M. U. Awan, N. Akhtar S. Iftikhar, M. A. Noor and Y.-M. Chu: New Hermite–Hadamard type inequalities for n polynomial harmonically convex functions, J. Inequal Appl., 2020:125 (2020).
W. W. Breckner: Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numér. Théor. Approx., 22 (1993), 39–51.
Y.-M. Chu, M. Adil Khan, T. U. Khan and T. Ali: Generalizations of Hermite–Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 4305–4316.
Y.-M. Chu, M. Adil Khan, T. U. Khan and J. Khan: Some new inequalities of Hermite–Hadamard type for s-convex
functions with applications, Open Math.,15 (2017), 1414–1430.
Y. Chalco–Cano, A. Flores–Franulic and H. Román–Flores: ˘ Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457–472.
Y. Chalco–Cano, W. A. Lodwick and W. Condori–Equice: Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput., 19 (2015), 3293–3300.
T. M. Costa, H. Román–Flores: Some integral inequalities for fuzzy-interval-valued functions, Inf. Sci., 420 (2017), 110–125.
T. M. Costa: Jensens inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets Syst., 327 (2017), 31–47.
M. R. Delavar, M. De La Sen: Some generalizations of Hermite–Hadamard type inequalities, SpringerPlus, 5:1661 (2016).
A. Guessab, G. Schmeisser: Sharp integral inequalities of the Hermite–Hadamard type, J. Approx. Theory, 115 (2)
(2002), 260–288.
[11] I. Iscan: Hermite–Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935–942.
A. Iqbal, M. Adil Khan, Sana Ullah and Y.-M. Chu: Some new Hermite–Hadamard type inequalities associated with conformable fractional integrals and their applications, Journal of Function Spaces, 2020 (2020), Art. ID 9845407.
M. A. Khan, Y. Khurshid and T. Ali: Hermite–Hadamard Inequality for fractional integrals via η-convex functions, Acta Math. Univ. Comenianae., LXXXVI (1) (2017), 153–164.
M. Adil Khan, T. Ali and T. U. Khan: Hermite–Hadamard Type Inequalities with Applications, Fasciculi Mathematici, 59 (2017), 57–74.
M. Adil Khan, N. Mohammad, E. R. Nwaeze and Y.-M. Chu: Quantum Hermite–Hadamard inequality by means of a green function, Adv. Diff. Equ., 2020:99 (2020).
J. Matkowski, K. Nikodem: An integral Jensen inequality for convex multifunctions, Results Math., 26 (1994), 348–353.
F.-C. Mitroi, K. Nikodem, S. Wa¸sowicz: Hermite–Hadamard inequalities for convex set-valued functions, Demonstratio Math., 46 (2013), 655–662.
R. E. Moore: Interval Analysis, Prentice-Hall: Englewood Cliffs, NJ, USA, (1966).
R. E. Moore: Method and Applications of Interval Analysis, SIAM: Philadelphia, PA, USA, (1979).
R. E. Moore, R. B. Kearfott and M. J. Cloud: Introduction to Interval Analysis, SIAM: Philadelphia, PA, USA, (2009).
K. Nikodem, J. L. Sánchez and L. Sánchez: Jensen and Hermite–Hadamard inequalities for strongly convex set-valued maps, Math. Æterna, 4 (2014), 979–987.
E. R. Nwaeze: Inequalities of the Hermite–Hadamard type for Quasi-convex functions via the (k, s)-Riemann–Liouville fractional integrals, Fractional Differ. Calc., 8(2) (2018), 327–336.
E. R. Nwaeze, D. F. M. Torres: Novel results on the Hermite–Hadamard kind inequality for η-convex functions by
means of the (k, r)-fractional integral operators. In: Silvestru Sever Dragomir, Praveen Agarwal, Mohamed Jleli and Bessem Samet (eds.) Advances in Mathematical Inequalities and Applications (AMIA). Trends in Mathematics. Birkhäuser, Singapore, (2018), 311–321.
E. R. Nwaeze, Muhammad Adil Khan and Yu-Ming Chu: Fractional inclusions of the Hermite–Hadamard type for
m-polynomial convex interval-valued functions, Adv. Diff. Equ., 2020:507 (2020).
