American Institute of Mathematical Sciences

Advanced
May  2021, 4(2): 89-103. doi: 10.3934/mfc.2021005

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

Ruonan Liu and  Run Xu ,

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

* Corresponding author: Run Xu

Received  December 2020 Revised  March 2021 Published  May 2021 Early access  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.

Keywords: Hermite-Hadamard inequality, interval analysis, harmonical $ (h1,h2)- $convex, harmonical $ (h1,h2)- $convex interval-valued functions, Riemann integrable.
Mathematics Subject Classification: 26D15, 26E25, 28B20.
Citation: Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, 2021, 4 (2) : 89-103. 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. 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.  Google Scholar

[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.   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 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.  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. Latif, S. 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. 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.  Google Scholar

[18]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.   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 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.  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. Latif, S. 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. 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.  Google Scholar

[18]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  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. 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.  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. 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.  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

[1]

Qiang Tu. A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5397-5407. doi: 10.3934/dcds.2021081
[2]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $ H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004
[3]

Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094
[4]

Lakehal Belarbi. Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group. Electronic Research Archive, 2020, 28 (1) : 157-163. doi: 10.3934/era.2020010
[5]

Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $. Advances in Mathematics of Communications, 2021, 15 (3) : 423-440. doi: 10.3934/amc.2020075
[6]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067
[7]

Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072
[8]

Hui Liu, Ling Zhang. Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021172
[9]

Xinliang An, Avy Soffer. Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013
[10]

Min Zhao, Changzheng Qu. The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2857-2883. doi: 10.3934/cpaa.2020245
[11]

Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189
[12]

Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035
[13]

Kailu Yang, Xiaomiao Wang, Menglong Zhang, Lidong Wang. Some progress on optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021012
[14]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378
[15]

Jordi-Lluís Figueras, Thomas Ohlson Timoudas. Sharp $ \frac12 $-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4519-4531. doi: 10.3934/dcds.2020189
[16]

Liqiang Jin, Yanyan Yin, Kok Lay Teo, Fei Liu. Event-triggered mixed $ H_\infty $ and passive control for Markov jump systems with bounded inputs. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1343-1355. doi: 10.3934/jimo.2020024
[17]

Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $ h\mathbb{Z} $. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131
[18]

Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368
[19]

Junlin Xiong, Wenjie Liu. $ H_{\infty} $ observer-based control for large-scale systems with sparse observer communication network. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 331-343. doi: 10.3934/naco.2020005
[20]

Canghua Jiang, Dongming Zhang, Chi Yuan, Kok Ley Teo. An active set solver for constrained $ H_\infty $ optimal control problems with state and input constraints. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 135-157. doi: 10.3934/naco.2021056

 Impact Factor: 

Tools

Metrics

  • PDF downloads (243)
  • HTML views (266)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy