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Wirtinger type inequalities for conformable fractional integrals

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MATHEMATICA, 61 (84), No 1, 2019, pp. 79–84

WIRTINGER TYPE INEQUALITIES

FOR CONFORMABLE FRACTIONAL INTEGRALS

MEHMET ZEKI SARIKAYA and CANDAN CAN B˙IL˙IS¸ ˙IK

Abstract. The aim of this paper is to establish a generalization and a refinement of Wirtinger’s inequality for conformable fractional integrals.

MSC 2010. 26D15, 26A51, 26A33, 26A42.

Key words. Conformable fractional integrals, H¨older’s inequality, Wirtinger inequality.

REFERENCES

[1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015) 57–66.

[2] E.F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin, 1983.

[3] D.R. Anderson, Taylor’s formula and integral inequalities for conformable fractional derivatives, in Contributions in Mathematics and Engineering. In Honor of Constantin Carath´eodory, Springer, 2016.

[4] P.R. Beesack, Integral inequalities involving a function and its derivative, Amer. Math. Monthly, 78 (1971), 705–741.

[5] D.S. Mitrinovi´c, J.E. Pecari´c and A.M. Fink, Inequalities involving functions and their integrals and derivatives, Springer, Dordrecht, 1991.

[6] M.A. Hammad and R. Khalil, Conformable fractional heat differential equations, Inter-national Journal of Pure and Applied Mathematics, 94 (2014), 215–221.

[7] M.A. Hammad and R. Khalil, Abel’s formula and wronskian for conformable fractional differential equations, International Journal of Differential Equations and Applications, 13 (2014), 177–183.

[8] O.S. Iyiola and E.R.Nwaeze, Some new results on the new conformable fractional calcu-lus with application using D’Alambert approach, Progress in Fractional Differentiation and Applications, 2 (2016), 115–122.

[9] R. Khalil, M. Al horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70.

[10] P.R. Beesack, Integral inequalities of the Wirtinger type, Duke Math. J., 25 (1958), 477–498.

[11] C.A. Swanson, Wirtinger’s inequality, SIAM J. Math. Anal., 9 (1978), 484–491. [12] J. Jaroˇs, On an integral inequality of the Wirtinger type, Appl. Math. Lett., 24 (2011),

1389–1392.

[13] C.F. Lee, C.C. Yeh, C.H. Hong and R.P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett., 17 (2004), 847–853.

[14] S.-E. Takahasi and T. Miura, A note on Wirtinger-Beesack’s integral inequalities, Math. Inequal. Appl., 6 (2003), 277–282.

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Received April 24, 2018 Accepted September 12, 2018 D¨uzce University Department of Mathematics D¨uzce, Turkey E-mail: sarikayamz@gmail.com E-mail: candancan@duzce.edu.tr

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