Ostrowski ’ s inequalities for functions whose first derivatives are s-logarithmically preinvex in the second sense

In this paper, some Ostrowski’s inequalities for functions whose first derivatives are s-logarithmically preinvex in the second sense are established.


Introduction
In 1938, A. M. Ostrowski proved an interesting integral inequality, estimating the absolute value of the derivative of a differentiable function by its integral mean as follows This is well-known Ostrowski's inequality.In recent years, a number of authors have written about generalizations, extensions and variants of such inequalities one can see [4,5,6,7,8,15] and the reference cited therein.
In recent years, lot of efforts have been made by many mathematicians to generalize the classical convexity.Hanson in [3], introduced a new class of generalized convex functions, called invex functions.In [2], the authors gave the concept of preinvex functions which is special case of invexity.Pini [13], Noor [10,11], Yang and Li [18] and Weir [17], have studied the basic properties of the preinvex functions and their role in optimization, variational inequalities and equilibrium problems.
Theorem 1.5.Let K ⊆ R be an open invex subset with respect to η : for every a, b ∈ K the following inequality holds: Theorem 1.6.Let K ⊆ R be an open invex subset with respect to η : is log-preinvex on K then, for every a, b ∈ K the following inequality holds: . Motivated by the above results, in this paper we establish some new Ostrowski type inequalities for functions whose first derivatives are logarithmically s-preinvex in the second sense.

Preliminaries
In this section we recall some concepts of convexity that are well known in the literature.Throughout this section I is an interval of R.

Definition 2.3 ([17]
).A set K is said to be invex at x with respect to η, if K is said to be an invex set with respect to η if K is invex at each x ∈ K.

Definition 2.4 ([10]
).A positive function f on the invex set K is said to be logarithmically preinvex function with respect to η, if is said to be s-logarithmically preinvex function in the second sense with respect to η, if holds for all x, y ∈ K and t ∈ [0, 1].
The following lemmas are essential to establishing our main results.

Main Results
In what follows we assume that K ⊆ [0, ∞) be an invex subset with respect to the bifunction η : is s-logarithmically preinvex function in the second sense for some fixed s ∈ (0, 1] with |f (a)| = 0, then for all x ∈ [a, a + η(b, a)] we have the following inequality s ln λ where Proof.From Lemma 2.2, and property of modulus, we have Since |f | is s-logarithmically preinvex function, we deduce From Lemma 2.1, we have where λ and N (s,λ) are defined by ( 1) and ( 2) respectively.Substituting (4) into (3), we obtain Clearly, in the case where λ = 1, we have (6) x−a η(b,a) 0 s ln λ .
Corollary 3.2.In Theorem 3.1, if we choose η (b, a) = b − a, we have the following inequality Moreover if we choose s = 1 we get the following inequality  we have the following inequality Moreover if we choose s = 1 we get the following inequality Theorem 3.2.Let f : K → (0, ∞) be a differentiable function such that f ∈ L ([a, a + η (b, a)]), and let q > 1 with 1 p + 1 q = 1.If |f | q is s-logarithmically preinvex function in the second sense for some fixed s ∈ (0, 1] with |f (a)| = 0, we have the following inequality (s,q,λ) x−a η(b,a) where λ is defined as in (1), and Proof.From Lemma 2.2, property of modulus, and Hölder's inequality, we have x−a η(b,a) Using the fact that |f | q is s-logarithmically preinvex and Lemma 2.1, we obtain x−a η(b,a) For λ = 1, (11) becomes qs ln λ where λ and N (s,q,λ) are defined as in ( 1) and ( 10) respectively, and the fact that qs ln λ .

2
, then we obtain the following midpoint inequality Remark 3.3.Theorem 3.2 will be reduces to Theorem 11 from [8], and Corollary 3.4 will be reduces to Corollary 12 from [8] and Theorem 4.2 from [14] if we put s = 1.
Corollary 3.5.In Theorem 3.2, if we choose η (b, a) = b−a, then we obtain the following inequality Ostrowski's inequalities for functions whose first derivatives. . .
Moreover if we choose s = 1 we get the following inequality Moreover if we choose s = 1 we get the following inequality  (qs ln λ) 2   1 q where λ and N (s,q,λ) are defined as in (1) and (10) respectively.
Proof.From Lemma 2.2, property of modulus, power mean inequality, slogarithmically preinvexity of |f | q , and Lemma 1, we get Ostrowski's inequalities for functions whose first derivatives. . .

Theorem 1 . 1 ( 4 + x − a+b 2 2 (
[9]).Let I ⊆ R be an interval.Let f : I → R, be a differentiable mapping in the interiorI • of I, and a, b ∈ I • with a < b.If |f | ≤ M for all x ∈ [a, b], then f (x) − 1 b − a b a f (t) d t ≤ M (b − a) 1 b − a) 2 , x ∈ [a, b] .