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Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals

Year 2022, Volume: 7 Issue: 3, 219 - 230, 30.12.2022

Recep Türker Havva Kavurmacı Önalan

Abstract

We establish some new Generalized Hermite-Hadamard-type inequalities involving generalized fractional integrals for quasi-convex functions. Our
results are suitable with the literature. The analysis used in the proofs is fairly elementary and based on the use of Hölder inequality and the power inequality.

Keywords

Quasi convex functions Generalized Fractional Integrals Hölder Inequalities

Project Number

Yok

References

  • Ali, A., Gulshan, G., Hussain, R., Latif, A. Muddassar, M. and Park, J., Generalized Inequalities of the type of Hermite-Hadamard-Fejer with Quasi-Convex Functions by way of k-Fractional Derivatives, J. Computational Analysis and applications, 22(7) (2017), 1208-1219.
  • Alomari, M., Darus, M., Dragomir, S.S., Inequalities of Hermite-Hadamard's type for functions whose derivatives absolute values are quasi-convex. RGMIA Res. Rep. Coll., 12: Supplement, Article 14 2009, 1-11.
  • Budak, H., Ertuğral, F., Sarıkaya, M.Z., New Generalization of Hermite-Hadamard Type Inequalities via Generalized Fractional Integrals. ResearchGate Article (2017). https://www.researchgate.net/publication/321760465.
  • Belarbi S., Dahmani Z., On some new fractional integral inequalities, J. Ineq. Pure Appl. Math., 10(3) (2009), Art. 86.
  • Carter, M., Brunt, B.V.: The Lebesgue-Stieljies Integral: A Practical Introduction. New York, Springer (2000).
  • Dahmani Z., New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9(4) (2010), 493-497.
  • Dragomir, S.S, Agarwal, R.P.: Two Inequalities for Di erentiable Mappings and Applications To Special Meansof Real Numbers and to Trapezoidal Formula. Appl. Math. Lett. 11 (1999), 91-95.
  • Dragomir, S.S., Pearce, C.E.M., Quasi-Convex Functions and Hadamard's Inequality. B. Aust. Math. Soc. 57 (1998), 377-385.
  • Dragomir, S.S., Pearce, C.E.M., Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA, Monographs. Victoria University, 2000.
  • Dragomir, S.S., Pecaric, J., and Persson, L.E., Some Inequalities of Hadamard Type. Soochow J. of Math. 21 (1995), 335-341.
  • Ertuğral, F. Sarıkaya, M.Z., Budak, H., On Hermite-Hadamard Type Inequalities Associated With The Generalized Fractional Integrals. ResearchGate Article (2019). https://www.researchgate.net/publication/334634529.
  • Gidergelmez, H.F., Akkurt, A., Yıldırım, H., Hermite-Hadamard Type Inequalities for Generalized Fractional Integrals via Strongly Convex Functions. KJM. 7 (2019), 268-273.
  • Gorenflo R., Mainardi F., Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer Verlag, Wien., (1997), 223-276.
  • Hadamard, J.: ėtude sur les propri ėtės des fonctions entiėres en particulier dűne fonction considėrėe par Riemann. Journal de mathėmatiques pures et appliquėes, 4e sėrie. 9 (1893), 171-216.
  • Hussain, R., Ali, A., Latif, A. and Gulshan, G., Some k-Fractional associates of Hermite-Hadamard's Inequality for Quasi-Convex Functions and Applications to Special Means, Fractional Differential Calculus, 7(2) (2017),301-309.
  • 10.. : Ion, D. A., Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Sci. Ser., 34 (2007), 82--87. Iscan I., New general integral inequalities for quasi-geometrically convex functions via fractional integrals, J. Inequal. Appl., (491) (2013), 1-15.
  • I scan, I.: Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals. Stud. Univ. Babes -Bolyai Math. 60 (2015), 355-366.
  • Khan, M.A., Khurshid, Y., Ali, T.: Hermite{Hadamard Inequality for Fractional Integrals via -Convex Functions. Acta Math. Univ. Comen. 86 (2017), 153-164.
