doi: 10.3934/mfc.2021005

Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

* Corresponding author: Run Xu

Received  December 2020 Revised  March 2021 Published  April 2021

Fund Project: This research is supported by NationalScience Foundation of China (11671227, 11971015) and the Natural Science Foundation of Shandong Province (ZR2019MA034)

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.

Citation: Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021005
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.  Google Scholar

[2]

M. U. AwanM. A. NoorK. 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.  Google Scholar

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Y. Chalco-CanoA. Flores-Franulič and H. Román-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457-472.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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E. de WeerdtQ. 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.  Google Scholar

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A. Dinghas, Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Z., 66 (1956), 173-188.  doi: 10.1007/BF01186606.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[10]

A. G. Ghazanfari, The Hermite-Hadamard type inequalities for operator $s$-convex functions, J. Adv. Res. Pure Math., 6 (2014), 52-61.   Google Scholar

[11]

İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[15]

M. A. LatifS. S. Dragomir and E. Momoniat, type inequalities for harmonically-convex functions with applications, J. Appl. Anal. Comput., 7 (2017), 795-813.  doi: 10.11948/2017050.  Google Scholar

[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.  Google Scholar

[17]

M. V. MihaiM. A. NoorK. 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.  Google Scholar

[18]

R. E. Moore, Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1996.  Google Scholar

[19]

M. A. NoorK. I. NoorM. 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.   Google Scholar

[20]

M. A. NoorK. I. NoorS. 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.   Google Scholar

[21]

H. Román-FloresY. 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.  Google Scholar

[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.  Google Scholar

[23]

M. Z. SarikayaA. 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.  Google Scholar

[24]

J. M. Snyder, Interval analysis for computer graphics, SIGGRAPH Comput. Graph., 26 (1992), 121-130.  doi: 10.1145/142920.134024.  Google Scholar

[25]

S. Varošanec, On $h-$convexity, J. Math. Anal. Appl., 326 (2007), 303-311.  doi: 10.1016/j.jmaa.2006.02.086.  Google Scholar

[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.   Google Scholar

[27]

D. ZhaoT. AnG. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. U. AwanM. A. NoorK. 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.  Google Scholar

[3]

Y. Chalco-CanoA. Flores-Franulič and H. Román-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457-472.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. de WeerdtQ. 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.  Google Scholar

[7]

A. Dinghas, Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Z., 66 (1956), 173-188.  doi: 10.1007/BF01186606.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. G. Ghazanfari, The Hermite-Hadamard type inequalities for operator $s$-convex functions, J. Adv. Res. Pure Math., 6 (2014), 52-61.   Google Scholar

[11]

İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

M. A. LatifS. S. Dragomir and E. Momoniat, type inequalities for harmonically-convex functions with applications, J. Appl. Anal. Comput., 7 (2017), 795-813.  doi: 10.11948/2017050.  Google Scholar

[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.  Google Scholar

[17]

M. V. MihaiM. A. NoorK. 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.  Google Scholar

[18]

R. E. Moore, Interval Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1996.  Google Scholar

[19]

M. A. NoorK. I. NoorM. 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.   Google Scholar

[20]

M. A. NoorK. I. NoorS. 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.   Google Scholar

[21]

H. Román-FloresY. 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.  Google Scholar

[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.  Google Scholar

[23]

M. Z. SarikayaA. 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.  Google Scholar

[24]

J. M. Snyder, Interval analysis for computer graphics, SIGGRAPH Comput. Graph., 26 (1992), 121-130.  doi: 10.1145/142920.134024.  Google Scholar

[25]

S. Varošanec, On $h-$convexity, J. Math. Anal. Appl., 326 (2007), 303-311.  doi: 10.1016/j.jmaa.2006.02.086.  Google Scholar

[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.   Google Scholar

[27]

D. ZhaoT. AnG. 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.  Google Scholar

[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.  Google Scholar

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