Abstract
In the present research, we develop some integral inequalities of Hermite–Hadamard type for differentiable η-convex functions. Moreover, our results include several new and known results as particular cases.
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1 Introduction
Throughout this paper, let I be an interval in \(\mathbb{R}\). Also consider \(\eta: A\times A \rightarrow B\) for appropriate \(A, B \subseteq\mathbb{R}\).
Let \(f:I \subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a convex function, and let \(a_{1}\), \({a_{2}} \in I\) with \(a_{1}< {a_{2}}\). The following double inequality
is known in the literature as the Hadamard inequality for convex functions. Fejer [1] gave a generalization of (1) as follows. If \(f:[a_{1}, {a_{2}}] \rightarrow\mathbb{R}\) is a convex function and \(g:[a_{1}, {a_{2}}] \rightarrow\mathbb{R}\) is nonnegative, integrable, and symmetric about \(\frac{a_{1} + {a_{2}}}{2}\), then
Since the Hermite–Hadamard inequality and fractional integrals have a wide range of applications, many researchers extend their studies to Hermite–Hadamard-type inequalities involving fractional integrals.
In 2015, Iscan [2] obtained Hermite–Hadamard–Fejér-type inequalities for convex functions via fractional integrals. In 2017, Farid and Tariq [3] developed fractional integral inequalities for m-convex functions. Also, Farid and Abbas [4] established Hermite–Hadamard–Fejér-type inequalities for p-convex functions via generalized fractional integrals. For recent generalizations, we refer to [5,6,7], and [8].
Xi and Qi [9], Ozdemir et al. [10], and Sarikaya et al. [5] established Hermite–Hadamard-type inequalities for convex functions. Gordji et al. [11] introduced an important generalization of convexity known as η-convexity.
Definition 1.1
([11])
A function \(f: I \rightarrow\mathbb{R}\) is called η-convex if
for all \(x, y \in I\) and \(\alpha\in[0, 1]\).
Theorem 1.1
([10])
Let \(f:I\subseteq[0, \infty) \rightarrow\mathbb{R}\) be a differentiable mapping on the interior \(I^{\circ}\) of I such that \(f'' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}\), \({a_{2}} \in I\) with \({a_{1}}< {a_{2}}\). If \(|f|\) is convex on \([{a_{1}}, {a_{2}}]\), then
Theorem 1.2
([10])
Let \(f:I\subseteq[0, \infty) \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\) such that \(f'' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}\), \({a_{2}} \in I\) with \({a_{1}}< {a_{2}}\). If \(|f''|^{q}\) for \(q \geq1\) is convex on \([{a_{1}}, {a_{2}}]\), then
Lemma 1.1
([9])
Let \(f:I\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a differentiable function on \(I^{\circ}\) such that \(f' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}\), \({a_{2}} \in I\) with \({a_{1}}< {a_{2}}\). If α, \(\beta\in\mathbb{R}\), then
Lemma 1.2
([9])
For \(s> 0\) and \(0 \leq\epsilon\leq1\), we have
The paper is organized as follows. In Sect. 2, we establish Hermite–Hadamard- and Fejer-type inequalities for η-convex functions. In the last section, we derive Fractional integral inequalities for η-convex functions.
2 Hermite–Hadamard- and Fejer-type inequalities
Theorem 2.1
Let \(f:I\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be an η-convex function with \(f \in L^{1}[a_{1}, {a_{2}}]\), where \(a_{1}\), \({a_{2}} \in I\) with \(a_{1}< {a_{2}}\),Then
Proof
According to (3), with \(x = ta_{1} + (1 - t){a_{2}}\), \(y = (1 - t)a_{1} + t{a_{2}}\), and \(\alpha= \frac{1}{2}\), where \(t \in[0, 1]\), we find that
Thus by integrating we obtain
so that
and the first inequality is proved. Taking \(x = a_{1}\) and \(y = {a_{2}}\) in (3), we get
Integrating this inequality with respect to α over \([0, 1]\), we get
Clearly, (9) and (10) yield (8). □
Remark 2.1
Taking \(\eta(x , y) = x - y\), we reduce (8) to inequality (1).
