Inequalities for the Finite Hilbert Transform of convex functions

Keywords: Finite Hilbert Transform, convex functions

Abstract

In this paper we obtain some new inequalities for the finite Hilbert transform of convex functions.

Applications for some particular functions of interest are provided as well.

References

N.M. Dragomir, S.S. Dragomir, P.M. Farrell, Some inequalities for the finite Hilbert transform, Inequality Theory and Applications, vol. I, pp. 113–122, Nova Sci. Publ., Huntington, NY, 2001.

N.M. Dragomir, S.S. Dragomir, P.M. Farrell, Approximating the finite Hilbert transform via trapezoid type inequalities, Comput. Math. Appl., 43 (2002) nos. 10–11, pp. 1359–1369. https://https//doi.org/10.1016/s0898-1221(02)00104-9

N.M. Dragomir, S.S. Dragomir, P.M. Farrell, G.W. Baxter, On some new estimates of the finite Hilbert transform, Libertas Math., 22 (2002), 65–75.

N.M. Dragomir, S.S. Dragomir, P.M. Farrell, G.W. Baxter, A quadrature rule for the finite Hilbert transform via trapezoid type inequalities, J. Appl. Math. Comput., 13 (2003) nos. 1–2, 67–84. https://doi.org/10.1007/bf02936075

N.M. Dragomir, S.S. Dragomir, P.M. Farrell, G.W. Baxter, A quadrature rule for the finite Hilbert transform via midpoint type inequalities, Fixed point theory and applications. vol. 5, pp. 11–22, Nova Sci. Publ., Hauppauge, NY, 2004.

S.S. Dragomir, Inequalities for the Hilbert transform of functions whose derivatives are convex, J. Korean Math. Soc., 39 (2002) no. 5, pp. 709–729. https://doi.org/10.4134/jkms.2002.39.5.709

S.S. Dragomir, Approximating the finite Hilbert transform via an Ostrowski type inequality for functions of bounded variation, J. Inequal. Pure Appl. Math., 3 (2002) no. 4, art. 51, 19 pp.

S.S. Dragomir, Approximating the finite Hilbert transform via Ostrowski type inequalities for absolutely continuous functions, Bull. Korean Math. Soc., 39 (2002) no. 4, pp. 543–559. https://doi.org/10.4134/bkms.2002.39.4.543

S.S. Dragomir, Some inequalities for the finite Hilbert transform of a product, Commun. Korean Math. Soc., 18 (2003) no. 1, pp. 39–57. https://doi.org/10.4134/ckms.2003.18.1.039

S.S. Dragomir, Sharp error bounds of a quadrature rule with one multiple node for the finite Hilbert transform in some classes of continuous differentiable functions, Taiwanese J. Math., 9 (2005) no. 1, pp. 95–109. https://doi.org/10.11650/twjm/1500407748

S.S. Dragomir, Inequalities and approximations for the Finite Hilbert transform: a survey of recent results, Preprint RGMIA Res. Rep. Coll., 21 (2018), art. 30, 90 pp. http://rgmia.org/papers/v21/v21a30.pdf

D. Gakhov, Boundary Value Problems (English translation), Pergamon Press, Oxford, 1966.

W. Liu, X. Gao, Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications, Appl. Math. Comput., 247 (2014), pp. 373–385. https://doi.org/10.1016/j.amc.2014.08.099

W. Liu, X. Gao, Y. Wen, Approximating the finite Hilberttransform via some companions of Ostrowski’s inequalities, Bull. Malays. Math. Sci. Soc., 39 (2016) no. 4, pp. 1499–1513. https://doi.org/10.1007/s40840-015-0251-9

W. Liu, N. Lu, Approximating the finite Hilbert transform via Simpson type inequalities and applications, Sci. Bull. Ser. A Appl. Math. Phys., Politehn. Univ. Bucharest, 77 (2015) no. 3, 107–122.

S.G. Mikhlin, S. Prossdorf, Singular Integral Operators (English translation), Springer Verlag, Berlin, 1986.

S. Wang, X. Gao, N. Lu, A quadrature formula in approximating the finite Hilbert transform via perturbed trapezoid type inequalities, J. Comput. Anal. Appl., 22 (2017) no. 2, pp. 239–246.

S. Wang, N. Lu, X. Gao, A quadrature rule for the finite Hilberttransform via Simpson type inequalities and applications, J. Comput. Anal. Appl., 22 (2017) no. 2, pp. 229–238.

Published
2020-01-21
How to Cite
Dragomir, S. S. (2020). Inequalities for the Finite Hilbert Transform of convex functions. J. Numer. Anal. Approx. Theory, 48(2), 148-158. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1177
Section
Articles