×
Awan, Muhammad Uzair; Noor, Muhammad Aslam; Mihai, Marcela V.; Noor, Khalida Inayat

On bounds involving \(k\)-Appell’s hypergeometric functions. (English) Zbl 1370.26037

J. Inequal. Appl. 2017, Paper No. 118, 15 p. (2017).
Summary: In this paper, we derive a new extension of Hermite-Hadamard’s inequality via \(k\)-Riemann-Liouville fractional integrals. Two new \(k\)-fractional integral identities are also derived. Then, using these identities as an auxiliary result, we obtain some new \(k\)-fractional bounds which involve \(k\)-Appell’s hypergeometric functions. These bounds can be viewed as new \(k\)-fractional estimations of trapezoidal and mid-point type inequalities. These results are obtained for the functions which have the harmonic convexity property. We also discuss some special cases which can be deduced from the main results of the paper.

Cited in 7 Documents

MSC:

26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
33B15 Gamma, beta and polygamma functions
33C65 Appell, Horn and Lauricella functions

Keywords:

convex functions; harmonic convex functions; \(k\)-fractional integrals; \(k\)-Appell’s hypergeometric functions; inequalities
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Breckner, WW: Stetigkeitsaussagen für eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen. Pupl. Inst. Math. 23, 13-20 (1978) · Zbl 0416.46029
[2] Cristescu, G, Lupsa, L: Non-connected Convexities and Applications. Kluwer Academic Publishers, Dordrecht, Holland (2002) · Zbl 1037.52008 · doi:10.1007/978-1-4615-0003-2
[3] Cristescu, G, Noor, MA, Awan, MU: Bounds of the second degree cumulative frontier gaps of functions with generalized convexity. Carpath. J. Math. 31(2), 173-180 (2015) · Zbl 1349.26011
[4] Dragomir, SS: Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces. RGMIA Res. Rep. Coll., Article ID 72 (2013) · Zbl 1163.33300
[5] Dragomir, SS, Pečarić, J, Persson, LE: Some inequalities of Hadamard type. Soochow J. Math. 21, 335-341 (1995) · Zbl 0834.26009
[6] Dragomir, SS, Pearce, CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications. Victoria University Australia (2000) · Zbl 1204.26036
[7] Godunova, EK, Levin, VI: Neravenstva dlja funkcii sirokogo klassa, soderzascego vypuklye, monotonnye i nekotorye drugie vidy funkii. Vycislitel. Mat. i. Fiz. Mezvuzov. Sb. Nauc. Trudov, MGPI, Moskva., 138-142 (1985) (in Russian)
[8] Iscan, I: Hermite-Hadamard type inequalities for harmonically convex functions. Hacet. J. Math. Stat. 43(6), 935-942 (2014) · Zbl 1321.26048
[9] Noor, MA, Noor, KI, Awan, MU: Integral inequalities for coordinated harmonically convex functions. Complex Var. Elliptic Equ. 60(6), 776-786 (2015) · Zbl 1317.26019 · doi:10.1080/17476933.2014.976814
[10] Noor, MA, Noor, KI, Awan, MU, Costache, S: Some integral inequalities for harmonically h-convex functions. UPB Sci. Bull., Series A. 77(1), 5-16 (2015) · Zbl 1349.26063
[11] Shi, H-N, Zhang, J: Some new judgement theorems of Schur geometric and Schur harmonic convexities for a class of symmetric functions. J. Inequal. Appl. 2013, 527 (2013) · Zbl 1297.26024 · doi:10.1186/1029-242X-2013-527
[12] Varočanec, S: On h-convexity. J. Math. Anal. Appl. 326, 303-311 (2007) · Zbl 1111.26015 · doi:10.1016/j.jmaa.2006.02.086
[13] Iscan, I: Hermite-Hadamard and Simpson-like type inequalities for differentiable harmonically convex functions. J. Math. 2014, Article ID 346305 (2014) · Zbl 1489.26034 · doi:10.1155/2014/346305
[14] Mihai, MV, Noor, MA, Noor, KI, Awan, MU: Some integral inequalities for h-convex functions involving hypergeometric functions. Appl. Math. Comput. 252(1), 257-262 (2015) · Zbl 1338.26016 · doi:10.1016/j.amc.2014.12.018
[15] Park, J: Hermite-Hadamard-like and Simpson-like type inequalities for harmonically convex functions. Int. J. Math. Anal. 8(27), 1321-1337 (2014) · doi:10.12988/ijma.2014.45136
[16] Antczak, T, Pitea, A: Parametric approach to multitime multiobjective fractional variational problems under (F,ρ)\((F,\rho)\)-convexity. Optim. Control Appl. Methods 37(5), 831-847 (2016) · Zbl 1348.90560 · doi:10.1002/oca.2192
[17] Pitea, A, Antczak, T: Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems. J. Inequal. Appl. 2014, Art. No. 333 (2014) · Zbl 1308.65107 · doi:10.1186/1029-242X-2014-333
[18] Pitea, A, Postolache, M: Duality theorems for a new class of multitime multiobjective variational problems. J. Glob. Optim. 54(1), 47-58 (2012) · Zbl 1250.65084 · doi:10.1007/s10898-011-9740-z
[19] Pitea, A, Postolache, M: Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions. Optim. Lett. 6(3), 459-470 (2012) · Zbl 1280.90109 · doi:10.1007/s11590-010-0272-0
[20] Pitea, A, Postolache, M: Minimization of vectors of curvilinear functionals on the second order jet bundle. Sufficient efficiency conditions. Optim. Lett. 6(8), 1657-1669 (2012) · Zbl 1258.49028 · doi:10.1007/s11590-011-0357-4
[21] Khattri, SK: Three proofs of the inequality e<(1+1n)n+\(0.5e<(1+\frac{1}{n})^{n+0.5}. Am\). Math. Mon. 117(3), 273-277 (2010) · Zbl 1204.26036 · doi:10.4169/000298910x480126
[22] Kilbas, A, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, Netherlands (2006) · Zbl 1092.45003
[23] Sarikaya, MZ, Set, E, Yaldiz, H, Basak, N: Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57, 2403-2407 (2013) · Zbl 1286.26018 · doi:10.1016/j.mcm.2011.12.048
[24] Diaz, R: On hypergeometric functions and Pochhammer k-symbol. Divulg. Mat. 2, 179-192 (2007) · Zbl 1163.33300
[25] Mubeen, S, Habibullah, GM: k-fractional integrals and applications. Int. J. Contemp. Math. Sci. 7(2), 89-94 (2012) · Zbl 1248.33005
[26] Sarikaya, MZ, Karaca, A: On the k-Riemann-Liouville fractional integral and applications. Int. J. Math. Stat. 1(3), 33-43 (2014)
[27] Mubeen, S, Iqbal, S, Rahman, G: Contiguous function relations and an integral representation for Appell k-series F1,k \(F_{1,k}\). Inter. J. Math. Research 4(2), 53-63 (2015) · doi:10.18488/journal.24/2015.4.2/24.2.53.63
[28] Set, E, Iscan, I, Zehir, F: On some new inequalities of Hermite-Hadamard type involving harmonically convex functions via fractional integrals. Konuralp J. Math. 3(1), 42-55 (2015) · Zbl 1337.26016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.