TITLE MARCH 2012 - Pakistan Academy of Sciences

Inequalities for Differentiable Functions 15
By solving (2.48), we have
Proof. From Lemma 2, we have
(2.44)
(2.49)
Relations (2.46), (2.47), and (2.49) together imply
(2.44).
Corollary 16. From theorem 15, Let
be differentiable function of , a, b, with a
< b, and if the mapping is s-
convex on for then
(2.45)
By applying H lder inequality on (2.45), we
follow as
Proof. The proof is similar to that of corollary 5.
Theorem 17. Let
be differentiable
function on , a , b with a < b, and
If the mapping is s-concave
on for then
Here
And
(2.46)
(2.47)
(2.50)
Proof. We proceed in a similar way as in theorem
12.
By
, we obtain
Since
(2.48)
(2.51)
Now (2.50) immediately follows from theorem 1.