Some new bounds оf Gauss – Jacobi аnd Hermite – Hadamard type integral inequalities
Abstract
UDC 517.5
In this paper, authors discover two interesting identities regarding Gauss–Jacobi and Hermite–Hadamard type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss–Jacobi type integral inequalities are established. Also, using the second lemma, some new estimates with respect to Hermite–Hadamard type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different positive real numbers and new error estimates for the trapezoidal are provided as well. These results give us the generalizations, refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.
References
S. M. Aslani, M. R. Delavar, S. M. Vaezpour, Inequalities of Fejer type related to generalized convex functions with ´applications, Int. J. Anal. and Appl., 16, № 1, 38 – 49 (2018).
F. X. Chen, S. H. Wu, Several complementary inequalities to inequalities of Hermite – Hadamard type for $s$-convex functions, J. Nonlinear Sci. and Appl., 9 , № 2, 705 – 716 (2016), https://doi.org/10.22436/jnsa.009.02.32 DOI: https://doi.org/10.22436/jnsa.009.02.32
Y. -M. Chu, M. A. Khan, T. U. Khan, T. Ali, Generalizations of Hermite – Hadamard type inequalities for $MT$ -convex functions, J. Nonlinear Sci. and Appl., 9 , № 6, 4305 – 4316 (2016), https://doi.org/10.22436/jnsa.009.06.72 DOI: https://doi.org/10.22436/jnsa.009.06.72
Z. Dahmani, On Minkowski and Hermite – Hadamard integral inequalities via fractional integration, Ann. Funct. Anal., 1, № 1, 51 – 58 (2010), https://doi.org/10.15352/afa/1399900993 DOI: https://doi.org/10.15352/afa/1399900993
M. R. Delavar, S. S. Dragomir, On $η$ -convexity, Math. Inequal. Appl., 20, № 1, 203 – 216 (2017), https://doi.org/10.7153/mia-20-14 DOI: https://doi.org/10.7153/mia-20-14
M. R. Delavar, M. De La Sen, Some generalizations of Hermite – Hadamard type inequalities, SpringerPlus, 5 , Article 1661 (2016). DOI: https://doi.org/10.1186/s40064-016-3301-3
S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, Appl. Math. Lett., 11, № 5, 91 – 95 (1998), https://doi.org/10.1016/S0893-9659(98)00086-X DOI: https://doi.org/10.1016/S0893-9659(98)00086-X
T. S. Du, J. G. Liao, Y. J. Li, Properties and integral inequalities of Hadamard – Simpson type for the generalized $(s, m)$-preinvex functions, J. Nonlinear Sci. and Appl., 9, 3112 – 3126 (2016), https://doi.org/10.22436/jnsa.009.05.102 DOI: https://doi.org/10.22436/jnsa.009.05.102
G. Farid, A. U. Rehman, Generalizations of some integral inequalities for fractional integrals, Ann. Math. Sil., 32, № 1, 201 – 214 (2018), https://doi.org/10.1515/amsil-2017-0010 DOI: https://doi.org/10.1515/amsil-2017-0010
M. E. Gordji, M. R. Delavar, M. De La Sen, On $φ $-convex functions, J. Math. Inequal., 10 , № 1, 173 – 183 (2016), https://doi.org/10.7153/jmi-10-15 DOI: https://doi.org/10.7153/jmi-10-15
M. E. Gordji, S. S. Dragomir, M. R. Delavar, An inequality related to $η$ -convex functions (II), Int. J. Nonlinear Anal. and Appl., 6 , № 2, 26 – 32 (2016).
R. Hussain, A. Ali, G. Gulshan, A. Latif, M. Muddassar, Generalized co-ordinated integral inequalities for convex functions by way of $k$-fractional derivatives, Miskolc Math. Notes (to appear).
A. Kashuri, R. Liko, Hermite – Hadamard type fractional integral inequalities for generalized $(r; s, m, φ)$-preinvex functions, Eur. J. Pure and Appl. Math., 10, № 3, 495 – 505 (2017).
A. Kashuri, R. Liko, Hermite – Hadamard type inequalities for generalized $(s, m, φ)$-preinvex functions via $k$-fractional integrals, Tbilisi Math. J., 10, № 4, 73 – 82 (2017), https://doi.org/10.1515/tmj-2017-0046 DOI: https://doi.org/10.1515/tmj-2017-0046
A. Kashuri, R. Liko, Hermite – Hadamard type fractional integral inequalities for $MT_{(m, φ) }$-preinvex functions, Stud. Univ. Babe¸s-Bolyai Math., 62, № 4, 439 – 450 (2017), https://doi.org/10.24193/subbmath.2017.4.03 DOI: https://doi.org/10.24193/subbmath.2017.4.03
A. Kashuri, R. Liko, S. S. Dragomir, Some new Gauss – Jacobi and Hermite – Hadamard type inequalities concerning $(n + 1)$-differentiable generalized $((h_1,h_2);(η_1,η_2))$-convex mappings, Tamkang J. Math., 49, № 4, 317 – 337 (2018), https://doi.org/10.5556/j.tkjm.49.2018.2772 DOI: https://doi.org/10.5556/j.tkjm.49.2018.2772
M. A. Khan, Y.-M. Chu, A. Kashuri, R. Liko, Hermite – Hadamard type fractional integral inequalities for
${rm MT}_{(g,m,φ)}$ -preinvex functions, J. Comput. Anal. and Appl., 26, № 8, 1487 – 1503 (2019).
