- Research
- Open access
- Published:
New Hadamard-type inequalities for E-convex functions involving generalized fractional integrals
Journal of Inequalities and Applications volume 2022, Article number: 35 (2022)
Abstract
In this article, we establish some new Hadamard-type inequalities for E-convex functions involving generalized fractional integrals. These inequalities include a generalized Hadamard-type inequality and the corresponding right Hadamard-type inequalities for E-convex functions. The results presented here are generalizations of some of the results discussed in the recent literature.
1 Introduction and basic definitions
The theory of convexity is not only important in itself but also it contributes to almost all areas of mathematics. Convexity gives rise to inequalities, the Hadamard inequality is the first consequence of convex functions. The book by Hardy [1] has played a key role in popularizing the subject of convex analysis. Over the years, the idea of convex sets and convex functions has been largely generalized. Today, the study of convex functions has evolved into a broader theory of functions including quasiconvex functions [2, 3], coordinated convex functions [4, 5], preinvex functions [6], GA-convex functions [7], strongly convex functions [8], \((g, \varphi _{h}) \)-convex functions [9], E-convex functions [10] and so on. Youness [10] defined the E-convex set and the corresponding function as follows:
Definition 1
A set \(S\subset \mathbb{R} \) is called E-convex if and only if there is a function \(E:\mathbb{R} \longrightarrow \mathbb{R} \) such that \(t E(\zeta) + (1-t) E(\eta)\in S \) for each \(\zeta,\eta \in S\) and \(t\in [0,1]\).
Definition 2
A function \(f: \mathbb{R}\longrightarrow \mathbb{R} \) is called E-convex on a set \(S \subseteq \mathbb{R}\) if and only if there is a map \(E: \mathbb{R}\longrightarrow \mathbb{R} \) such that S is an E-convex set and
holds for each \(\zeta,\eta \in S\) and \(t\in [0,1]\). On the other hand, if the inequality sign in the inequality (1.1) is reversed then f is called E-concave on the set S.
Every convex function f on a convex set S is an E-convex function provided that E is an identity function. For a detailed explanation of E-convex functions see [10]. The Hadamard-type inequality for E-convex given in [11] is as follows:
Theorem 1
Let \(E : J\subset \mathbb{R}\longrightarrow \mathbb{R} \) be a continuous increasing function and \(\zeta , \eta \in J\) with \(\zeta <\eta \). Let \(f:I\subseteq \mathbb{R} \longrightarrow \mathbb{R}\) be an E-convex function on \([\zeta , \eta ]\), then we have
inequality (1.2) is Hadamard’s inequality for E-convex functions.
Convexity is mixed with other mathematical concepts such as; optimization [12], time scale [13, 14], quantum and postquantum calculus [15, 16], and fractional calculus [3, 11, 17–19]. Fractional calculus is basically a generalization of integer-order calculus. Strictly speaking, it is a generalization of operators beyond the integral order to real or complex order. Many fractional models have been proposed so far [20–27]. The key drivers behind such proposals are identified with the various real data corresponding to different systems under consideration requiring different kernels. Raina [27] and Agarwal [26] defined the following generalized fractional operators:
Definition 3
Let \(f\in L(\zeta , \eta ) \), then for \(\sigma , \rho >0\), \(\omega \in \mathbb{R} \) the right-handed and left-handed generalized fractional integrals of f are, respectively, defined as follows:
and
where \(\mathcal{F}_{\sigma , \rho }^{\alpha }(s) \) is defined in [27] as follows:
where R is a real positive constant. The coefficients \(\alpha (n)\) (\(n\in N_{0} = N\cup \{0\}\)) are terms of a bounded sequence of positive real numbers and \(\mathbb{R} \) is the set of real numbers. Moreover, the operators \(\mathcal{J}_{\sigma ,\rho ,\zeta +;\omega }^{\alpha }f \) and \(\mathcal{J}_{\sigma ,\rho , \eta +;\omega }^{\alpha }f \) are bounded on \(L(\zeta , \eta ) \), i.e.,
and
where \(\mathfrak{P}:= \mathcal{F}^{\alpha }_{\sigma ,\rho }[\omega (t-s)^{ \sigma }]<\infty \) and \(\| f\| _{1}= \int _{\zeta }^{\eta }|f(t)|\,dt\).
