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Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China |
We introduce the concept of interval harmonical $ (h1,h2)- $convex functions, establish some new Hermite-Hadamard type inequalities on interval Riemann integrable functions, and generalize the results of Noor et al. 2015 and Zhao Dafang et al. 2019.
References:
| [1] |
Y. An, G. Ye, D. Zhao and W. Liu, Hermite-Hadamard type inequalities for interval $(h_{1}, h_{2})$- convex functions, Mathematics, 7 (2019).
doi: 10.3390/math7050436. |
| [2] |
M. U. Awan, M. A. Noor, K. I. Noor and A. G. Khan,
Some new classes of convex functions and inequalities, Miskolc Math. Notes, 19 (2018), 77-94.
doi: 10.18514/MMN.2018.2179. |
| [3] |
Y. Chalco-Cano, A. Flores-Franulič and H. Román-Flores,
Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457-472.
|
| [4] |
T. M. Costa,
Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets and Systems, 327 (2017), 31-47.
doi: 10.1016/j.fss.2017.02.001. |
| [5] |
T. M. Costa and H. Román-Flores,
Some integral inequalities for fuzzy-interval-valued functions, Inform. Sci., 420 (2017), 110-125.
doi: 10.1016/j.ins.2017.08.055. |
| [6] |
E. de Weerdt, Q. P. Chu and J. A. Mulder,
Neural network output optimization using interval analysis, IEEE Trans. Neural Netw., 20 (2009), 638-653.
doi: 10.1109/TNN.2008.2011267. |
| [7] |
A. Dinghas,
Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Z., 66 (1956), 173-188.
doi: 10.1007/BF01186606. |
| [8] |
S. S. Dragomir,
Inequalities of Hermite-Hadamard type for $h$-convex functions on linear speaces, Proyecciones, 32 (2015), 323-341.
doi: 10.4067/S0716-09172015000400002. |
| [9] |
N. A. Gasilov and Ş. E. Amrahov,
Solving a nonhomogeneous linear system of interval differential equations, Soft Comput., 22 (2018), 3817-3828.
doi: 10.1007/s00500-017-2818-x. |
| [10] |
A. G. Ghazanfari,
The Hermite-Hadamard type inequalities for operator $s$-convex functions, J. Adv. Res. Pure Math., 6 (2014), 52-61.
|
| [11] |
İ. İşcan,
Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.
|
| [12] |
İ. İşcan,
On generalization of different type inequalities for harmonically quasi-convex functions via fractional integrals, Appl. Math. Comput., 275 (2016), 287-298.
doi: 10.1016/j.amc.2015.11.074. |
| [13] |
M. Kunt and İ Işcan,
Hermite-Hadamard-Féjer type inequalities for $p$-convex functions via fractional integrals, Iranian J. Sci. Tech., 42 (2018), 2079-2089.
doi: 10.1007/s40995-017-0352-4. |
| [14] |
M. A. Latif and M. Alomari,
On Hadamard-type inequalities for $h$-convex functions on the co-ordinates, Int. J. Math. Anal. (Ruse), 3 (2009), 1645-1656.
|
| [15] |
M. A. Latif, S. S. Dragomir and E. Momoniat,
|
| [16] |
Y. Li and T. Wang,
Interval analysis of the wing divergence, Aerosp. Sci. Technol., 74 (2018), 17-21.
doi: 10.1016/j.ast.2018.01.001. |
| [17] |
M. V. Mihai, M. A. Noor, K. I. Noor and M. U. Awan,
Some integral inequalities for harmonic $h$-convex functions involving hypergeometric functions, Appl. Math. Comput., 252 (2015), 257-262.
doi: 10.1016/j.amc.2014.12.018. |
| [18] |
R. E. Moore, Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1996. |
| [19] |
M. A. Noor, K. I. Noor, M. U. Awan and S. Costache,
Some integral inequalities for harmonically $h$-convex functions, Politehm. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77 (2015), 5-16.
|
| [20] |
M. A. Noor, K. I. Noor, S. Iftikhar and C. Ionescu,
Hermite-Hadamard inequalities for co-ordinated harmonic convex functions., Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2017), 25-34.
|
| [21] |
H. Román-Flores, Y. Chalco-Cano and W. A. Lodwick,
Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2018), 1306-1318.
doi: 10.1007/s40314-016-0396-7. |
| [22] |
E. J. Rothwell and M. J. Cloud,
Automatic error analysis using intervals, IEEE Trans. Ed., 55 (2012), 9-15.
doi: 10.1109/TE.2011.2109722. |
| [23] |
M. Z. Sarikaya, A. Saglam and H. Yildirim,
On some Hadamard-type inequalities for $h$-convex functions, J. Math. Inequal., 2 (2008), 335-341.
doi: 10.7153/jmi-02-30. |
| [24] |
J. M. Snyder,
Interval analysis for computer graphics, SIGGRAPH Comput. Graph., 26 (1992), 121-130.
doi: 10.1145/142920.134024. |
| [25] |
S. Varošanec,
On $h-$convexity, J. Math. Anal. Appl., 326 (2007), 303-311.
doi: 10.1016/j.jmaa.2006.02.086. |
| [26] |
S.-H. Wang and F. Qi,
Hermite-Hadamard type inequalities for $s$-convex functions via Riemann-Liouville fraction integrals, J. Comput. Anal. Appl., 22 (2017), 1124-1334.
|
| [27] |
D. Zhao, T. An, G. Ye and F. M. Delfim,
On Hermite-Hadamard type inequalities for harmonical $h$-convex interval-valued functions, Math. Inequal. Appl., 23 (2020), 95-105.
doi: 10.7153/mia-2020-23-08. |
| [28] |
D. Zhao, T. An, G. Ye and W. Liu, New Jensen and Hermite-Hadamard type inequalities for $h$-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 14pp.