Theorem 2.2
Let f and g be nonnegative η-convex functions with \(fg \in L^{1}[a_{1}, {a_{2}}]\), where \(a_{1}, {a_{2}} \in I\), \(a_{1}< {a_{2}}\). Then
where
Proof
Since f and g are η-convex functions, we have
for all \(t \in[0, 1]\). Since f and g are nonnegative, we have
Integrating both sides of the inequality over \([0, 1]\), we obtain
Then
□
Remark 2.2
By taking \(\eta(x , y) = x - y\) inequality (11) becomes inequality (1.4) in [5].
Theorem 2.3
Let f be an η-convex function with \(f \in L^{1}[a_{1}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in I\), \(a_{1}< {a_{2}}\), and let \(g: [a_{1}, {a_{2}}] \rightarrow\mathbb{R}\) be nonnegative, integrable, and symmetric about \(\frac{(a_{1} + {a_{2}})}{2}\). Then
Proof
Since, f is an η-convex function and g is nonnegative, integrable. and symmetric about \(\frac{(a_{1} + {a_{2}})}{2}\), we find that
□
Remark 2.3
If we choose \(\eta(x, y) = x - y\) and \(g(x) = 1\), then (12) reduces to the second inequality in (1), and if we take \(\eta (x, y) = x - y\), then (12) reduces to the second inequality in (2).
3 Fractional integral inequalities
Theorem 3.1
Let \(f:I \rightarrow\mathbb{R}\), \(I\subseteq\mathbb{R}\) be a differentiable mapping on \(I^{0}\) with \(f' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}\), \({a_{2}} \in I\), \({a_{1}}< {a_{2}}\). If \(|f' (x)|^{q}\) for \(q \geq1\) is η-convex on \([{a_{1}}, {a_{2}}]\) and \(0 \leq\alpha\), \(\beta\leq1\), then
Proof
For \(q> 1\), by Lemma 1.1, the η-convexity of \(|f' (x)|^{q}\) on \([{a_{1}}, {a_{2}}]\), and the Hölder integral inequality, we have
Using Lemma 1.2, by a direct calculation we get
and
Substituting these two inequalities into inequality (14) and using Lemma 1.2 result in inequality (13) for \(q> 1\).
For \(q = 1\), from Lemmas 1.1 and 1.2 it follows that
□
Remark 3.1
If we take \(\eta(x , y) = x - y\), then inequality (13) reduces to inequality (3.1) in [9].
Taking \(\alpha= \beta\) in Theorem 3.1, we derive the following corollary.
Corollary 3.1
Let \(f:I \rightarrow\mathbb{R}\), \(I\subseteq\mathbb{R}\) be a differentiable mapping on \(I^{0}\) with \(f'\in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in I\), \({a_{1}}< {a_{2}}\). If \(|f' (x)|^{q}\) for \(q \geq1\) is η-convex on \([{a_{1}}, {a_{2}}]\) and \(0 \leq\alpha\leq1\), then
Remark 3.2
If we take \(\eta(x , y) = x - y\), then inequality (16) reduces to inequality (3.5) in [9].
By choosing \(\alpha= \beta= \frac{1}{2}, \frac{1}{3}\), respectively, in Theorem 3.1 we can deduce the following inequalities.
Corollary 3.2
Let \(f:I \rightarrow\mathbb{R}\), \(I\subseteq\mathbb{R}\), be a differentiable mapping on \(I^{0}\) with \(f' \in L^{1}[{a_{1}} , {a_{2}}]\), where \({a_{1}}, {a_{2}} \in\) I, \({a_{1}}<{a_{2}}\). If \(|f' (x)|^{q}\) for \(q \geq1\) is η-convex on \([{a_{1}} , {a_{2}}]\) and \(0 \leq\alpha, \beta\leq1\), then
Setting \(q = 1\) in Corollary 3.2, we have the following:
Corollary 3.3
Let \(f:I \rightarrow\mathbb{R}\), \(I\subseteq\mathbb{R}\), be a differentiable mapping on \(I^{0}\) with \(f' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in I\), \({a_{1}}<{a_{2}}\). If \(|f' (x)|\) is η-convex on \([{a_{1}}, {a_{2}}]\), then
Remark 3.3
If we take \(\eta(x , y) = x - y\), then inequalities (17) and (18) reduce to inequalities (3.6) and (3.7) in [9].