These fractional integrals are really important because of their generality. Many other fractional operators can be obtained by specifying the coefficients \(\alpha (n) \). For instance, if we set \(n=0\), \(\alpha (0)=1 \) and \(\omega =0\), we obtain the well-known Riemann–Liouville fractional operators
and
Lemma 1
For \(0<\alpha \leq 1\) and \(0\leq x < y\), we have
Fractional calculus has useful applications in almost all areas of applied mathematics and other sciences, see [30] and the references therein. In the present work, notions of E-convexity and generalized fractional operators are joined together. These ideas are independently utilized before, however, in combined form we obtain even more generalized results.
2 Main outcomes
In this section, mainly the Hadamard inequality for E-convex function (1.2) is extended using Definition 3 of generalized fractional integrals. Then, an identity is established for differentiable functions that is used to develop right Hadamard-type inequalities for the said extended Hadamard-type inequality. Likewise, another important identity is developed for twice-differentiable functions that is further used to develop more right Hadamard-type inequalities for the said extended Hadamard-type inequality for E-convex functions.
In the following, we use J to represent the interval of nonnegative real numbers and I to represent the interval of real numbers. Moreover, we use the following notations for brevity;
Theorem 2
Let \(E: J \longrightarrow \mathbb{R}\) be a continuous increasing function and \(\zeta , \eta \in J\) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a function such that \(f\in L[E(\zeta ),E(\eta )] \), where \(E(\zeta ),E(\eta )\in I \). If f is an E-convex function on \([\zeta , \eta ]\), then the following inequality holds for generalized fractional integral operators
for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\).
Proof
Since f is an E-convex function on \([\zeta , \eta ]\), therefore for \(E(x), E(y) \in I\) we have
and we let \(E(x)=tE(\zeta )+(1-t)E(\eta )\) and \(E(y)=(1-t)E(\zeta )+tE(\eta )\), so that we have
On multiplying both sides of inequality (2.2) by \(t^{\rho -1}\mathcal{F}^{\alpha }_{\sigma ,\rho }[\omega (E(\Delta ))^{ \sigma }t^{\sigma }]\) and then integrating the resultant inequality with respect to t over \([0,1]\), we have
Further suppose that \(u=tE(\zeta )+(1-t)E(\eta )\) and \(v=(1-t)E(\zeta )+tE(\zeta )\) and using the definition of generalized fractional integrals
Considering again the E-convexity of f over the interval \([\zeta , \eta ] \), we have
and on adding inequality (2.4) and inequality (2.5), we have
On multiplying both sides of inequality (2.6) by \(t^{\rho -1}\mathcal{F}^{\alpha }_{\sigma ,\rho }[\omega (E(\Delta ))^{ \sigma }t^{\sigma }]\), integrating with respect to t over the interval \([0,1] \) and finally using the definition of generalized fractional integrals, we have
on letting \(u=tE(\zeta )+(1-t)E(\eta )\) and \(v=(1-t)E(\zeta )+tE(\zeta )\) and then using the definition of generalized fractional integrals, we have
On combining inequality (2.3) and inequality (2.7), we obtain the required result. Hence it is proved. □
Remark 1
If in Theorem 2, the function E is chosen to be an identity function, then the following inequality holds for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\):
which was given in [31].
Remark 2
If in Theorem 2, the function E is chosen to be an identity function, \(\alpha (0)=1\), \(\rho=\lambda\) and \(\omega =0\), then the following inequality holds:
which was given in [32].
Lemma 2
Let \(E: J \longrightarrow \mathbb{R} \) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a differentiable function on \(I^{o} \). If \(f^{\prime }\in L([E(\zeta ),E(\eta )])\) for \(E(\zeta ),E(\eta )\in I \), then the following identity holds for generalized fractional operators:
Proof
Solving the subsequent integral by integration by parts, then using a change of variable and finally the definition of the left generalized fractional integral operator
Similarly,
and on subtracting inequality (2.9) and inequality (2.10), then multiplying by \(\frac{E(\Delta )}{2\mathcal{F}^{\alpha }_{\sigma ,\rho +1}[\omega (E(\Delta ))^{\sigma }]} \), we obtain
and on submitting the expressions for \(I_{1} \) and \(I_{2} \), we obtain the required result. □
Theorem 3
Let \(E: J \longrightarrow \mathbb{R} \) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a differentiable function on \(I^{o} \) and \(f^{\prime }\in L([E(\zeta ),E(\eta )])\) for \(E(\zeta ),E(\eta )\in I \). If \(|f^{\prime }| \) is an E-convex function on \([\zeta , \eta ]\), then the following inequality holds for generalized fractional integral operators:
for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\), where
Proof
Using Lemma 2, the properties of modulus, and the E-convexity of \(|f^{\prime }|\), respectively, we have
Consider the following integral
Similarly,
and on submitting values of integrals \(I_{3} \) and \(I_{4}\) into inequality (2.12), we have
where \(\alpha _{1}\) is as defined in (2.11). On rearranging we obtain the required result.
