Abstract
In this paper, we derive new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are (α,m)-convex.
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Introduction
Let be a convex function defined on the interval I of real numbers and a,b ∈ I with a < b. The following double inequality is well known in the literature as Hermite-Hadamard integral inequality
The class of (α,m)-convex functions was first introduced in [1], and it is defined as follows:
The function , b > 0, is said to be (α,m)-convex where (α,m) ∈ [0,1]2, if we have
for all x,y ∈ [0,b] and t ∈ [0,1]. It can be easily deduced 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 star-shaped, star-shaped, m-convex, convex, and α-convex functions, respectively.
As denoted by , the set of all (α,m)-convex functions is on [0,b] for which f(0) ≤ 0. For recent results and generalizations concerning (α,m)-convex functions, see [1–8].
In [7], Set et al. proved the following Hadamard type inequality for (α,m)-convex functions.
Theorem 1
Let be an (α,m)-convex function with (α,m) ∈ (0,1]2. If 0 ≤ a < b < ∞ and f ∈ L[a,b], then one has the inequality
The following inequality is well known in the literature as Simpson’s inequality.
Let be a four-time continuously differentiable mapping on (a,b) and ∥f(4)∥ ∞ = supx ∈ (a,b)|f(4)(x)| < ∞. Then the following inequality holds:
In recent years, many authors have studied error estimations for Simpson’s inequality; for refinements, counterparts, generalizations, and new Simpson type inequalities, see [9–12].
In this paper, in order to provide a unified approach to establish midpoint, trapezoid, and Simpson type inequality for functions whose derivatives in absolute value at certain power are (α,m)-convex, we derive a general integral identity for convex functions.
Main results
In order to generalize the classical Trapezoid, midpoint and Simpson type inequalities and prove them, we need the following Lemma:
Lemma 1
Let be a differentiable mapping on I∘ such that f′ ∈ L[a,b], where a,b ∈ I with a < b, λ,μ ∈ [0,1] and m ∈ (0,1]. Then the following equality holds:
A simple proof of the equality can be done by performing an integration by parts in the integrals from the right side and changing the variable. The details are left to the interested reader.
Theorem 2
Let be a differentiable mapping on I∘ such that f′ ∈ L[a,b], where a,b ∈ I∘ with a < b and λ,μ ∈ [0,1]. If |f′|q is (α,m)-convex on [a,b], for (α,m) ∈ (0,1]2, m b > a, q ≥ 1, then the following inequality holds:
where
Proof
From Lemma 1 and using the properties of modulus and the well known power mean inequality, we have
Since |f′|q is (α,m)-convex on [a,b], we know that for t ∈ [0,1]
hence, by simple computation
Thus, using (6)-(9) in (5), we obtain the inequality (4). This completes the proof. □
Corollary 2.1
Under the assumptions of Theorem 2 with q = 1, we have
Remark 1
In Corollary 2.1,
-
(i)
If we choose , and α = 1, we have the following inequality
which is the same of the Simpson type inequality in Corollary 2.3 (ii) in [9].
-
(ii)
If we choose , λ = 1 and α = 1, we have the following inequality
which is the same of the trapezoid type inequality in Corollary 2.3 (i) in [9].
-
(iii)
If we choose , λ = 0 and α = 1, we have the following midpoint type inequality
Corollary 2.2
Under the assumptions of Theorem 2 with and , we have the following Simpson type inequality
where
and
Remark 2
In Corollary 2.2, if we take α = m = 1, we obtain the following Simpson type inequality
which is the same of the inequality in Theorem 10 in [11] for s = 1.
Corollary 2.3
Under the assumptions of Theorem 2 with and λ = 1, we have the following trapezoid type inequality
Corollary 2.4
Under the assumptions of Theorem 2 with and λ = 0, we have the following midpoint type inequality
Theorem 3
Let be a differentiable mapping on I∘ such that f′ ∈ L[a,b], where a,b ∈ I∘ with a < b and λ,μ ∈ [0,1]. If |f′|q is (α,m)-convex on [a,b], for (α,m) ∈ (0,1]2, m b > a, q > 1, then the following inequality holds:
where
and
Proof
From Lemma 1 and by Hölder’s integral inequality, we have
Since |f′|q is (α,m)-convex on [a,b], for (α,m) ∈ (0,1]2 and μ ∈ (0,1] by the inequality (2), we get
The inequality (12) also holds for μ = 0. Similarly, for μ ∈ [0,1) by the inequality (2), we have
The inequality (12) also holds for μ = 1. By simple computation
and
thus, using (12)-(15) in (11), we obtain the inequality (10). This completes the proof. □
Corollary 2.5
Under the assumptions of Theorem 3 with and , we have the following Simpson type inequality
where
and
Remark 3
In Corollary 2.5, if we take α=m=1, then we obtain the following Simpson type inequality
which is the same of the inequality in Corollary 3 in [11].
Corollary 2.6
Under the assumptions of Theorem 3 with and λ=1, we have the following trapezoid type inequality
Corollary 2.7
Under the assumptions of Theorem 3 with and λ=0, we obtain the following midpoint type inequality
Theorem 4
Let be a differentiable mapping on I∘ such that f′∈L[a,b], where a,b∈I∘ with a<b and λ,μ∈[0,1]. If |f′|q is (α,m)-convex on [a,b], for (α,m)∈(0,1]2, m b>a, q>1, then the following inequality holds:
where
and
Proof
From Lemma 1 and by Hölder’s integral inequality, we have inequality (11). Since |f′|q is (α,m)-convex on [a,b], we know that for t ∈ [0,μ] and t ∈ [μ,1]
Hence,
Thus, using (14), (15) in (17), we obtain the inequality (16). This completes the proof. □
Corollary 2.8
Let the assumptions of Theorem 4 hold. Then for and from the inequality (16), we get the following Simpson type inequality
where
and
Corollary 2.9
Let the assumptions of Theorem 4 hold. Then for and λ = 1 from the inequality (16), we get the following trapezoid type inequality:
Corollary 2.10
In Corollary 2.9, if we take α = 1 we obtain the following trapezoid type inequality
where we have used the fact that 1/2 < (1/(p + 1))1/p<1. We note that the inequality (18) is the same of the inequality in Corollary 2.7 (i) [9].
Corollary 2.11
Under the assumptions of Theorem 4 with and λ = 0, we have the following midpoint type inequality:
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İşcan, İ. A new generalization of some integral inequalities for (α,m)-convex functions. Math Sci 7, 22 (2013). https://doi.org/10.1186/2251-7456-7-22
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DOI: https://doi.org/10.1186/2251-7456-7-22