Theorem 3.2
Let \(f:I \rightarrow\mathbb{R}\), \(I\subseteq\mathbb{R}\), be a differentiable mapping on \(I^{0}\) with \(f' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in\) I, \({a_{1}}<{a_{2}}\). If \(|f' (x)|^{q}\) for \(q \geq1\) is η-convex on \([{a_{1}}, {a_{2}}]\) and \(0 \leq\alpha, \beta\leq1\), then
Proof
For \(q> 1\), by the η-convexity of \(|f'(x)|^{q}\) on \([{a_{1}}, {a_{2}}]\) and Hölder’s integral inequality it follows that
By Lemma 1.2 we have
and
Substituting the last two equalities into inequality (20) yields inequality (19) for \(q> 1\).
For \(q = 1\), the proof is the same as that of (15), and the theorem is proved. □
Remark 3.4
If we take \(\eta(x , y) = x - y\), then inequality (19) reduces to inequality (3.8) in [9].
Similarly to corollaries of Theorem 3.1, we can obtain the following corollaries of Theorem 3.2.
Corollary 3.4
Let \(f:I\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\) with \(f' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in I\), \({a_{1}}< {a_{2}}\). If \(|f' (x)|^{q}\) for \(q \geq1\) is η-convex on \([{a_{1}}, {a_{2}}]\) and \(0 \leq\alpha\leq 1\), then
Remark 3.5
If we take \(\eta(x , y) = x - y\), then inequality (21) reduces to inequality (3.11) in [9].
Corollary 3.5
Let \(f:I\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\) with \(f' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in I\), \({a_{1}}< {a_{2}}\). If \(|f' (x)|^{q}\) for \(q \geq1\) is η-convex on \([{a_{1}}, {a_{2}}]\) and \(0 \leq\alpha\), \(\beta\leq1\), then
Remark 3.6
If we take \(\eta(x , y) = x - y\), then inequality (22) reduces to inequality (3.12) in [9] respectively.
If we take \(q = 1\) in Corollary 3.5, then we get Corollary 3.3.
To prove our next results, we consider the following lemma proved in [10].
Lemma 3.1
Let \(f:I \rightarrow\mathbb{R}\), \(I\subseteq\mathbb{R}\) be a differentiable mapping on \(I^{0}\) with \(f'' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in\) I and \({a_{1}}<{a_{2}}\). Then
Theorem 3.3
Let \(f : I \subset[0, \infty) \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\) with \(f'' \in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in I\) and \({a_{1}}< {a_{2}}\). If \(|f''|\) is η-convex on \([{a_{1}}, {a_{2}}]\), then
Proof
From Lemma 3.1 we have
This proves inequality (24). □
Remark 3.7
If we take \(\eta(x, y) = x - y\), then inequality (24) reduces to inequality (4).
Theorem 3.4
Let \(f : I \subset[0, \infty) \rightarrow\mathbb{R}\) be a differentiable mapping on \(I^{\circ}\) with \(f''\in L^{1}[{a_{1}}, {a_{2}}]\), where \({a_{1}}, {a_{2}} \in I\) and \({a_{1}}< {a_{2}}\). If \(|f''|^{q}\) for \(q \geq1\) with \(\frac{1}{p} + \frac{1}{q}= 1\) is η-convex on \([{a_{1}}, {a_{2}}]\), then
Proof
Suppose that \(p \geq1\). From Lemma 3.1, using the power mean inequality, we have
Because \(|f''|^{q}\) is η-convex, we have
and
Therefore we have
□
Remark 3.8
If we take \(\eta(x, y) = x - y\), then inequality (25) reduces to inequality (5).
4 Application to means
For two positive numbers \(a_{1} > 0\) and \(a_{2} > 0\), define
for \(0 \leq w < \infty\). These means are respectively called the arithmetic, geometric, harmonic, generalized logarithmic, identric, and Heronian means of two positive numbers \(a_{1}\) and \(a_{2}\).