Hence it is proved. □
Remark 3
If in Theorem 3, the function E is chosen to be an identity function, then the following inequality holds for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\):
which was given in [31].
Remark 4
If in Theorem 3, the function E is chosen to be an identity function, \(\alpha (0)=1\), \(\rho=\lambda\) and \(\omega =0\), then the following inequality holds:
which was given in [32].
Theorem 4
Let \(E: J\longrightarrow \mathbb{R}\) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a differentiable function on \(I^{o} \) and \(f^{\prime }\in L([E(\zeta ),E(\eta )])\) for \(E(\zeta ),E(\eta )\in I \). If \(\vert f^{\prime } \vert ^{q}, q>1 \), is an E-convex function on \([\zeta , \eta ]\), then the following inequality holds for generalized fractional integral operators:
for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\), where p, q are conjugate indices and
Proof
Using Lemma 2, the properties of modulus, and the well-known Hölder’s inequality, respectively,
Solving the first integral from the right side of inequality (2.14) and using Lemma 1, we have
Solving the second integral from the right side of inequality (2.14) by using the fact that \(\vert f^{\prime } \vert ^{q}\), for any \(q>1 \), is E-convex, therefore we have
On submitting the values of integrals \(I_{5} \) and \(I_{6} \) on the right side of inequality (2.14), we have
where \(\alpha _{2} \) is as defined in (2.13). Hence it is proved. □
Remark 5
If in Theorem 4, the function E is chosen to be an identity function, then the following inequality holds for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\):
which was given in [33].
Remark 6
If in Theorem 4, the function E is chosen to be an identity function, \(\alpha (0)=1\), \(\rho=\lambda\) and \(\omega =0\), then the following inequality holds:
which was given in [33].
Theorem 5
Let \(E: J\longrightarrow \mathbb{R}\) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a differentiable function on \(I^{o} \) and \(f^{\prime }\in L([E(\zeta ),E(\eta )])\) for \(E(\zeta ),E(\eta )\in I \). If \(\vert f^{\prime } \vert ^{q} , q\geq 1 \), is an E-convex function on \([\zeta , \eta ]\), then the following inequality holds for generalized fractional integral operators:
for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\), where
Proof
Using Lemma 2, the well-known power mean inequality, and the E-convexity of \(\vert f^{\prime } \vert ^{q}\), respectively, we have
and on rearranging we obtain
where \(\alpha _{3} \) is as defined in (2.16).
Hence it is proved. □
Remark 7
If in Theorem 5, the function E is chosen to be an identity function, then the following inequality holds for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\):
which was given in [33].
Remark 8
If in Theorem 5, the function E is chosen to be an identity function, \(\alpha (0)=1\), \(\rho=\lambda\) and \(\omega =0\), then the following inequality holds:
which was given in [33].
Lemma 3
Let \(E: J\subset \mathbb{R}^{+}\cup \{0\} \longrightarrow \mathbb{R} \) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a twice differentiable function on \(I^{o} \). If \(f^{\prime \prime }\in L([E(\zeta ),E(\eta )])\) for \(E(\zeta ),E(\eta )\in I \), then the following identity holds for generalized fractional operators:
for all \(\sigma , \rho > 0 \) and \(\omega \geq 0\).
