Some Ostrowski's Type Inequalities for Functions whose Second Derivatives are s-Convex in the Second Sense and Applications

Some new inequalities of Ostrowski type for twice differentiable mappings whose derivatives in absolute value are s-convex in the second sense are given.Applications for special means are also provided.


INTRODUCTION
In 1938, Ostrowski proved the following integral inequality [12]: Then, the inequality holds: for all x ∈ [a, b] . The constant 1 4 is sharp in the sense that it cannot be replaced by a smaller one.
The class of s−convexity in the second sense is defined in the following way [9]: a function f : [0, ∞) → R is said to be s−convex in the second sense if for all x, y ∈ [0, ∞), t ∈ [0, 1] and some fixed s ∈ (0, 1].This class is usually denoted by K 2 s . In [10], Dragomir and Fitzpatrick proved te Hadamard's inequality for s−convex functions in the second sense: is an s-convex function in the second sense, where s ∈ (0, 1), and let a, b ∈ [0, ∞), a < b. If f ∈ L 1 ([a, b]), then the following inequalities hold: The constant k = 1 s+1 is the best possible in the second inequality in (1.1). In [3], Cerone et.al. proved the following inequalities of Ostrowski type and Hadamard type, respectively.
|f ′′ (t)| < ∞. Then we have the inequality: Corollary 1. Under the above assumptions, we have the mid-point inequality: In this article, we establish new Ostrowski's type inequalities for s−convex functions in the second sense and using this results we note some applications to special means.

Main Results
In order to establish our main results we need the following Lemma.
Proof. By integration by parts, we have the following identity Using the change of the variable u = tx+(1−t)a for t ∈ [0, 1] and by multiplying the both sides (2.2) by (x−a) 3 2(b−a) , we obtain Similarly, we observe that Thus, adding (2.3) and (2.4) we get the required identity (2.1).
The following result may be stated: for some fixed s ∈ (0, 1], then the following inequality holds: Proof. From Lemma 1 and since |f ′′ | is s−convex, then we have where we have used the fact that .
This completes the proof. .
Here, by simple computation shows that Remark 1. If in Corollary 2 we choose s = 1, then we recapture the inequality (1.2).
Corollary 3. If in Corollary 2 we choose x = a+b 2 , then we get the mid-point inequality .
for some fixed s ∈ (0, 1], p, q > 1 and 1 p + 1 q = 1, then the following inequality holds: Proof. Suppose that p > 1. From Lemma 1 and using the Hölder inequality, we have Since |f ′′ | q is s−convex in the second sense, then we have Therefore, we have where 1 p + 1 q = 1, which is required. Corollary 4. Under the above assumptions we have the following inequality: This follows by Theorem 5, choosing |f ′′ (x)| ≤ M, M > 0.
Corollary 5. With the assumptions in Corollary 4, one has the mid-point inequality:

M.
This follows by Corollary 4, choosing x = a+b 2 .
Corollary 6. With the assumptions in Corollary 4, one has the following perturbed trapezoid like inequality: This follows using Corollary 4 with x = a, x = b, adding the results and using the triangle inequality for the modulus.
for some fixed s ∈ (0, 1] and q ≥ 1, then the following inequality holds: Proof. Suppose that q ≥ 1. From Lemma 1 and using the well known power mean inequality, we have Since |f ′′ | q is s−convex in the second sense, we have Therefore, we have Corollary 7. Under the above assumptions we have the following inequality This follows by Theorem 6, choosing |f ′′ (x)| ≤ M, M > 0.
Corollary 8. With the assuptions in Corollary 7, one has the mid-point inequality: This follows by Corollary 7, choosing x = a+b 2 . Remark 2. If in Corollary 8 we choose s = 1 and q = 1, then we have the following inequality: which is the inequality (1.3).

M.
This follows using Corollary 7 with x = a, x = b, adding the results and using the triangle inequality for the modulus.
Remark 3. All of the above inequalities obviously hold for convex functions. Simply choose s = 1 in each of those results to get desired results.
The following result holds for s−concave.
q is s−concave in the second sense on [a, b] for some fixed s ∈ (0, 1], p, q > 1 and 1 p + 1 q = 1, then the following inequality holds: Proof. Suppose that q > 1. From Lemma 1 and using the Hölder inequality, we have Since |f ′′ | q is s−concave in the second sense, using the inequality (1.1) A combination of (2.11) and (2.12) inequalities, we get This completes the proof.
Corollary 10. If in (2.10), we choose x = a+b 2 , then we have For instance, if s = 1, then we have ) .

Applications to Some Special Means
Let us recall the following special means: (1) The arithmetic mean: (2) The Identric mean: (c) The generalized log-mean: The following simple relationship is well known in the literature It is known that L p is monotonic nondecreasing in p ∈ R with L o := I. Now, using the results of Section 2, we give some applications to special means of positive real numbers.