Inequalities of Hardy type for Jackson Nörlund Integrals


  • Naheed Ashraf Department of Mathematics, Govt. Post Graduate College for Women, Sargodha 40100
  • Khuram Ali Khan Department of Mathematics, University of Sargodha, Sargodha 40100
  • Shakila Mobeen Department of Mathematics, Govt. Post Graduate College for Women, Sargodha 40100
  • Ammara Nosheen Department of Mathematics, The University of Lahore, Sargodha Campus 40100



Convex function, Jackson integral, Nörlund sums, Jensen's inequality, Hardy inequality


In the paper, we prove Jensen's inequality for Jackson Nörlund integrals, and by using Jensen's inequality, Hardy type inequalities with general kernels as well as choosing special kernels are proved. In seek of applications to these inequalities we give Hilbert-Hardy inequality and Polya-Knop type inequalities.


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S. Abramovich, K. Krulic, J. Pecaric and L.E. Persson, Some new refined Hardy type inequaliteis with general kernels and measures, Aequat. Math. 79 (2010), 157 – 172.

M.H. Annaby, A.E. Hamza, K.A. Aldwoah, Hahn difference operator and associated Jackson-Nörlund integral, J. Optim. Theory Appl. 154 (2012), 133 – 153.

A.O. Baiarystanov, L.E. Persson, S. Shaimardan and A. Temirkhanova, Some new Hardy-type inequalities in q-analysis, Jour. Math. Ineq. 10(3) (2016), 761 – 781.

M.Bohner and A. Peterson, Dynamic Equation on Time Scales: An Introduction with Application, Birkhäuser, Boston (2001).

M. Bohner, A. Nosheen, J. Peˇcari`c and A. Younus, Some dynamic Hardy-type inequalities with general kernal, Jour. Math. Ineq. 8(1) (2014), 185 – 199.

A.M.C. Brito da Cruz, N. Martins and D.F.M. Torres, A symmetric quantum calculus, Springer Proceedings in Math. Stat. 47 (2013), 359 – 366.

P. Cheung, V. Kac, Quntum Calculus, Edwards Brothers Inc., Ann Arbor, MI, USA (2000).

G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge University Press (1934).

A. Kufner, L. Maligranda and L.-E. Persson, The Hardy Inequality. About Its History and Some Related Result, Vydavatelsky Servis Publishing House, Pilsen (2007).

A. Kufner, L. Maligranda and L.-E. Persson, The prehistory of the Hardy inequality, Springer, Amer. Math. Monthly 113 (2006), 715–732.

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

A.B. Malinowska and D.F.M. Torres, The Hahn quantum variational calculus, J. Optim. Theory Appl. 147(3) (2010), 419 – 442.

C. Niculescu and L.E. Persson, Convex Functions and Their Applications, A Contemporary Approach, Springer-Verlag (2006).

U.M. í–zkan, M.Z. Sarikaya and H. Yildirim, Existences of certain integral inequalities on time scales, Appl. Math. Letters 21(10) (2008), 993 – 1000.




How to Cite

Ashraf, N., Khan, K. A., Mobeen, S., & Nosheen, A. (2018). Inequalities of Hardy type for Jackson Nörlund Integrals. Communications in Mathematics and Applications, 9(3), 411–419.



Research Article