Hermite-Hadamard and Simpson-Like Type Inequalities for Differentiable ( α , m )-Convex Mappings

The author establish several Hermite-Hadamard and Simpson-like type inequalities for mappings whose first derivative in absolute value aroused to the 𝑞th (𝑞≥1) power are (𝛼,𝑚)-convex. Some applications to special means of positive real numbers are also given.


Introduction
Recall that, for some fixed α ∈ 0, 1 and m ∈ 0, 1 , a mapping f : I ⊆ 0, ∞ → R is said to be α, m -convex on an interval I if the inequality In 1, 3 , Klaričić Bakula andÖzdemir et al., proved the following Hadamard's inequalities for mappings whose second derivative in absolute value aroused to the q-th q ≥ 1 power are α, m -convex. Theorem 1.1. Let f : I ⊆ 0, b * → R be a twice differentiable mapping on the interior I 0 of an interval I such that f ∈ L 1 a, b , where a, b ∈ I with a < b and b * > 0. If |f | q is α, m -convex on a, b for α, m ∈ 0, 1 2 and q ≥ 1 with 1/p 1/q 1, then the following inequality holds: Theorem 1.2. Under the same notations in Theorem 2.2, if |f | q is α, m -convex on a, b for α, m ∈ 0, 1 2 and q > 1 with 1/p 1/q 1, then the following inequality holds: Note that for α, m ∈ { 0, 0 , α, 0 , 1, 0 , 1, m , 1, 1 , α, 1 } one obtains the following classes of functions: increasing, α-starshaped, starshaped, m-convex, convex, and α-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 α, m -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.
International Journal of Mathematics and Mathematical Sciences 3 Theorem 1.3. Let f : I ⊂ 0, ∞ → R be an absolutely continuous mapping on a, b such that f ∈ L p a, b , where a, b ∈ I with a < b. Then the following inequality holds: The readers can estimate the error f 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 α, m -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.

Generalizations of Simpson-Like Type Inequalities on K α m I
In order to generalize the classical Simpson-like type inequalities and prove them, we need the following lemma 6 .

2.2
By the similar way as Theorems 1.1-1.3, we obtain the following theorems.

International Journal of Mathematics and Mathematical Sciences
where

2.4
Proof. From Lemma 2.1 and using the properties of the modulus, we have the following:

2.5
Since |f | is α, m -convex on a, b , we know that for any t ∈ 0, 1 By 2.5 and 2.6 , we get the following: International Journal of Mathematics and Mathematical Sciences which completes the proof.

2.8
and ii if we choose α 1 and r 6, then we have the following mb − a f a m f b .

2.9
Theorem 2.4. Under the same notations in Theorem 2.2, if |f | q ∈ K α m a, b , for some α, m ∈ 0, 1 2 , mb > a and q > 1 with 1/p 1/q 1, then, for any r ≥ 2, the following inequality holds: Proof. From Lemma 2.1 and using the properties of modulus, we have the following:

2.11
Using the power-mean integral inequality and α, m -convexity of |f | q for any t ∈ 0, 1 , we have the following By the similar way as the above inequalities a -d , we have the following:

2.23
Proof. Suppose that q > 1. From Lemma 2.1, using the Hölder integral inequality, we get the following: S b a f α, m, r ≤

3.3
Proof. The assertions follow from Corollary 2.3 for f x x n . 3.4 Proof. The assertions follow from Corollary 2.3 for f x 1/x.