Some refinements of the Hermite–Hadamard inequality with the help of weighted integrals

B.  Bayraktar; J. E.  Nápoles; F.  Rabossi

doi:10.37863/umzh.v75i6.7126

B. Bayraktar Bursa Uludag University, Faculty of Education, Gorukle Campus, Turkey
J. E. Nápoles UNNE, FaCENA, Corrientes and UTN-FRRE, Resistencia, Chaco, Argentina
F. Rabossi UTN-FRRE, Resistencia, Chaco, Argentina

DOI: https://doi.org/10.37863/umzh.v75i6.7126

Keywords: Hermite-Hadamard integral inequality, integral operators weighted, (h,m)-convex modified functions

Abstract

UDC 517.5

By using the definition of modified $(h,m, s)$-convex functions of the second type, we present various refinements of the classical Hermite–Hadamard inequality obtained within the framework of weighted integrals. Throughout the paper, we show that various known results available from the literature can be obtained as particular cases of our results.

References

P. Agarwal, M. Jleli, M. Tomar, Certain Hermite–Hadamard type inequalities via generalized $k$-fractional integrals, J. Inequal. and Appl., 2017, Article 55 (2017); DOI 10.1186/s13660-017-1318-y. DOI: https://doi.org/10.1186/s13660-017-1318-y

A. O. Akdemir, E. Deniz, E. Yukse, On some integral inequalities via conformable fractionals integrals, Appl. Math. and Nonlinear Sci., 6, № 1, 489–498 (2021). DOI: https://doi.org/10.2478/amns.2020.2.00071

A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for $(k,h)$-Riemann–Liouville fractional integral, New Trends Math. Sci., 4, № 1, 138–146 (2016); http://dx.doi.org/10.20852/ntmsci.2016217824. DOI: https://doi.org/10.20852/ntmsci.2016217824

M. A. Ali, J. E. Nápoles Valdés, A. Kashuri, Z. Zhang, Fractional non conformable Hermite–Hadamard inequalities for generalized $phi$-convex functions, Fasc. Math., 64, 5–16 (2020); DOI: 10.21008/j.0044-4413.2020.0007.

M. Alomari, M. Darus, Some Ostrowski type inequalities for convex functions with applications, RGMIA Res. Rep. Coll., 13, № 2, Article 3 (2010); http://ajmaa.org/RGMIA/v13n2.php.

M. U. Awan, M. A. Noor, F. Safdar, A. Islam, M. V. Mihai, K. I. Noor, Hermite–Hadamard type inequalities with applications, Miskolc Math. Notes, 21, № 2, 593–614 (2020). DOI: https://doi.org/10.18514/MMN.2020.2837

M. K. Bakula, M. E. Özdemir, J. Pecaric, Inequalities for $m$-convex and $(alpha,m)$-convex functions, J. Inequal. Pure and Appl. Math., 9, № 4, Article 96 (2008).

S. Bermudo, P. K'orus, J. E. Nápoles Valdés, On $q$-Hermite–Hadamard inequalities for general convex functions, Acta Math. Hungar., 162, 364–374; (2020); https://doi.org/10.1007/s10474-020-01025-6. DOI: https://doi.org/10.1007/s10474-020-01025-6

M. Bessenyei, Z. Páles, On generalized higher-order convexity and Hermite–Hadamard-type inequalities, Acta Sci. Math. (Szeged), 70, № 1-2, 13–24 (2004).

W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen R{"a}umen, Publ. Inst. Math., 23, 13–20 (1978).

A. M. Bruckner, E. Ostrow, Some function classes related to the class of convex functions, Pacif. J. Math., 12, 1203–1215 (1962). DOI: https://doi.org/10.2140/pjm.1962.12.1203

R. Díaz, E. Pariguan, On hypergeometric functions and Pochhammer $k$-symbol, Divulg. Mat., 15, № 2, 179–192 (2007).

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). DOI: https://doi.org/10.1016/S0893-9659(98)00086-X

S. S. Dragomir, S. Fitzpatrik, The Hadamard inequality for $s$-convex functions in the second sense, Demonstr. Math., 32, № 4, 687–696 (1999). DOI: https://doi.org/10.1515/dema-1999-0403

S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite–Hadamard inequalities, RGMIA Monographs, Victoria Univ. (2000); http://rgmia.vu.edu.au/monographs/hermite_hadamard.html.