Applying Theorems 3.1 and 3.2 to \(f(x) = x^{s}\) for \(s \neq0\) and \(x > 0\) results in the following inequalities for means.
Theorem 4.1
Let \(a_{1} > 0\), \(a_{2} > 0\), \(a_{1} \neq a_{2}\), \(q\geq1 \), and either \(s> 1\) and \((s -1)q \geq1\) or \(s < 0\). Then
Theorem 4.2
Let \(a_{1} > 0\), \(a_{2} > 0\), \(a_{1} \neq a_{2}\), \(q\geq1 \), and either \(s> 1\) and \((s -1)q \geq1\) or \(s < 0\). Then
Taking \(f(x) = \ln x\) for \(x>0\) in Theorems 3.1 and 3.2 results in the following inequalities for means.
Theorem 4.3
For \(a_{1} > 0\), \(a_{2} > 0\), \(a_{1} \neq a_{2}\) and \(q\geq1\), we have
Theorem 4.4
For \(a_{1} > 0\), \(a_{2} > 0\), \(a_{1} \neq a_{2}\) and \(q\geq1\), we have
Finally, we can establish an inequality for the Heronian mean as follows.
Theorem 4.5
For \(a_{2}> a_{1}> 0\), \(a_{1} \neq a_{2}\), \(w \geq0\), and \(s \geq4\) or \(0 \neq s<1\), we have
Proof
Let \(f(x) = \frac{x^{s} + wx^{\frac{s}{2}} + 1}{w + 2}\) for \(x>0\) and \(s \notin(1, 4)\). Then
By Corollary 3.3 it follows that
On the other hand, we have
References
Fejer, L.: Uber die Fourierreihen II. Math. Naturwiss., Anz. Ungar. Akad. Wiss. 24, 369–390 (1960) (in Hungarian)
Iscan, I.: Hermite–Hadamard–Fejér type inequalities for convex functions via fractional integrals. Stud. Univ. Babeş–Bolyai, Math. 60(3), 355–366 (2015)
Farid, G., Tariq, B.: Some integral inequalities for m-convex functions via fractional integrals. J. Inequal. Spec. Funct. 8(1), 170–185 (2017)
Farid, G., Abbas, G.: Generalizations of some Hermite–Hadamard–Fejér type inequalities for p-convex functions via generalized fractional integrals. J. Fract. Calc. Appl. 9(2), 56–76 (2018)
Sarikaya, M.Z., Saglam, A., Yildirim, H.: On some Hadamard type inequalities for h-convex functions. J. Math. Anal. 2(3), 335–341 (2008)
Kirmaci, U.S., Bakula, M.K., Ozdemir, M.E., Pecaric, J.: Hadamard type inequalities for s-convex functions. Appl. Math. Comput. 193, 26–35 (2007)
Dragomir, S.S., Pecaric, J., Persson, L.E.: Some inequalities of Hadamard type. Soochow J. Math. 21, 335–341 (1995)
Dragomir, S.S., Fitzpatrik, S.: The Hadamard’s inequality for s-convex functions in the second sense. Demonstr. Math. 32(4), 687–696 (1999)
Xi, B.Y., Qi, F.: Some integral inequalities of Hermite–Hadamard type for convex functions with applications to means. J. Funct. Spaces Appl. 2012, Article ID 980438 (2012). https://doi.org/10.1155/2012/980438
Ozdemir, M.E., Yildiz, C., Akdemir, A.C., Set, E.: On some inequalities for s-convex functions and applications. J. Inequal. Appl. 2013, Article ID 333 (2013)
Gordji, M.E., Delavar, M.R., De La Sen, M.: On ϕ-convex functions. J. Math. Inequal. 10(1), 173–183 (2016)
Acknowledgements
This work was presented at the 6th International Conference on Education (ICE), which was organized by Division of Science and Technology, University of Education, Lahore, Pakistan.
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Kwun, Y.C., Saleem, M.S., Ghafoor, M. et al. Hermite–Hadamard-type inequalities for functions whose derivatives are η-convex via fractional integrals. J Inequal Appl 2019, 44 (2019). https://doi.org/10.1186/s13660-019-1993-y
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DOI: https://doi.org/10.1186/s13660-019-1993-y