M. A. Khan, Y.-M. Chu, A. Kashuri, R. Liko, G. Ali, Conformable fractional integrals versions of Hermite – Hadamard inequalities and their generalizations, J. Funct. Spaces, Article ID 6928130, (2018), 9 p., https://doi.org/10.1155/2018/6928130 DOI: https://doi.org/10.1155/2018/6928130
M. A. Khan, Y. Khurshid, T. Ali, Hermite – Hadamard inequality for fractional integrals via $η$ -convex functions, Acta Math. Univ. Comenian. (N.S.), 79, № 1, 153 – 164 (2017).
W. Liu, New integral inequalities involving beta function via $P$ -convexity, Miskolc Math. Notes, 15, № 2, 585 – 591 (2014), https://doi.org/10.18514/mmn.2014.660 DOI: https://doi.org/10.18514/MMN.2014.660
W. J. Liu, Some Simpson type inequalities for $h$-convex and $(α , m)$-convex functions, J. Comput. Anal. and Appl., 16, № 5, 1005 – 1012 (2014).
W. Liu, W. Wen, J. Park, Hermite – Hadamard type inequalities for $MT$ -convex functions via classical integrals and fractional integrals, J. Nonlinear Sci. and Appl., 9, 766 – 777 (2016), https://doi.org/10.22436/jnsa.009.03.05 DOI: https://doi.org/10.22436/jnsa.009.03.05
C. Luo, T. S. Du, M. A. Khan, A. Kashuri, Y. Shen, Some $k$-fractional integrals inequalities through generalized $λ φ m-MT $-preinvexity, J. Comput. Anal. and Appl., 27, № 4, 690 – 705 (2019).
S. Mubeen, G. M. Habibullah, $k$-Fractional integrals and applications, Int. J. Contemp. Math. Sci., 7, 89 – 94 (2012).
M. A. Noor, K. I. Noor, M. U. Awan, S. Khan, Hermite – Hadamard inequalities for $s$-Godunova – Levin preinvex functions, J. Adv. Math. Stud., 7, № 2, 12 – 19 (2014). DOI: https://doi.org/10.1080/23311835.2015.1035856
O. Omotoyinbo, A. Mogbodemu, Some new Hermite – Hadamard integral inequalities for convex functions, Int. J. Sci. Innov. Tech., 1, № 1, 1 – 12 (2014).
M. E. Özdemir, S. S. Dragomir, C. Yildiz, The Hadamard’s inequality for convex function via fractional integrals, Acta Math. Sci., 33, № 5, 153 – 164 (2013), https://doi.org/10.1016/S0252-9602(13)60081-8 DOI: https://doi.org/10.1016/S0252-9602(13)60081-8
M. E. Özdemir, E. Set, M. Alomari, ¨ Integral inequalities via several kinds of convexity, Creat. Math. Inform., 20, № 1, 62 – 73 (2011). DOI: https://doi.org/10.37193/CMI.2011.01.08
C. Peng, C. Zhou, T. S. Du, Riemann – Liouville fractional Simpson’s inequalities through generalized $(m,h_1,h_2)$-preinvexity, Ital. J. Pure and Appl. Math., 38, 345 – 367 (2017). DOI: https://doi.org/10.4067/S0716-09172018000200345
M. Z. Sarikaya, F. Ertugral, On the generalized Hermite – Hadamard inequalities (2017);
https://www.researchgate.net/publication/321760443.
M. Z. Sarikaya, H. Yildirim, On generalization of the Riesz potential, Indian J. Math. and Math. Sci., 3, № 2, 231 – 235 (2007).
E. Set, M. A. Noor, M. U. Awan, A. Gozpinar, Generalized Hermite – Hadamard type inequalities involving fractional integral operators, J. Inequal. and Appl., 169, 1 – 10 (2017), https://doi.org/10.1186/s13660-017-1444-6 DOI: https://doi.org/10.1186/s13660-017-1444-6
H. N. Shi, Two Schur-convex functions related to Hadamard-type integral inequalities, Publ. Math. Debrecen, 78, № 2, 393 – 403 (2011), https://doi.org/10.5486/PMD.2011.4777 DOI: https://doi.org/10.5486/PMD.2011.4777
D. D. Stancu, G. Coman, P. Blaga, Analiză numerică şi teoria aproximării. Vol. II. (Romanian) [[Numerical analysis and approximation theory. Vol. II]], Presa Univ. Clujeana., 2, Cluj-Napoca (2002).
H. Wang, T. S. Du, Y. Zhang, $k$-Fractional integral trapezium-like inequalities through $(h, m)$-convex and $(alpha , m)$-convex mappings, J. Inequal. and Appl., 2017, Article ID 311 (2017), 20 p., https://doi.org/10.1186/s13660-017-1586-6 DOI: https://doi.org/10.1186/s13660-017-1586-6
T. Weir, B. Mond, Preinvex functions in multiple objective optimization, J. Math. Anal. and Appl., 136, 29 – 38 (1988), https://doi.org/10.1016/0022-247X(88)90113-8 DOI: https://doi.org/10.1016/0022-247X(88)90113-8
X. M. Zhang, Y.-M. Chu, X. H. Zhang, The Hermite – Hadamard type inequality of GA-convex functions and its applications, J. Inequal. and Appl., Article ID 507560, (2010), 11 p., https://doi.org/10.1155/2010/507560 DOI: https://doi.org/10.1155/2010/507560
Y. Zhang, T. S. Du, H. Wang, Y. J. Shen, A. Kashuri, Extensions of different type parameterized inequalities for generalized $(m, h)$-preinvex mappings via $k$-fractional integrals, J. Inequal. and Appl.,2008, Article 49 (2018), 30 p., https://doi.org/10.1186/s13660-018-1639-5 DOI: https://doi.org/10.1186/s13660-018-1639-5
Copyright (c) 2021 Artion Kashuri, Miftar Ramosacaj, Rozana Liko
This work is licensed under a Creative Commons Attribution 4.0 International License.
