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304 Muhammad Iqbal et al
q
Theorem 8: Let f ' be h convex function and
the assumptions of Lemma 1 hold, then
f( a) f( b) 1
ba
a
1
1
p
b
a
1/ p
( p 1)
0
b
f( x)
dx
q
q
f '( a) h(1 t) f '( b) h( t)
dt
1
1
1
q
+ p {| f '( a) | h(1 t)
q
| f '( b)| h( t)}
dt
1
q
Proof: By applying Hölder’s inequality and
q
convexity on f ' , we have
1
q
(2.6)
h
Putting the above inequalities in (2.5) we get
(2.6).
Corollary 1: For , and convex function
inequality (2.6) reduces as:
f ( a)
f ( b)
1
2 b a
b
a
f ( x)
dx
3 f '( a) f '( b)
b a
4
4( p 1)
f '( a) 3 f '( b)
4
1
q q q
1/ p
1
q q q
Theorem 8 may be extended to be as follows:
q
Theorem 9: Let f ' be h convex function and
the assumptions of Lemma 1 hold, then
t
0
t t
f ' a
b
dt
f ( a)
f ( b)
1
b a
b
a
f ( x)
dx
1
q
p
p t t
t dt
f ' a b dt
0
0
( )
1
q
q
f '( a) h(1 t)
dt
11/ p
1/ q
0
1/ p
( p 1)
q
Analogously
0
f '( b) h( t)
dt
0
t
t
t f ' a
b
dt
( )
q
f '( a) h(1 t)
dt
11/ p
1/ q
0
1/ p
( p 1)
q
f '( b) h( t)
dt
0
1
q
1
q
1
q
q
b a
t {| f '( a) | h(1 t)
1/ p
2
+
0
q
| f '( b)| h( t)}
1
q
1 q
t {| f '( a) | h(1 t)
(2.7)
q
| f '( b)| h( t)}
Proof. The proof of this theorem is same as proof
2
of Theorem 8 and using facts that 1
and
2
( )
2
( )
reader.
2
1. However, the details are left to the
Now, we give the following Hadamard-type
inequality for h concave mappings.
Theorem 10: Let
q
f ' be h concave function
and the assumptions of Lemma 1 hold, then