k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications
Abstract
:1. Introduction and Preliminaries
- Arithmetic Mean
- Logarithmic Mean
- Generalized Log-Mean
2. Auxiliary Result
3. Main Results
4. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
- Sarikaya, M.Z.; Set, E.; Yaldiz, H.; Basak, N. Hermite–Hadamards inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 2013, 57, 2403–2407. [Google Scholar] [CrossRef]
- Kilbas, A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier B.V.: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Diaz, R. On hypergeometric functions and Pochhammer k-symbol. Divulgaciones Matema´ticas 2007, 2, 179–192. [Google Scholar]
- Sarikaya, M.Z.; Karaca, A. On the k-Riemann–Liouville fractional integral and applications. Int. J. Stat. Math. 2014, 1, 33–43. [Google Scholar] [CrossRef]
- Mubeen, S.; Iqbal, S.; Rahman, G. Contiguous function relations and an integral reprezentation for Appell k-series F1,k. Int. J. Math. Res. 2015, 4, 53–63. [Google Scholar] [CrossRef]
- Breckner, W.W. Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer Funktionen in topologischen linearen Räumen. Publ. Inst. Math. 1978, 23, 13–20. [Google Scholar]
- Dragomir, S.S.; Pearce, C.E.M. Selected Topics on Hermite–Hadamard Inequalities and Applications; Victoria University: Melbourne, Australia, 2000. [Google Scholar]
- Noor, M.A.; Noor, K.I.; Awan, M.U.; Khan, S. Fractional Hermite–Hadamard inequalities for some new classes of Godunova–Levin functions. Appl. Math. Inf. Sci. 2014, 8, 2865–2872. [Google Scholar] [CrossRef]
- Set, E. New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals. Comput. Math. Appl. 2012, 63, 1147–1154. [Google Scholar] [CrossRef]
- Niculescu, C.P.; Persson, L.-E. Convex Functions and Their Applications. A Contemporary Approach, 2nd ed.; Springer: New York, NY, USA, 2018. [Google Scholar]
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Awan, M.U.; Noor, M.A.; Mihai, M.V.; Noor, K.I. k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications. Fractal Fract. 2019, 3, 38. https://doi.org/10.3390/fractalfract3030038
Awan MU, Noor MA, Mihai MV, Noor KI. k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications. Fractal and Fractional. 2019; 3(3):38. https://doi.org/10.3390/fractalfract3030038
Chicago/Turabian StyleAwan, Muhammad Uzair, Muhammad Aslam Noor, Marcela V. Mihai, and Khalida Inayat Noor. 2019. "k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications" Fractal and Fractional 3, no. 3: 38. https://doi.org/10.3390/fractalfract3030038