
f
f ta
m
t b
f
t a
mtb
+
+
−
−
+
=
+
+
−
+
−
+
( , )
m
* + ' $ + $ ,
-and
(1
)
(1
)
2
2
2
[
(1
) ]
[(1
)
]
.
2
a
mb
ta
m
t b
t a
mtb
g
g
g ta
m
t b
g
t a
mtb
+
+
−
−
+
=
+
+
−
+
−
+
=
Again, using the inequality
ER
+
FP
≤
EP
+
FR
, which holds forE
≤
F
andP
≤
R
, we obtain[
(1
) ]
[(1
)
]
2
2
[
(1
) ]
[(1
)
]
2
2
[
(1
) ]
[(1
)
] [
(1
) ]
[(1
)
]
2
2
2
2
a
mb g ta
m
t b
g
t a
mtb
f
a
mb
f ta
m
t b
f
t a
mtb
g
f ta
m
t b
f
t a
mtb g ta
m
t b
g
t a
mtb
a
mb
a
mb
f
g
+
+
−
+
−
+
+
+
−
+
−
+
+
+
−
+
−
+
+
−
+
−
+
≤
+
+
+
or
1
{ [
(1
) ]
[(1
)
]}
2
2
1
{ [
(1
) ]
[(1
)
]}
2
2
1
{ [
(1
) ] [
(1
) ]
[(1
)
] [(1
)
]}
4
1
{ [
(1
) ] [(1
)
]
[(1
)
] [
(1
)
4
a
mb
f
g ta
m
t b
g
t a
mtb
a
mb
g
f ta
m
t b
f
t a
mtb
f ta
m
t b g ta
m
t b
f
t a
mtb g
t a
mtb
f ta
m
t b g
t a
mtb
f
t a
mtb g ta
m
t b
+
+
−
+
−
+
+
+
+
−
+
−
+
≤
+
−
+
−
+
−
+
−
+
+
−
−
+
+
−
+
+
−
]}
2
2
a
mb
a
mb
f
+
g
+
+
1
{ [
(1
) ] [
(1
) ]
[(1
)
] [(1
)
]}
4
1
{[ ( )
(1
) ( )][(1
) ( )
( )]}
4
1
{[(1
) ( )
( )][ ( )
(1
) ( )]}
4
2
2
1
{ [
(1
) ] [
(1
) ]
[(1
4
f ta
m
t b g ta
m
t b
f
t a
mtb g
t a
mtb
tf a
m
t f b
t g a
mtg b
t f a
mtf b
tg a
m
t g b
a
mb
a
mb
f
g
f ta
m
t b g ta
m
t b
f
≤
+
−
+
−
+
−
+
−
+
+
+
−
−
+
+
−
+
+
−
+
+
+
=
+
−
+
−
+
2
2 2
)
] [(1
)
]}
1
2 (1
)[ ( ) ( )
( ) ( )]
4
1
[
(1
) ][ ( ) ( )
( ) ( )]
4
.
2
2
t a
mtb g
t a
mtb
t
t
f a g a
m f b g b
m t
t
f a g b
f b g a
a
mb
a
mb
f
g
−
+
−
+
+
−
+
+
+
−
+
+
+
* + ' $ + $ , ) Hence, integrating the above inequality over
[0,1]
, we get1 0 1 0 1
0
1 2
0
1
{ [
(1
) ]
[(1
)
]}
2
2
1
{ [
(1
) ]
[(1
)
]}
2
2
1
{ [
(1
) ] [
(1
) ]
[(1
)
] [(1
)
]}
4
1
[ ( ) ( )
( ) ( )]2
(1
)
4
1
4
a
mb
f
g ta
m
t b
g
t a
mtb dt
a
mb
g
f ta
m
t b
f
t a
mtb dt
f ta
m
t b g ta
m
t b
f
t a
mtb g
t a
mtb dt
f a g a
m f b g b
t
t dt
m
+
+
−
+
−
+
+
+
+
−
+
−
+
≤
+
−
+
−
+
−
+
−
+
+
+
−
+
1 2 20
[ ( ) ( )
( ) ( )] [
(1
) ]
.
2
2
f a g b
f b g a
t
t
dt
a
mb
a
mb
f
g
+
+
−
+
+
+
Now, we substitute
ta
+
m
(1
−
t b
)
=
x
and(1
−
t a
)
+
mtb
=
x
to the above integrals, and taking into account that1 2 2 1
0 0
2
1
[
(1
) ]
,
2 (1
)
3
3
t
+
−
t
dt
=
t
−
t dt
=
1 0 1 0
[
(1
) ] [
(1
) ]
1
[(1
)
] [(1
)
]
mb( ) ( )
a
f ta
m
t b g ta
m
t b dt
f
t a
mtb g
t a
mtb dt
f x g x dx
mb
a
+
−
+
−
=
−
+
−
+
=
−
1 1
0 0
1 1
0 0
1
[
(1
) ]
[(1
)
]
( )
1
[
(1
) ]
[(1
)
]
( )
mb
a
mb
a
f ta
m
t b dt
f
t a
mtb dt
f x dx
mb
a
g ta
m
t b dt
g
t a
mtb dt
g x dx
mb
a
+
−
=
−
+
=
−
+
−
=
−
+
=
−
we finally obtain
2
2
( )
( )
1
( , ; )
( , ; )
( ) ( )
.
2(
)
12
6
2
2
mb mb
a a
mb
a
a
mb
a
mb
f
g
g x dx
f x dx
mb
a
mb a
M a b m
N a b m
a
mb
a
mb
f x g x dx
f
g
mb a
+
+
+
−
−
+
+
≤
+
+
+
−
With this the proof of the theorem is completed.
REFERENCES:
[1] G. H. Toader, Some generalizations of the convexity, Proc. Colloq. Approx. Optim., Cluj - Napoca (Romania), 1984, 329--338.
[2] S. S. Dragomir, On some new inequalities of Hermite-Hadamard type for
m
−
convex functions, Tamkang Jour. of Math. Vol. 33, No. 1, 2002, 45--55.[3] S. Toader, The order of a star--convex function, Bull. Applied and Comp. Math., 85-B (1998), BAM--1473, 347–-350.
[4] P.T. Mocanu, I. erb and G. Toader, Real star-convex functions, Studia Univ. Babes-Bolyai, Math., 42(3) (1997), 65–-80.
[5] V.G. Mihe an, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca (Romania) (1993).