S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21, 335–241 (1995).

G. Farid, A. U. Rehman, Q. U. Ain, $k$-Fractional integral inequalities of Hadamard type for $(h,m)$-convex functions, Comput. Methods Different. Equat., 8, № 1, 119–140 (2020); DOI:10.22034/cmde.2019.9462.

G. Farid, A. Rehman, M. Zahra, On Hadamard inequalities for $k$-fractional integrals, Nonlinear Funct. Anal. and Appl., 21, № 3, 463–478 (2016); http://nfaa.kyungnam.ac.kr/journal-nfaa.

P. M. Guzm'{a}n, J. E. N'{a}poles Valdés, Y. Gasimov, Integral inequalities within the framework of generalized fractional integrals, Fract. Different. Calc. (to appear).

J. Hadamard, étude sur les propriétés des fonctions entiéres et en particulier d'une fonction considérée par Riemann, J. Math. Pures Appl., 9, 171–216 (1893).

C. Hermite, Sur deux limites d'une intégrale définie, Mathesis, 3, 82–83 (1883).

J. E. Hernández Hern'{a}ndez, On some new integral inequalities related with the Hermite–Hadamard inequality via $h$-convex functions, MAYFEB J. Math., 4, 1–12 (2017).

H. Hudzik, L. Maligranda, Some remarks on $s$-convex functions, Aequationes Math., 48, № 1, 100–111 (1994). DOI: https://doi.org/10.1007/BF01837981

R. Hussain, A. Ali, G. Gulshani, A. Latif, K. Rauf, Hermite–Hadamard type inequalities for $k$-Riemann–Liouville fractional integrals via two kinds of convexity, Austral. J. Math. Anal. and Appl., 13, № 1, 1–12 (2016).

D. A. Ion, Some estimates on the Hermite–Hadamard inequality through quasi-convex functions, Ann. Univ. Craiova, Math. Comp. Sci. Ser., 34, 82–87 (2007).

H. Kadakal, On refinements of some integral inequalities using improved power-mean integral inequalities, Numer. Methods Partial Different. Equat., 36, № 6, 1–11 (2020);

https://doi.org/10.1002/num.22491. DOI: https://doi.org/10.1002/num.22491

M. A. Khan, Y. Khurshid, Hermite–Hadamar's inequalities for $eta$-convex functions via conformable fractional integrals and related inequalities, Acta Math. Univ. Comenian., 90, № 2, 157–169 (2021).

U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. and Comput., 147, 137–146 (2004). DOI: https://doi.org/10.1016/S0096-3003(02)00657-4

M. Klaričić, E. Neuman, J. Pečarić, V. Šimić, Hermite–Hadamard's inequalities for multivariate $g$-convex functions, Math. Inequal. Appl., 8, № 2, 305–316 (2005). DOI: https://doi.org/10.7153/mia-08-28

M. Matloka, On some integral inequalities for $(h,m)$-convex functions, Math. Econ., 9, № 16, 55–70 (2013).

V. G. Mihesan, A generalization of the convexity, Semin. Funct. Equat., Approx. and Convex., Cluj-Napoca, Romania (1993).

M. S. Moslehian, Matrix Hermite–Hadamard type inequalities, Houston J. Math., 39, № 1, 177–189 (2013).

S. Mubeen, G. M. Habibullah, $k$-Fractional integrals and application, Int. J. Contemp. Math. Sci., 7, № 2, 89–94 (2012).

