TITLE MARCH 2012 - Pakistan Academy of Sciences

12 Muhammad Muddassar et al
Proof. From Lemma 1,
Proof. The proof is similar to that of corollary 5.
(2.21)
By applying H lder’s inequality on (2.21) for q >
1, we have
Theorem 9. Let the assumptions of theorem 2 are
satisfied with p > 1 such that . If the
mapping
is s-concave on [a, b], then
(2.22)
By s-convexity of on [a, b] for all .
(2.22) can be written as:
Proof. We proceed similarly as in theorem 6.
By of we obtain
(2.25)
(2.26)
Now (2.25) immediately follows from theorem 1.
Here,
(2.23)
Theorem 10. Let the assumptions of theorem 2
are satisfied, then
(2.24)
By (2.23) and (2.24) in (2.21), we get (2.20).
Proof. From Lemma 2.
(2.27)
Corollary 8. From theorem 7, the assumptions of
theorem 4 are satisfied with p > 1 such that
. If the mapping is -convex on
then