  • K lbas, A.A., Srivastava, H.M., Trujillo J.J.: Theory and Applications of Fractional Di erential Equations. Elsevier, Amsterdam (2006).
  • Kunt, M., Karap nar, D., Turhan, S., İscan, İ.: The Left Riemann-Liouville Fractional Hermite-Hadamard Type Inequalities for Convex Functions. Math. Slovaca. 69 (2019), 773-784.
  • Miller S., Ross B., An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley. Soons. USA., 1993.
  • Mitrinovi c, D.S.: Analytic Inequalities. Springer, Berlin (1970).
  • Mitrinovi c, D.S., Pe cari c, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer, Dordrecht (1993).
  • Niculescu, C.P., Persson, L.E.: Convex Functions and Their Applications: A Contemporary Approach. Springer, New York (2006). Özdemir, M. E., Yıldız, Ç., Annals of the University of Craiova, Mathematics and Computer Science., 40 (2), (2013),167-173.
  • Pearce C. E. M., Quasi-convexity, fractional programming and extrenal traffic congestion, in "Frontiers in Global Optimization", Kluwer, Dordrecht, "Nonlinear Optimization and its Applications", 74 (2004), 403-409.
  • Pearce C. E. M. and Rubinov A. M., P- functions, quasi-convex functions and Hadamard type inequalities, J. Math. Anal. Applic., 240 (1999), 92-104.
  • Pe cari c, J.E., Proschan, F.,Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Boston, Academic Press (1992).
  • Ponstein, J., Seven types of convexity, SIAM Review 9 (1967), 115-119.
  • Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973).
  • Sar kaya, M.Z., Ertuğral, F.: On The Generalized Hermite-Hadamard Inequalities. Annals of University of Craiova, Math. Comp. Sci. Ser.,47 (1) (2020),193-213.
  • Sarıkaya M.Z., Ogunmez H., On new inequalities via Riemann-Liouville fractional integration, Abst. Appl. Anal., Art.ID 428983, (2012), 10 pages. http://dx.doi.org./10.1155/2012/428983.
  • Sar kaya, M.Z., Set, E., Yaldiz, H., Ba sak, N.: Hermite-Hadamard's Inequalities for Fractional Integrals and Related Fractional Inequalities. Math. Comput. Modell. 57 (2013), 2403-2407. Sarikaya, M.Z. and Yildirim, H., On generalization of the Riesz potential, Indian Jour. of Math. and Mathematical Sci. 3 (2007), no. 2, 231-235.
  • Set, E. and Çelik, B, Fractional Hermite-Hadamard Type Inequalities for Quasi-convex functions, Ordu Univ. J. Sci. Tech. 6, 1 (2016), 137--149.
  • Set, E., Karata s S.S., Khan, M.A.: Hermite{Hadamard Type Inequalities Obtained via Fractional Integrals for Di erentiable m-Convex and ( ;m)-Convex Functions. Int. J. Anal (2016). https://doi.org/10.1155/2016/4765691.
  • Set, E., Sar kaya, M.Z., Ozdemir, M.E., Yıldırım, H.: The Hermite-Hadamard's Inequality for Some Convex Functions via Fractional Integrals and Related Results. JAMSI. 10 (2014), 69-83.
  • Shi, D.-P., Xi, B.-Y., Qi, F.: Hermite-Hadamard Type Inequalities for Reimann-Liouville Fractional Integrals of (alpha;m)-Convex Functions. Fractional Di er. Calc. 4 (2014), 31-43.
  • Tun c, M.: On New Inequalities for h-Convex Functions via Riemann-Liouville Fractional Integration. Filomat, 27(2013), 559-565.
  • Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326 (2007), 303-311.
  • Yaldiz, H., Set, E.: Re nements Hermite-Hadamard-Fejer Type Inequalities For Generalized Fractional Integrals.ResearchGate Article (2018). https://www.researchgate.net/publication/323357856
  • Yıldırım, M.E., Sarıkaya, M.Z., Yldırım, H.: The Generalized Hermite-Hadamard-Fejer Type Inequalities For Generalized Fractional Integrals, ResearchGate Article (2018) https://www.researchgate.net/publication/322592667.
  • Zhao, D.M., Ali, A., Kashuri, A., Budak, H.: Generalized Fractional Integral Inequalities of Hermite-Hadamard Type for Harmonically Convex Functions. Adv. Di er. Equ. 1 (2020),1-14.