M. Muddassar, M. I. Bhatti, W. Irshad, Generalisation of integral inequalities of Hermite–Hadamard type through convexity, Bull. Aust. Math. Soc., 88, № 2, 320–330 (2014). DOI: https://doi.org/10.1017/S0004972712000937

J. E. Nápoles Valdés, F. Rabossi, A. D. Samaniego, Convex functions: Ariadne's thread or Charlotte's spiderweb?, Adv. Math. Models Appl., 5, № 2, 176–191 (2020).

B. Bayraktar, J. E. Nápoles Valdés, Integral inequalities for mappings whose derivatives are $(h,m,s)$-convex modified of second type via Katugampola integrals, Ann. Univ. Craiova, Math. and Comput. Sci., 49, № 2, 371–383 (2022); DOI: 10.52846/ami.v.49i2.1596. DOI: https://doi.org/10.52846/ami.v49i2.1596

J. E. Nápoles Valdés, J. M. Rodríguez, J. M. Sigarreta, On Hermite–Hadamard type inequalities for non-conformable integral operators, Symmetry, 11, № 9, Article 1108 (2019). DOI: https://doi.org/10.3390/sym11091108

M. A. Noor, K. I. Noor, M. U. Awan, Generalized fractional Hermite–Hadamard inequalities, Miskolc Math. Notes, 21, № 2, 1001–101 (2020); DOI: 10.18514/MMN.2020.1143. DOI: https://doi.org/10.18514/MMN.2020.1143

M. E. Özdemir, M.Avci, H. Kavurmaci, Hermite–Hadamard type inequalities for $s$-convex and $s$- concave functions via fractional integrals, arxiv:1202,0380v1, 2 feb 2012.

M. E. Özdemir, S. S. Dragomir, ç Yıldız, The Hadamard's inequality for convex function via fractional integrals, Acta Math. Sci. Ser. B, 33, № 5, 1293–1299 (2013). DOI: https://doi.org/10.1016/S0252-9602(13)60081-8

M. E. Özdemir, H. Kavurmaci, M. Avci, Ostrowski type inequalities for convex functions, Tamkang J. Math., 45, № 4, 335–340; (2014); DOI:10.5556/j.tkjm.45.2014.1143. DOI: https://doi.org/10.5556/j.tkjm.45.2014.1143

J. Park, Some Hermite–Hadamard type inequalities for MT-convex functions via classical and Riemann–Liouville fractional integrals, Appl. Math. Sci., 9, № 101, 5011–5026 (2015). DOI: https://doi.org/10.12988/ams.2015.56425

C. E. M. Pearce, J. Pečarić, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13, № 2, 51–55 (2000). DOI: https://doi.org/10.1016/S0893-9659(99)00164-0

F. Qi, B. N. Guo, Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 111, № 2, 425–434 (2017); https://doi.org/10.1007/s13398-016-0302-6. DOI: https://doi.org/10.1007/s13398-016-0302-6

F. Qi, T. Y. Zhang, B. Y. Xi, Hermite–Hadamard type integral inequalities for functions whose first derivatives are of convexity, Ukr. Math. J., 67, № 4, 555–567 (2015); DOI:10.1007/s11253-015-1103-3. DOI: https://doi.org/10.1007/s11253-015-1103-3

E. D. Rainville, Special functions, Macmillan Co., New York (1960).

M. Rostamian Delavar, S. S. Dragomir, M. De La Sen, Estimation type results related to Fejér inequality with applications, J. Inequal. and Appl., 2018, Article 85 (2018); https://doi.org/10.1186/s13660-018-1677-z. DOI: https://doi.org/10.1186/s13660-018-1677-z

M. Z. Sarikaya, On new Hermite–Hadamard Fejer type integral inequalities, Stud. Univ. Babeş-Bolyai Math., 57, № 3, 377–386 (2012).

