In this study, the concept of $ (m, n)- $polynomial $ (p_{1}, p_{2}) $- convex functions on the co-ordinates has been established with some basic properties. Dependent on this new concept, a new Hermite-Hadamard type inequality has been proved, then some new integral inequalities have been obtained for partial differentiable $ (m, n)- $polynomial $ (p_{1}, p_{2}) $- convex functions on the co-ordinates. Several special cases that some of them proved in earlier works have been considered.
Citation: Ahmet Ocak Akdemir, Saad Ihsan Butt, Muhammad Nadeem, Maria Alessandra Ragusa. Some new integral inequalities for a general variant of polynomial convex functions[J]. AIMS Mathematics, 2022, 7(12): 20461-20489. doi: 10.3934/math.20221121
In this study, the concept of $ (m, n)- $polynomial $ (p_{1}, p_{2}) $- convex functions on the co-ordinates has been established with some basic properties. Dependent on this new concept, a new Hermite-Hadamard type inequality has been proved, then some new integral inequalities have been obtained for partial differentiable $ (m, n)- $polynomial $ (p_{1}, p_{2}) $- convex functions on the co-ordinates. Several special cases that some of them proved in earlier works have been considered.
| [1] |
S. S. Dragomir, On Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math., 5 (2001), 775–788. https://doi.org/10.11650/twjm/1500574995 doi: 10.11650/twjm/1500574995
|
| [2] |
M. E. Özdemir, M. A. Latif, A. O. Akdemir, On some Hadamard-type inequalities for product of two $h-$convex functions on the co-ordinates, J. Inequal. Appl., 2012 (2012), 21. https://doi.org/10.1186/1029-242X-2012-21 doi: 10.1186/1029-242X-2012-21
|
| [3] |
M. K. Bakula, J. Pečarić, On the Jensen's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math., 10 (2006), 1271–1292. https://doi.org/10.11650/twjm/1500557302 doi: 10.11650/twjm/1500557302
|
| [4] |
S. Banić, M. K. Bakula, Jensen's inequality for functions superquadratic on the co-ordinates, J. Math. Inequal., 9 (2015), 1365–1375. https://doi.org/10.7153/JMI-09-104 doi: 10.7153/JMI-09-104
|
| [5] |
S. I. Butt, A. Kashuri, M. Nadeem, A. Aslam, W. Gao, Approximately two-dimensional harmonic $(p_1, h_1)-(p_2, h_2)$-convex functions and related integral inequalities, J. Inequal. Appl., 2020 (2020), 230. https://doi.org/10.1186/s13660-020-02495-6 doi: 10.1186/s13660-020-02495-6
|
| [6] |
M. E. Özdemir, A. O. Akdemir, C. Yildiz, On Co-ordinated quasi-convex functions, Czech. Math. J., 62 (2012), 889–900. https://doi.org/10.1007/s10587-012-0072-z doi: 10.1007/s10587-012-0072-z
|
| [7] | M. Alomari, M. Darus, The Hadamard's inequality for $ s- $convex functions of $2-$variables on the co-ordinates, Int. J. Math. Anal., 2 (2008), 629–638. |
| [8] |
T. Du, T. Zhou, On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings, Chaos Soliton. Fract., 156 (2022), 111846. https://doi.org/10.1016/j.chaos.2022.111846 doi: 10.1016/j.chaos.2022.111846
|
| [9] |
M. V. Mihai, M. U. Awan, M. A. Noor, T. S. Du, A. G. Khan, Two dimensional operator preinvex functions and associated Hermite-Hadamard type inequalities. Filomat, 32 (2018), 2825–2836. https://doi.org/10.2298/FIL1808825M doi: 10.2298/FIL1808825M
|
| [10] |
M. Z. Sarıkaya, E. Set, M. Emin Özdemir, S. S. Dragomir, New some Hadamard's type inequalities for co-ordinated convex functions, Tamsui Oxford J. Inf. Math. Sci., 28 (2012), 137–152. https://doi.org/10.48550/arXiv.1005.0700 doi: 10.48550/arXiv.1005.0700
|
| [11] |
M. E. Özdemir, H. Kavurmaci, A. O. Akdemir, M. Avci, Inequalities for convex and $s-$convex functions on $\Delta = \left[ a, b \right] \times \left[ c, d\right]$, J. Inequal. Appl., 2012 (2012), 20. https://doi.org/10.1186/1029-242X-2012-20 doi: 10.1186/1029-242X-2012-20
|
| [12] |
T. Toplu, M. Kadakal, I. Iscan, On $n-$polynomial convexity and some related inequalities, AIMS Mathematics, 5 (2020), 1304–1318. https://doi.org/10.3934/math.2020089 doi: 10.3934/math.2020089
|
| [13] |
S. I. Butt, S. Rashid, M. Tariq, M. K. Wang, Novel refinements via polynomial harmonically type convex functions and application in special functions, J. Funct. Space., 2021 (2021), 1–17. https://doi.org/10.1155/2021/6615948 doi: 10.1155/2021/6615948
|
| [14] |
S. I. Butt, A. Kashuri, M. Tariq, J. Nasir, A. Aslam, W. Gao, Hermite Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications, Adv. Differ. Equ., 2020 (2020), 508. https://doi.org/10.1186/s13662-020-02967-5 doi: 10.1186/s13662-020-02967-5
|
| [15] |
L. Xu, S. Yu, T. Du, Properties and integral inequalities arising from the generalized n-polynomial convexity in the frame of fractal space. Fractals, 30 (2022), 2250084. https://doi.org/10.1142/S0218348X22500840 doi: 10.1142/S0218348X22500840
|
| [16] |
C. Park, Y. Chu, M. S. Saleem, N. Jahangir, N. Rehman, On $ n- $polynomial $p-$convex functions and some related inequalities, Adv. Differ. Equ., 2020 (2020), 666. https://doi.org/10.1186/s13662-020-03123-9 doi: 10.1186/s13662-020-03123-9
|
| [17] | M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, New York: Dover Publications, 1965. |
| [18] |
S. I. Butt, A. O. Akdemir, M. Nadeem, N. Mlaiki, I. Iscan, T. Abdeljawad, $(m, n)-$Harmonically polynomial convex functions and some Hadamard inequalities on co-ordinates, AIMS Mathematics, 6 (2021), 4677–4690. https://doi.org/10.3934/math.2021275 doi: 10.3934/math.2021275
|
Ahmet Ocak Akdemir, Saad Ihsan Butt, Muhammad Nadeem, Maria Alessandra Ragusa. Some new integral inequalities for a general variant of polynomial convex functions[J]. AIMS Mathematics, 2022, 7(12): 20461-20489. doi: 10.3934/math.20221121
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