doi: 10.1186/s13660-018-1896-3. |
show all references
References:
| [1] |
Y. An, G. Ye, D. Zhao and W. Liu, Hermite-Hadamard type inequalities for interval $(h_{1}, h_{2})$- convex functions, Mathematics, 7 (2019).
doi: 10.3390/math7050436. |
| [2] |
M. U. Awan, M. A. Noor, K. I. Noor and A. G. Khan,
Some new classes of convex functions and inequalities, Miskolc Math. Notes, 19 (2018), 77-94.
doi: 10.18514/MMN.2018.2179. |
| [3] |
Y. Chalco-Cano, A. Flores-Franulič and H. Román-Flores,
Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457-472.
|
| [4] |
T. M. Costa,
Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets and Systems, 327 (2017), 31-47.
doi: 10.1016/j.fss.2017.02.001. |
| [5] |
T. M. Costa and H. Román-Flores,
Some integral inequalities for fuzzy-interval-valued functions, Inform. Sci., 420 (2017), 110-125.
doi: 10.1016/j.ins.2017.08.055. |
| [6] |
E. de Weerdt, Q. P. Chu and J. A. Mulder,
Neural network output optimization using interval analysis, IEEE Trans. Neural Netw., 20 (2009), 638-653.
doi: 10.1109/TNN.2008.2011267. |
| [7] |
A. Dinghas,
Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Z., 66 (1956), 173-188.
doi: 10.1007/BF01186606. |
| [8] |
S. S. Dragomir,
Inequalities of Hermite-Hadamard type for $h$-convex functions on linear speaces, Proyecciones, 32 (2015), 323-341.
doi: 10.4067/S0716-09172015000400002. |
| [9] |
N. A. Gasilov and Ş. E. Amrahov,
Solving a nonhomogeneous linear system of interval differential equations, Soft Comput., 22 (2018), 3817-3828.
doi: 10.1007/s00500-017-2818-x. |
| [10] |
A. G. Ghazanfari,
The Hermite-Hadamard type inequalities for operator $s$-convex functions, J. Adv. Res. Pure Math., 6 (2014), 52-61.
|
| [11] |
İ. İşcan,
Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.
|
| [12] |
İ. İşcan,
On generalization of different type inequalities for harmonically quasi-convex functions via fractional integrals, Appl. Math. Comput., 275 (2016), 287-298.
doi: 10.1016/j.amc.2015.11.074. |
| [13] |
M. Kunt and İ Işcan,
Hermite-Hadamard-Féjer type inequalities for $p$-convex functions via fractional integrals, Iranian J. Sci. Tech., 42 (2018), 2079-2089.
doi: 10.1007/s40995-017-0352-4. |
| [14] |
M. A. Latif and M. Alomari,
On Hadamard-type inequalities for $h$-convex functions on the co-ordinates, Int. J. Math. Anal. (Ruse), 3 (2009), 1645-1656.
|
| [15] |
M. A. Latif, S. S. Dragomir and E. Momoniat,
|
| [16] |
Y. Li and T. Wang,
Interval analysis of the wing divergence, Aerosp. Sci. Technol., 74 (2018), 17-21.
doi: 10.1016/j.ast.2018.01.001. |
| [17] |
M. V. Mihai, M. A. Noor, K. I. Noor and M. U. Awan,
Some integral inequalities for harmonic $h$-convex functions involving hypergeometric functions, Appl. Math. Comput., 252 (2015), 257-262.
doi: 10.1016/j.amc.2014.12.018. |
| [18] |
R. E. Moore, Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1996. |
| [19] |
M. A. Noor, K. I. Noor, M. U. Awan and S. Costache,
Some integral inequalities for harmonically $h$-convex functions, Politehm. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77 (2015), 5-16.
|
| [20] |
M. A. Noor, K. I. Noor, S. Iftikhar and C. Ionescu,
Hermite-Hadamard inequalities for co-ordinated harmonic convex functions., Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2017), 25-34.
|
| [21] |
H. Román-Flores, Y. Chalco-Cano and W. A. Lodwick,
Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2018), 1306-1318.
doi: 10.1007/s40314-016-0396-7. |
| [22] |
E. J. Rothwell and M. J. Cloud,
Automatic error analysis using intervals, IEEE Trans. Ed., 55 (2012), 9-15.
doi: 10.1109/TE.2011.2109722. |
| [23] |
M. Z. Sarikaya, A. Saglam and H. Yildirim,
On some Hadamard-type inequalities for $h$-convex functions, J. Math. Inequal., 2 (2008), 335-341.
doi: 10.7153/jmi-02-30. |
| [24] |
J. M. Snyder,
Interval analysis for computer graphics, SIGGRAPH Comput. Graph., 26 (1992), 121-130.
doi: 10.1145/142920.134024. |
| [25] |
S. Varošanec,
On $h-$convexity, J. Math. Anal. Appl., 326 (2007), 303-311.
doi: 10.1016/j.jmaa.2006.02.086. |
| [26] |
S.-H. Wang and F. Qi,
Hermite-Hadamard type inequalities for $s$-convex functions via Riemann-Liouville fraction integrals, J. Comput. Anal. Appl., 22 (2017), 1124-1334.
|
| [27] |
D. Zhao, T. An, G. Ye and F. M. Delfim,
On Hermite-Hadamard type inequalities for harmonical $h$-convex interval-valued functions, Math. Inequal. Appl., 23 (2020), 95-105.
doi: 10.7153/mia-2020-23-08. |
| [28] |
D. Zhao, T. An, G. Ye and W. Liu, New Jensen and Hermite-Hadamard type inequalities for $h$-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 14pp.
doi: 10.1186/s13660-018-1896-3. |
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