Improvement of an inequality of G. H. Hardy

Article Sidebar

PDF
Published: Sep 30, 2012
DOI: https://doi.org/10.5556/j.tkjm.43.2012.834
Keywords:
Convex function kernel fractional derivative fractional integrals means

Main Article Content

Sajid Iqbal
Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore, 54600, PAKISTAN.
Kristina Krulić
Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia
Josip Pečarić
Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia

Abstract

In this paper, we give an improvement of an inequality of G. H. Hardy using fractional integrals and fractional derivatives. We also obtain means of Cauchy type and prove their monotonicity.

Article Details

How to Cite
Iqbal, S., Krulić, K., & Pečarić, J. (2012). Improvement of an inequality of G. H. Hardy. Tamkang Journal of Mathematics, 43(3), 399–416. https://doi.org/10.5556/j.tkjm.43.2012.834
Issue
Vol. 43 No. 3 (2012)
Section
Papers

References

G. A. Anastassiou, Fractional Differentiation Inequalities, Springer Science-Businness Media, LLC, Dordrecht, the Netherlands, 2009.

S. N. Bernstein, it Sur les fonctions absolument monotones, Acta Math., 52(1929), 1--66.

D. E. Edmunds, V. Kokilashvili and A. Meskhi, Bounded and Compact Integral Operators, Kluwer Academic Publishers, 101, Philip Drive, Norwell, MA02061, USA.

N. Elezovic, K. Krulic and J. Pev caric, it Bounds for Hardy type differences, Acta Mathematica Sinica, English Series, 27(4)(2011), 671--684.

G. D. Handley, J. J. Koliha and J. Pevcaric, it Hilbert-Pachpatte type integral inequalities for fractional derivatives, Fract. Calc. Appl. Anal., Fractional Calculus and Applied Analysis, 4(2001), 37--46.

G. H. Hardy, it Notes on some points in the integrral calculus, Messenger. Math., 47(10) (1918), 145--150.

S. Iqbal, J. Pevcaric and Y. Zhou, it Generalization of an inequality for integral transforms with kernel and related results, J. Inequal. Appl., vol. 2010. Artical ID

, 2010.

S. Iqbal, K. Krulic and J. Pevcaric, it On an inequality of H. G. Hardy, J. Inequal. Appl., vol. 2010. Artical ID 264347,2010.

J. Jakv setic and J.Pev cari c, it Exponential convexity method, J. Convex Anal., (accepted).

S.Kaijser, L.Nikolova, L-E.Persson and A.Wedestig, it Hardy type inequalities via convexity, Math. Inequal. Appl. 8(3)(2005), 403--417.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractinal Differential Equations, North-Holland Mathematics Sttudies, 204, Elsevier, New York-London, 2006.

K. Krulic, J. Pevcaric and L. E. Persson, it Some new Hardy type inequalities with general kernels, Math. Inequal. Appl., 12(2009), 473-485.

D. S. Mitrinovic, J. Pevcaric and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publisher, 1993.

D. S. Mitrinovic and J.E.Pevcaric, On Some Inequalities for Monotone Functions, Boll. Unione. Math. Ital., (7),5--13, 407--416, 1991.

C.Niculescu and L.-E.Persson, Convex Functions and Their Applications. A Contemporary Approach, CMC Books in Mathematics, Springer, New York, 2006.

J. Pevcaric, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Inc. 1992.

S. G. Samko, A. A. Kilbas and O. J. Marichev, Fractional Integral and Derivatives : Theory and Applications, Gordon and Breach Science Publishers, Switzerland, 1993.

J. Pevcaric, I. Peric and H. M. Srivastava, On the Cauchy type mean value theorems, J. Math. Anal. Appl., 306(2005), 730--739.

J. J. Trujillo, M. Rivero and B. Bonila, it On Riemann-Liouville generalized Taylor's formula, J. Math. Anal. Appl., 231(1999), 255--265.