M. Z. Sarikaya, A. Saglam, H. Yildirim, New inequalities of Hermite–Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex, Int. J. Open Problems Comput. Sci. and Math., 5, № 3 (2012); https://doi.org/10.12816/0006114. DOI: https://doi.org/10.12816/0006114

M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite–Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. and Comput. Modelling, 57, № 9-10, 2403–2407 (2013). DOI: https://doi.org/10.1016/j.mcm.2011.12.048

M. Z. Sarikaya, H. Yildirim, On Hermite–Hadamard type inequalities for Riemann–Liouville fractional integrals, Miskolc Math. Notes, 17, № 2, 1049–105 (2017); DOI:10.18514/MMN.2017.1197. DOI: https://doi.org/10.18514/MMN.2017.1197

E. Set, J. Choi, A. GÖzpinar, Hermite–Hadamard type inequalities involving nonlocal conformable fractional integrals, Malays. J. Math. Sci., 15, № 1, 33–43 (2021).

E. Set, A. GÖzpinar, A study on Hermite–Hadamard type inequalities for $s$-convex functions via conformable fractional integrals, Stud. Univ. Babeş-Bolyai Math., 62, № 3, 309–32 (2017); DOI: 10.24193/subbmath.2017.3.04. DOI: https://doi.org/10.24193/subbmath.2017.3.04

E. Set, A. Gǫzpinar, A. Ekinci, Hermfe–Hadamard type inequalities via conformable fractional integrals, Acta Math. Univ. Comenian., 86, № 2, 309–320 (2017).

G. Toader, Some generalizations of the convexity, Proc. Colloq. Approxim. and Optim., Univ. Cluj-Napoca, 329–338 (1985).

M. Tunç, On new inequalities for $H$-convex functions via Riemann–Liouville fractional integration, arXiv:1203.3318v1.

M. Tunç, S. Balgeçti, Some inequalities for differentiable convex functions with applications}; http://arxiv.org/pdf/ 1406.7217.pdf.

M. Tunç, S. Balgeçti, Integral inequalities for mappings whose derivatives are $s$-convex in the second sense and applications to special means for positive real numbers, Turkish J. Anal. and Number Theory, 4, № 2, 48–53 (2016); DOI:10.12691/tjant-4-2-5.

H. Wang, T. Du, Y. Zhang, $k$-Fractional integral trapezium-like inequalities through $(h,m)$-convex and $(alpha,m)$-convex mappings, J. Inequal. and Appl., 2017, Article 311 (2017); https://doi.org/10.1186/s13660-017-1586-6. DOI: https://doi.org/10.1186/s13660-017-1586-6

B. Y. Xi, F. Qi, Inequalities of Hermite–Hadamard type for extended $s$-convex functions and applications to means, J. Nonlinear Convex. Anal., 16, № 5, 873–890 (2015).

B. Y. Xi, D. D. Gao, F. Qi, Integral inequalities of Hermite–Hadamard type for $(alpha,s)$-convex and $(alpha,s,m)$-convex functions, Ital. J. Pure and Appl. Math., 44, 499–510 (2020). DOI: https://doi.org/10.1186/s13660-020-02442-5

Z. H. Yang, J. F. Tian, Monotonicity and inequalities for the gamma function, J. Inequal. and Appl., 2017, Article 317 (2017); https://doi.org/10.1186/s13660-017-1591-9. DOI: https://doi.org/10.1186/s13660-017-1591-9

Z. H. Yang, J. F. Tian, Monotonicity and sharp inequalities related to gamma function, J. Math. Inequal., 12, № 1, 1–22; (2018); https://doi.org/10.7153/jmi-2018-12-01. DOI: https://doi.org/10.7153/jmi-2018-12-01

C. Yildiz, M. E. Özdemir, H. Kavurmaci, Fractional integral inequalities via $s$-convex functions, Turkish J. Anal. and Number Theory, 5, № 1, 18–22 (2017); DOI: 10.12691/tjant-5-1-4.

C. Zhu, M. Feckan, J. Wang, Factional integral inequalities for differential convex mappings and applications to special means and a midpoint formula, J. Appl. Math. Stat. and Inform., 8, № 2, 21–28 (2012). DOI: https://doi.org/10.2478/v10294-012-0011-5