TITLE MARCH 2012 - Pakistan Academy of Sciences

Inequalities for Differentiable Functions 13
(2.32)
(2.28)
By applying H lder Inequality in (2.32), we have
By using s-convexity of on [a, b] for all
on right side of (2.28), we have
(2.33)
(2.29)
But
But
(2.30)
By (2.29) and (2.30) we get (2.27).
Theorem 11. Let the assumptions of Theorem 2
are satisfied. Furthermore, if the mapping is
concave on [a, b] for q > 1, then
(2.34)
Since is concave on [a, b] so by using
Jensen’s Integral Inequality on first integral in
R.H.S., we have
Proof. From Lemma 2, we have
(2.31)
= (2.35)
Hence (2.33), (2.34) and (2.35) together imply
(2.31).
Theorem 12. Let the assumptions of Theorem 2
are satisfied. Furthermore, if the mapping is
s-convex on [a, b] for then