E. Sadowska: Hadamard inequality and a refinement of Jensen inequality for set valued functions, Results Math., 32 (1997), 332–337.
J. Sun, B.-Y. Xi and F. Qi: Some new inequalities of the Hermite–Hadamard type for extended s-convex functions, J. Comput. Anal. Appl., 26(6) (2019), 985–996.
H. Román–Flores, Y. Chalco–Cano and W. A. Lodwick: Some integral inequalities for interval-valued functions, Comput. Appl. Math., 35 (2016), 1–13.
D. Zhao, T. An, G. Ye and W. Liu: New Jensen and HermiteHadamard type inequalities for h-convex interval-valued functions, J. Inequal Appl., 2018:302 (2018).
D. Zhao, T. An, G. Ye and D. F. M. Torres: On Hermite–Hadamard type inequalities for harmonical h-convex intervalvalued functions, Math. Inequal. Appl., 23 (1) (2020), 95–105.
Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions
Year 2021,
Volume: 4 Issue: 3, 260 - 273, 16.09.2021
We introduce the notion of $m$-polynomial harmonically convex interval-valued function. A relationship between a given interval-valued function and its component real-valued functions is pointed out. Moreover, some new Hermite--Hadamard type results are established for this class of functions. Our results complement and extend existing results in the literature. By taking $m\geq 2$, we derive loads of new and interesting inclusions. We anticipate that the idea outlined herein will trigger further investigations in this direction.
M. U. Awan, N. Akhtar S. Iftikhar, M. A. Noor and Y.-M. Chu: New Hermite–Hadamard type inequalities for n polynomial harmonically convex functions, J. Inequal Appl., 2020:125 (2020).
W. W. Breckner: Continuity of generalized convex and generalized concave set-valued functions, Rev. Anal. Numér. Théor. Approx., 22 (1993), 39–51.
Y.-M. Chu, M. Adil Khan, T. U. Khan and T. Ali: Generalizations of Hermite–Hadamard type inequalities for MT-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 4305–4316.
Y.-M. Chu, M. Adil Khan, T. U. Khan and J. Khan: Some new inequalities of Hermite–Hadamard type for s-convex
functions with applications, Open Math.,15 (2017), 1414–1430.
Y. Chalco–Cano, A. Flores–Franulic and H. Román–Flores: ˘ Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457–472.
Y. Chalco–Cano, W. A. Lodwick and W. Condori–Equice: Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput., 19 (2015), 3293–3300.
T. M. Costa, H. Román–Flores: Some integral inequalities for fuzzy-interval-valued functions, Inf. Sci., 420 (2017), 110–125.
T. M. Costa: Jensens inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets Syst., 327 (2017), 31–47.
M. R. Delavar, M. De La Sen: Some generalizations of Hermite–Hadamard type inequalities, SpringerPlus, 5:1661 (2016).
A. Guessab, G. Schmeisser: Sharp integral inequalities of the Hermite–Hadamard type, J. Approx. Theory, 115 (2)
(2002), 260–288.
[11] I. Iscan: Hermite–Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935–942.
A. Iqbal, M. Adil Khan, Sana Ullah and Y.-M. Chu: Some new Hermite–Hadamard type inequalities associated with conformable fractional integrals and their applications, Journal of Function Spaces, 2020 (2020), Art. ID 9845407.
M. A. Khan, Y. Khurshid and T. Ali: Hermite–Hadamard Inequality for fractional integrals via η-convex functions, Acta Math. Univ. Comenianae., LXXXVI (1) (2017), 153–164.
M. Adil Khan, T. Ali and T. U. Khan: Hermite–Hadamard Type Inequalities with Applications, Fasciculi Mathematici, 59 (2017), 57–74.
M. Adil Khan, N. Mohammad, E. R. Nwaeze and Y.-M. Chu: Quantum Hermite–Hadamard inequality by means of a green function, Adv. Diff. Equ., 2020:99 (2020).