Year 2022, Volume: 7 Issue: 3, 219 - 230, 30.12.2022

Recep Türker Havva Kavurmacı Önalan

Abstract

Supporting Institution

Yok

Project Number

Yok

Thanks

Recep Türker'in yüksek lisans tezinden yayınlanmıştır.

References

  • Ali, A., Gulshan, G., Hussain, R., Latif, A. Muddassar, M. and Park, J., Generalized Inequalities of the type of Hermite-Hadamard-Fejer with Quasi-Convex Functions by way of k-Fractional Derivatives, J. Computational Analysis and applications, 22(7) (2017), 1208-1219.
  • Alomari, M., Darus, M., Dragomir, S.S., Inequalities of Hermite-Hadamard's type for functions whose derivatives absolute values are quasi-convex. RGMIA Res. Rep. Coll., 12: Supplement, Article 14 2009, 1-11.
  • Budak, H., Ertuğral, F., Sarıkaya, M.Z., New Generalization of Hermite-Hadamard Type Inequalities via Generalized Fractional Integrals. ResearchGate Article (2017). https://www.researchgate.net/publication/321760465.
  • Belarbi S., Dahmani Z., On some new fractional integral inequalities, J. Ineq. Pure Appl. Math., 10(3) (2009), Art. 86.
  • Carter, M., Brunt, B.V.: The Lebesgue-Stieljies Integral: A Practical Introduction. New York, Springer (2000).
  • Dahmani Z., New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9(4) (2010), 493-497.
  • Dragomir, S.S, Agarwal, R.P.: Two Inequalities for Di erentiable Mappings and Applications To Special Meansof Real Numbers and to Trapezoidal Formula. Appl. Math. Lett. 11 (1999), 91-95.
  • Dragomir, S.S., Pearce, C.E.M., Quasi-Convex Functions and Hadamard's Inequality. B. Aust. Math. Soc. 57 (1998), 377-385.
  • Dragomir, S.S., Pearce, C.E.M., Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA, Monographs. Victoria University, 2000.
  • Dragomir, S.S., Pecaric, J., and Persson, L.E., Some Inequalities of Hadamard Type. Soochow J. of Math. 21 (1995), 335-341.
  • Ertuğral, F. Sarıkaya, M.Z., Budak, H., On Hermite-Hadamard Type Inequalities Associated With The Generalized Fractional Integrals. ResearchGate Article (2019). https://www.researchgate.net/publication/334634529.
  • Gidergelmez, H.F., Akkurt, A., Yıldırım, H., Hermite-Hadamard Type Inequalities for Generalized Fractional Integrals via Strongly Convex Functions. KJM. 7 (2019), 268-273.
  • Gorenflo R., Mainardi F., Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer Verlag, Wien., (1997), 223-276.
  • Hadamard, J.: ėtude sur les propri ėtės des fonctions entiėres en particulier dűne fonction considėrėe par Riemann. Journal de mathėmatiques pures et appliquėes, 4e sėrie. 9 (1893), 171-216.
  • Hussain, R., Ali, A., Latif, A. and Gulshan, G., Some k-Fractional associates of Hermite-Hadamard's Inequality for Quasi-Convex Functions and Applications to Special Means, Fractional Differential Calculus, 7(2) (2017),301-309.
  • 10.. : Ion, D. A., Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Sci. Ser., 34 (2007), 82--87. Iscan I., New general integral inequalities for quasi-geometrically convex functions via fractional integrals, J. Inequal. Appl., (491) (2013), 1-15.
  • I scan, I.: Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals. Stud. Univ. Babes -Bolyai Math. 60 (2015), 355-366.
  • Khan, M.A., Khurshid, Y., Ali, T.: Hermite{Hadamard Inequality for Fractional Integrals via -Convex Functions. Acta Math. Univ. Comen. 86 (2017), 153-164.
  • K lbas, A.A., Srivastava, H.M., Trujillo J.J.: Theory and Applications of Fractional Di erential Equations. Elsevier, Amsterdam (2006).
  • Kunt, M., Karap nar, D., Turhan, S., İscan, İ.: The Left Riemann-Liouville Fractional Hermite-Hadamard Type Inequalities for Convex Functions. Math. Slovaca. 69 (2019), 773-784.
  • Miller S., Ross B., An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley. Soons. USA., 1993.
  • Mitrinovi c, D.S.: Analytic Inequalities. Springer, Berlin (1970).
  • Mitrinovi c, D.S., Pe cari c, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer, Dordrecht (1993).
  • Niculescu, C.P., Persson, L.E.: Convex Functions and Their Applications: A Contemporary Approach. Springer, New York (2006). Özdemir, M. E., Yıldız, Ç., Annals of the University of Craiova, Mathematics and Computer Science., 40 (2), (2013),167-173.
  • Pearce C. E. M., Quasi-convexity, fractional programming and extrenal traffic congestion, in "Frontiers in Global Optimization", Kluwer, Dordrecht, "Nonlinear Optimization and its Applications", 74 (2004), 403-409.
  • Pearce C. E. M. and Rubinov A. M., P- functions, quasi-convex functions and Hadamard type inequalities, J. Math. Anal. Applic., 240 (1999), 92-104.
  • Pe cari c, J.E., Proschan, F.,Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Boston, Academic Press (1992).
  • Ponstein, J., Seven types of convexity, SIAM Review 9 (1967), 115-119.
  • Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973).
  • Sar kaya, M.Z., Ertuğral, F.: On The Generalized Hermite-Hadamard Inequalities. Annals of University of Craiova, Math. Comp. Sci. Ser.,47 (1) (2020),193-213.
  • Sarıkaya M.Z., Ogunmez H., On new inequalities via Riemann-Liouville fractional integration, Abst. Appl. Anal., Art.ID 428983, (2012), 10 pages. http://dx.doi.org./10.1155/2012/428983.
  • Sar kaya, M.Z., Set, E., Yaldiz, H., Ba sak, N.: Hermite-Hadamard's Inequalities for Fractional Integrals and Related Fractional Inequalities. Math. Comput. Modell. 57 (2013), 2403-2407. Sarikaya, M.Z. and Yildirim, H., On generalization of the Riesz potential, Indian Jour. of Math. and Mathematical Sci. 3 (2007), no. 2, 231-235.
  • Set, E. and Çelik, B, Fractional Hermite-Hadamard Type Inequalities for Quasi-convex functions, Ordu Univ. J. Sci. Tech. 6, 1 (2016), 137--149.
  • Set, E., Karata s S.S., Khan, M.A.: Hermite{Hadamard Type Inequalities Obtained via Fractional Integrals for Di erentiable m-Convex and ( ;m)-Convex Functions. Int. J. Anal (2016). https://doi.org/10.1155/2016/4765691.
  • Set, E., Sar kaya, M.Z., Ozdemir, M.E., Yıldırım, H.: The Hermite-Hadamard's Inequality for Some Convex Functions via Fractional Integrals and Related Results. JAMSI. 10 (2014), 69-83.
  • Shi, D.-P., Xi, B.-Y., Qi, F.: Hermite-Hadamard Type Inequalities for Reimann-Liouville Fractional Integrals of (alpha;m)-Convex Functions. Fractional Di er. Calc. 4 (2014), 31-43.
  • Tun c, M.: On New Inequalities for h-Convex Functions via Riemann-Liouville Fractional Integration. Filomat, 27(2013), 559-565.
  • Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326 (2007), 303-311.
  • Yaldiz, H., Set, E.: Re nements Hermite-Hadamard-Fejer Type Inequalities For Generalized Fractional Integrals.ResearchGate Article (2018). https://www.researchgate.net/publication/323357856
  • Yıldırım, M.E., Sarıkaya, M.Z., Yldırım, H.: The Generalized Hermite-Hadamard-Fejer Type Inequalities For Generalized Fractional Integrals, ResearchGate Article (2018) https://www.researchgate.net/publication/322592667.
  • Zhao, D.M., Ali, A., Kashuri, A., Budak, H.: Generalized Fractional Integral Inequalities of Hermite-Hadamard Type for Harmonically Convex Functions. Adv. Di er. Equ. 1 (2020),1-14.
There are 41 citations in total.