Proof
Solving the following integral by simple integration
Solving the next integral by applying integration by parts twice, we have
and similarly
on subtracting \(I_{8} \) and \(I_{9} \) from \(I_{7} \), then multiplying by \(\frac{(E(\Delta ))^{2}}{2\mathcal{F}^{\alpha }_{\sigma ,\rho +1}[\omega (E(\Delta ))^{\sigma }]} \), we obtain
and on submitting the values of \(I_{7} \), \(I_{8} \) and \(I_{9} \) we obtain the required result. □
Theorem 6
Let \(E: J\longrightarrow \mathbb{R}\) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a differentiable function on \(I^{o} \) and \(f^{\prime \prime }\in L([E(\zeta ),E(\eta )])\) for \(E(\zeta ),E(\eta )\in I \). If \(\vert f^{\prime \prime } \vert \) is an E-convex function on \([\zeta , \eta ]\), then the following inequality holds for generalized fractional integral operators:
for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\), where
Proof
Using Lemma 3, the properties of modulus, and the E-convexity of \(\vert f^{\prime \prime } \vert \), respectively,
where \(\alpha _{4}(n)\) is as defined in (2.18).
Hence it is proved. □
Remark 9
If in Theorem 6, the function E is chosen to be an identity function, then the following inequality holds for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\):
which was given in [33].
Remark 10
If in Theorem 6, the function E is chosen to be an identity function, \(\alpha (0)=1\), \(\rho=\lambda\) and \(\omega =0\), then the following inequality holds:
which was given in [33].
Theorem 7
Let \(E: J\longrightarrow \mathbb{R}\) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a differentiable function on \(I^{o} \) and \(f^{\prime \prime }\in L([E(\zeta ),E(\eta )])\) for \(E(\zeta ),E(\eta )\in I \). If \(|f^{\prime \prime }|^{q}, q>1 \), is an E-convex function on \([\zeta , \eta ]\), then the following inequality holds for generalized fractional integral operators:
for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\), where
Proof
Using Lemma 3 and applying Hölder’s inequality, respectively,
where \(\alpha _{5} \) is as defined in (2.19). Hence it is proved. □
Remark 11
If in Theorem 7, the function E is chosen to be an identity function, then the following inequality holds for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\):
which was given in [33].
Remark 12
If in Theorem 7, the function E is chosen to be an identity function, \(\alpha (0)=1\), \(\rho=\lambda\) and \(\omega =0\), then the following inequality holds:
which was given in [33].
Theorem 8
Let \(E: J\longrightarrow \mathbb{R}\) be a continuous increasing function and \(\zeta , \eta \in J \) with \(\zeta < \eta \). Let \(f : I\longrightarrow \mathbb{R}\) be a differentiable function on \(I^{o}\). If \(|f^{\prime \prime }|^{q}, q \geq 1 \), is an E-convex function on \([\zeta , \eta ]\), then the following inequality holds for generalized fractional integral operators:
for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\), where p, q are conjugate indices and
Proof
Using Lemma 3 and applying the well-known power mean inequality
where \(\alpha _{6} \) is as defined in (2.20). Hence it is proved. □
Corollary 1
If in Theorem 8, the function E is chosen to be an identity function, then the following inequality holds for all \(\sigma , \rho \in \mathbb{R}^{+}\) and \(\omega \in \mathbb{R}\):
Corollary 2
If in Theorem 8, the function E is chosen to be an identity function, \(\alpha (0)=1\), \(\rho=\lambda\) and \(\omega =0\), then the following inequality holds:
Availability of data and materials
This work does not involve any supplementary data or material, however, all related calculations can be supplied on demand.
