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302 Muhammad Iqbal et al
Kirmaci et al [7] established the following
new Hadamard-type inequality for concave
functions:
Theorem 4: Let f : I RR
be differentiable
function on I o where a, bI
with a b . If
q
f ' , q 1
is concave on [ a , b]
, then
b
f(
a)
f ( b)
1
b a q 1
f ( x)
dx
2 b a a 4 2q
1
1
1
q
3a
b a 3b
f ' f '
(1.5)
4 4
Alomari et al [2] established the following
Hadamard-type inequality for concave functions:
Theorem 5: Let f : I RR
be differentiable
function on I o where a, bI
with a b . If
q
f ' , q 1
is concave on [ a , b]
, then
1
b
a b
b
f ( x)
dx f
a
a 2
b a q 1
4 2q
1
1
1
q
3a
b a 3b
f ' f '
(1.6)
4 4
Definition 2: A function f : I RR
is said to be
Godunova-Levin function or f belongs to class
Q (I) if f is non-negative and for all x, yI
and
t (0,1), the inequality holds [3]:
1 1
f ( t x (1 t)
y)
f ( x)
f ( y).
t 1 t
Definition 3: A function f : R [0,
]
is said to
be a P function or that f belongs to the class
P (I) , if for all x, yR
and t (0,1)
[3]. we have :
f ( t x (1
t)
y)
f ( x)
f ( y).
Definition 4: [5] Let s be a real number,
s (0, 1]. A function f :[0,
)
[0,
)
is said to
be s function(in the second sense) or f belongs
2
to the class K s , if for all x , y[0,
)
and t[0,1] ,
the following inequality holds:
s
f ( tx (1 t)
y)
t
f ( x)
(1 t)
f ( y).
Varošane [13] defined a large class of nonnegative
functions, the so-called h convex
s
functions. This class contains several well-known
classes of functions such as non-negative convex
functions, s-convex in the second sense,
Godunova-Levin functions and P-functions. The
definition of an h-convex function is stated below:
Definition 5: Let h : JRR
be a non-negative
function. A non-negative function f : I RR
is
said to be h convex function (or f belongs to
class SX ( h,
I)
) if for all x, yI
and t (0,1)
, the
following inequality holds:
f ( t x (1 t)
y)
h(
t)
f ( x)
h(1
t)
f ( y).
If the inequality is reversed then f is said to be
h concave function (or f SV( h,
I)
).
Remark 1: It is noted that if:
h( t)
t , then all the non-negative convex
functions belong to the class SX ( h,
I)
and all
non-negative concave functions belong to the
class SV ( h,
I)
.
1
h( t)
, then SX( h,
I)
Q(
I)
.
t
h ( t)
1, then SX( h,
I)
P(
I)
.
s
2
( h,
I)
K s
h( t)
t , where s (0,1)
, then SX .
Some interesting and important inequalities for
h convex functions can be found in [11,12].
In [12], Sarikaya et. al. established a new
Hadamard-type inequality for h convex
functions.
Theorem 6: Let f SX( h,
I)
, a, bI
with a b
and f L[ a,
b]
, then
1 a
b 1
f
2 h(1/2) 2 b
a
[ f( a) f( b)] h( t)
dt
1
0
b
a
f( x)
dx
In this paper new results of Hadamard-type on the
basis of two established lemmas for h convex
functions will be presented.
2. MAIN RESULTS
Lemma 1: Let f : I RR
be differentiable
o
function on I where a, bI
with a b . If
f ' L[
a,
b]
and , [0,
)
with 0
, then