Details

Primary Language English
Journal Section Volume VII Issue III
Authors

Recep Türker This is me 0000-0001-6047-9007

Havva Kavurmacı Önalan 0000-0002-0034-778X

Project Number Yok
Publication Date December 30, 2022
Published in Issue Year 2022 Volume: 7 Issue: 3

Cite

APA Türker, R., & Kavurmacı Önalan, H. (2022). Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals. Turkish Journal of Science, 7(3), 219-230.
AMA Türker R, Kavurmacı Önalan H. Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals. TJOS. December 2022;7(3):219-230.
Chicago Türker, Recep, and Havva Kavurmacı Önalan. “Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals”. Turkish Journal of Science 7, no. 3 (December 2022): 219-30.
EndNote Türker R, Kavurmacı Önalan H (December 1, 2022) Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals. Turkish Journal of Science 7 3 219–230.
IEEE R. Türker and H. Kavurmacı Önalan, “Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals”, TJOS, vol. 7, no. 3, pp. 219–230, 2022.
ISNAD Türker, Recep - Kavurmacı Önalan, Havva. “Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals”. Turkish Journal of Science 7/3 (December 2022), 219-230.
JAMA Türker R, Kavurmacı Önalan H. Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals. TJOS. 2022;7:219–230.
MLA Türker, Recep and Havva Kavurmacı Önalan. “Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals”. Turkish Journal of Science, vol. 7, no. 3, 2022, pp. 219-30.
Vancouver Türker R, Kavurmacı Önalan H. Generalized Inequalities for Quasi-Convex Functions via Generalized Riemann-Liouville Fractional Integrals. TJOS. 2022;7(3):219-30.
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