Department of Mathematics, Hanseo University, Chungnam-do, Seosan-si 356-706, Republic of Korea
The author establish several Hermite-Hadamard and Simpson-like
type inequalities for mappings whose first derivative in absolute value aroused
to the th () power are ()-convex. Some applications to special means
of positive real numbers are also given.
1. Introduction
Recall that, for some fixed and , a mapping is said to be -convex on an interval if the inequality
holds for , and .
Denote by the set of all -convex mappings on . For recent results and generalizations concerning -convex and -convex mappings, see [1–4].
For the simplicities of notations, for , let us denote
In [1, 3], Klaričić Bakula and Özdemir et al., proved the following Hadamard’s inequalities for mappings whose second derivative in absolute value aroused to the -th power are -convex.
Theorem 1.1. Let be a twice differentiable mapping on the interior of an interval such that , where with and . If is -convex on for and with , then the following inequality holds:
where
where
where
Theorem 1.2. Under the same notations in Theorem 2.2, if is -convex on for and with , then the following inequality holds:
Note that for one obtains the following classes of functions: increasing, -starshaped, starshaped, -convex, convex, and -convex. For the definitions and elementary properties of these classes, see [4–8].
For recent years, many authors present some new results about Simpson’s inequality for -convex mappings and have established error estimations for the Simpson’s inequality: for refinements, counterparts, generalizations, and new Simpson’s type inequalities, see [1–3, 6].
In [9], Dragomir et al. proved the following theorem.
Theorem 1.3. Let be an absolutely continuous mapping on such that , where with . Then the following inequality holds:
The readers can estimate the in the generalized Simpson’s formula without going through its higher derivatives which may not exist, not be bounded, or may be hard to find.
In this paper, the author establishes some generalizations of Hermite-Hadamard and Simpson-like type inequalities based on differentiable -convex mappings by using the following new identity in Lemma 2.1 and by using these results, obtain some applications to special means of positive real numbers.
2. Generalizations of Simpson-Like Type Inequalities on
In order to generalize the classical Simpson-like type inequalities and prove them, we need the following lemma [6].
Lemma 2.1. Let be a differentiable mapping on the interior of an interval , where with and . If , then, for and with , the following equality holds:
for and each , where
By the similar way as Theorems 1.1–1.3, we obtain the following theorems.
Theorem 2.2. Let be a differentiable mapping on such that , where with and . If , for some and , then, for any , the following inequality holds:
where
Proof. From Lemma 2.1 and using the properties of the modulus, we have the following:
Since is -convex on , we know that for any
By (2.5) and (2.6), we get the following:
which completes the proof.
Corollary 2.3. In Theorem 2.2, (i) if we choose and , then we have the following:
and (ii) if we choose and , then we have the following
Theorem 2.4. Under the same notations in Theorem 2.2, if , for some , and with , then, for any , the following inequality holds:
Proof. From Lemma 2.1 and using the properties of modulus, we have the following:
Using the power-mean integral inequality and -convexity of for any , we have the following(a)(b)(c)(d)
By the similar way as the above inequalities (a)–(d), we have the following:(a′)(b′)(c′)(d′)
By (2.11) and (2.16)–(2.19) the assertion (2.10) holds.
Corollary 2.5. In Theorem 2.4, (i) if we choose and , then we have that
and (ii) if we choose and , then we have that
Theorem 2.6. Under the same notations in Theorem 2.2, if , for some , and with , then, for any , the following inequality holds:
where
Proof. Suppose that . From Lemma 2.1, using the Hölder integral inequality, we get the following:
where we have used the fact that .
Since for some fixed and , we have the followings:
Hence, if we combine the inequalities in (2.24)-(2.25), we get the desired result.
Corollary 2.7. In Theorem 2.6, (i) if we choose and , then we have that
and (ii) if we choose and , then we have
where we have used the fact that .
3. Applications to Special Means
Now using the results of Section 2, we give some applications to the following special means of positive real numbers with .(1) The arithmetic mean: .(2)The geometric mean: .(3)The logarithmic mean: ) for .(4)The harmonic mean: .(5)The power mean: , .(6)The generalized logarithmic mean: (7)The identric mean:
Proposition 3.1. For and with , we have the following inequalities:
Proof. The assertions follow from Corollary 2.3 for .
Proposition 3.2. For , we have the following inequalities:
Proof. The assertions follow from Corollary 2.3 for .
Proposition 3.3. For and , we have the following inequalities:
Proof. The assertions follow from Corollary 2.5 for .
Proposition 3.4. For and , we have the following inequalities:
Proof. The assertions follow from Corollary 2.7 for .
Proposition 3.5. For , we have the following inequalities:
Proof. The assertions follow from Corollary 2.7 for and .
Acknowledgment
The author is so indebted to the referee who read carefully through the paper very well and mentioned many scientifically and expressional mistakes.