References
Hardy, G.H., Litllewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1934)
Dragomir, S.S., Pearce, C.E.M.: Quasi-convex functions and Hadamards inequality. Bull. Aust. Math. Soc. 57, 377–385 (1998)
Hussain, R., Ali, A., Latif, A., Gulshan, G.: Some k-fractional associates of Hadamard’s inequality for quasi-convex functions and applications to special means. Fract. Differ. Calc. 7, 301–309 (2017)
Dragomir, S.S.: On the Hadamard’s inequlality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan. J. Math. 5, 775–788 (2001)
Hussain, R., Ali, A., Latif, A., Gulshan, G.: Co-ordinated convex function of three variables and some analogues inequalities with applications. J. Comput. Anal. Appl. 29, 505–551 (2021)
Wang, S.H., Qi, F.: Hermite-Hadamard-type inequalities for n-times differentiable and preinvex functions. J. Inequal. Appl. 2014, 49 (2014). https://doi.org/10.1186/1029-242X-2014-49
Zhang, X.M., Chu, Y.M., Zhang, X.H.: The Hermite-Hadamard type inequality of GA-convex functions and its application. J. Inequal. Appl. 2010, 507560 (2010). https://doi.org/10.1155/2010/507560
Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966)
z̈demir, M.E., Gürbüz, M., Kavurmacı, H.: Hermite-Hadamard-type inequalities for \((g, \varphi _{h}) \)-convex dominated functions. J. Inequal. Appl. 2013, 184 (2013). https://doi.org/10.1186/1029-242X-2013-184
Youness, E.A.: E-convex sets, E-convex functions and E-convex programming. J. Optim. Theory Appl. 102, 439–450 (1999)
Sarikaya, M.Z., Yaldiz, H.: On Hermite-Hadamard type inequalities for φ-convex functions via fractional integrals. Malaysian J. Math. Sci. 9, 243–258 (2015)
Bubeck, S.: Convex optimization: algorithms and complexity. Found. Trends Mach. Learn. 8, 231–357 (2015)
Dinu, C.: Hermite-Hadamard inequality on time scales. J. Inequal. Appl. 2008, 287947 (2008) https://doi.org/10.1155/2008/287947
Fagbemigun, B.O., Mogbademu, A.A.: Some classes of convex functions on time scales. RGMIA Research Report Collections 22, 1–12 (2019)
Alp, N., Sarıkaya, M.Z., Kunt, M., Iscan, I.: q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. J. King Saud Univ., Sci. 30, 193–203 (2018)
Sudsutad, W., Ntouyas, S.K., Tariboon, J.: Quantum integral inequalities for convex functions. J. Math. Inequal. 9, 781–793 (2015)
Farid, G., Mahreen, K., Chu, Y.M.: Study of inequalities for unified integral operators of generalized convex functions. Open J. Math. Sci. 5(1), 80–93 (2021)
Farid, G., Rehman, A.U., Bibi, S., Chu, Y.M.: Refinements of two fractional versions of Hadamard inequalities for Caputo fractional derivatives and related results. Open J. Math. Sci. 5(1), 1–10 (2021)
Set, E., Dragomir, S.S., Gözpinar, A.G.: Some generalized Hermite-Hadamard-type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex. Acta Math. Univ. Comen. 88, 87–100 (2019)
Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20, 763–769 (2016)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent–II. Geophys. J. Int. 13, 529–539 (1967)
Farid, G.: Existence of an integral operator and its consequences in fractional and conformable integrals. Open J. Math. Sci. 3(3), 210–216 (2019)
Farid, G.: A unified integral operator and further its consequences. Open J. Math. Anal. 4(1), 1–7 (2020)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Prabhakar, T.R.: A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)
Agarwal, R.P., Luo, M.J., Raina, R.K.: On Ostrowski type inequalities. Fasc. Math. 56, 5–27 (2016)
Raina, R.K.: On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Math. J. 21, 191–203 (2005)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integral and series. In: Elementary Functions, vol. 1. Nauka, Moscow (1981)
Wang, J., Zhu, C., Zhou, Y.: New generalized Hermite-Hadamard-type inequalities and applications to special means. J. Inequal. Appl. 2013, 325 (2013)
Ross, B. (ed.): Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974, vol. 457. Springer, Berlin (2006)
Yaldiz, H., Sarikaya, M.Z.: On the midpoint type inequalities via generalized fractional integral operators. In: Xth International Statistics Days Conference, pp. 181–189 (2016)
Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N.: Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403–2407 (2013)
Usta, F., Budak, H., Sarikaya, M.Z., Set, E.: On generalization of trapezoid type inequalities for s-convex functions with generalized fractional integral operators. Filomat 32, 2153–2171 (2018)
Acknowledgements
The authors are grateful to the Mirpur University of Science & Technology for allowing study leave and utilization of educational resources to undertake this work.
Funding
There is no special funding from any source for this work.
Author information
Authors and Affiliations
Contributions
AL: conceptualization, methodology, investigation, writing the original draft, writing, reviewing and editing. RH: problem statement, investigation, supervision, provision of study resources, reviewing and editing. The authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The publication of this work is approved by all authors, and the authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Latif, A., Hussain, R. New Hadamard-type inequalities for E-convex functions involving generalized fractional integrals. J Inequal Appl 2022, 35 (2022). https://doi.org/10.1186/s13660-022-02771-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-022-02771-7