J. Matkowski, K. Nikodem: An integral Jensen inequality for convex multifunctions, Results Math., 26 (1994), 348–353.
F.-C. Mitroi, K. Nikodem, S. Wa¸sowicz: Hermite–Hadamard inequalities for convex set-valued functions, Demonstratio Math., 46 (2013), 655–662.
R. E. Moore: Interval Analysis, Prentice-Hall: Englewood Cliffs, NJ, USA, (1966).
R. E. Moore: Method and Applications of Interval Analysis, SIAM: Philadelphia, PA, USA, (1979).
R. E. Moore, R. B. Kearfott and M. J. Cloud: Introduction to Interval Analysis, SIAM: Philadelphia, PA, USA, (2009).
K. Nikodem, J. L. Sánchez and L. Sánchez: Jensen and Hermite–Hadamard inequalities for strongly convex set-valued maps, Math. Æterna, 4 (2014), 979–987.
E. R. Nwaeze: Inequalities of the Hermite–Hadamard type for Quasi-convex functions via the (k, s)-Riemann–Liouville fractional integrals, Fractional Differ. Calc., 8(2) (2018), 327–336.
E. R. Nwaeze, D. F. M. Torres: Novel results on the Hermite–Hadamard kind inequality for η-convex functions by
means of the (k, r)-fractional integral operators. In: Silvestru Sever Dragomir, Praveen Agarwal, Mohamed Jleli and Bessem Samet (eds.) Advances in Mathematical Inequalities and Applications (AMIA). Trends in Mathematics. Birkhäuser, Singapore, (2018), 311–321.
E. R. Nwaeze, Muhammad Adil Khan and Yu-Ming Chu: Fractional inclusions of the Hermite–Hadamard type for
m-polynomial convex interval-valued functions, Adv. Diff. Equ., 2020:507 (2020).
E. Sadowska: Hadamard inequality and a refinement of Jensen inequality for set valued functions, Results Math., 32 (1997), 332–337.
J. Sun, B.-Y. Xi and F. Qi: Some new inequalities of the Hermite–Hadamard type for extended s-convex functions, J. Comput. Anal. Appl., 26(6) (2019), 985–996.
H. Román–Flores, Y. Chalco–Cano and W. A. Lodwick: Some integral inequalities for interval-valued functions, Comput. Appl. Math., 35 (2016), 1–13.
D. Zhao, T. An, G. Ye and W. Liu: New Jensen and HermiteHadamard type inequalities for h-convex interval-valued functions, J. Inequal Appl., 2018:302 (2018).
D. Zhao, T. An, G. Ye and D. F. M. Torres: On Hermite–Hadamard type inequalities for harmonical h-convex intervalvalued functions, Math. Inequal. Appl., 23 (1) (2020), 95–105.
Nwaeze, E. (2021). Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions. Constructive Mathematical Analysis, 4(3), 260-273. https://doi.org/10.33205/cma.793456
AMA
Nwaeze E. Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions. CMA. September 2021;4(3):260-273. doi:10.33205/cma.793456
Chicago
Nwaeze, Eze. “Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions”. Constructive Mathematical Analysis 4, no. 3 (September 2021): 260-73. https://doi.org/10.33205/cma.793456.
EndNote
Nwaeze E (September 1, 2021) Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions. Constructive Mathematical Analysis 4 3 260–273.
IEEE
E. Nwaeze, “Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions”, CMA, vol. 4, no. 3, pp. 260–273, 2021, doi: 10.33205/cma.793456.
ISNAD
Nwaeze, Eze. “Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions”. Constructive Mathematical Analysis 4/3 (September 2021), 260-273. https://doi.org/10.33205/cma.793456.
JAMA
Nwaeze E. Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions. CMA. 2021;4:260–273.
MLA
Nwaeze, Eze. “Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions”. Constructive Mathematical Analysis, vol. 4, no. 3, 2021, pp. 260-73, doi:10.33205/cma.793456.
Vancouver
Nwaeze E. Set Inclusions of the Hermite-Hadamard Type for $m$ Polynomial Harmonically Convex Interval Valued Functions. CMA. 2021;4(3):260-73.