Abstract
In the present article, we derive several new weighted dynamic inequalities for monotone functions involving kernels, some of which are the Hardy-type inequalities. The established inequalities are characterized by appropriate relations for the accompanying weight functions. Our results are time scale extensions of several classical weighted inequalities known from the literature. As an application, we obtain the corresponding discrete weighted inequalities for monotone sequences, which are essentially new.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agarwal, R.P., O’Regan, D., Saker, S.H.: Dynamic Inequalities on Time Scales. Springer, Berlin (2014)
Agarwal, R.P., O’Regan, D., Saker, S.H.: Hardy Type Inequalities on Time Scales. Springer, Berlin (2016)
Barza, S., Persson, L.-E., Soria, J.: Sharp weighted multidimensional integral inequalities for monotone functions. Math. Nachr. 210, 43–58 (2000)
Bennett, G., Grosse-Erdmann, K.-G.: Weighted Hardy inequalities for decreasing sequences and functions. Math. Ann. 334, 489–531 (2006)
Bibi, R., Bohner, M., Pečarić, J., Varošanec, S.: Minkowski and Beckenbach–Dresher inequalities and functionals on time scales. J. Math. Inequal. 7, 299–312 (2013)
Bohner, M., Georgiev, S.G.: Multiple Integration on Time Scales, Multivariable Dynamic Calculus on Time Scales, pp. 449–515. Springer, Cham (2016)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Springer Science and Business Media, New York (2002)
Fabelurin, O.O., Oguntuase, J.A., Persson, L.-E.: Multidimensional Hardy-type inequalities on time scales with variable exponents. J. Math. Inequal. 13, 725–736 (2019)
Heinig, H., Maligranda, L.: Weighted inequalities for monotone and concave functions. Studia Math. 116, 133–165 (1995)
Oguntuase, J.A., Persson, L.-E.: Time scales Hardy-type inequalities via superquadracity. Ann. Funct. Anal. 5, 61–73 (2014)
Opic, B., Kufner, A.: Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series. Longman Scientific and Technical, Harlow (1990)
Ou, B.: Some extended integral inequalities on time scales. J. Math. Inequal. 12, 1013–1027 (2018)
Řehak, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 2005(5), 495–507 (2005)
Saker, S.H., Mahmoud, R.R., Peterson, A.: Weighted Hardy-type inequalities on time scales with applications. Mediterr. J. Math. 13, 585–606 (2016)
Saker, S.H., O’Regan, D.: Hardy and Littlewood inequalities on time scales. Bull. Malays. Math. Soc. 39, 527–543 (2016)
Saker, S.H., Rezk, H.M., Krnić, M.: More accurate dynamic Hardy-type inequalities obtained via superquadraticity, Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales. Serie A. Matematicas 113, 2691–2713 (2019)
Stepanov, V.D.: Boundedness of linear integral operators on a class of monotone functions. Sib. Math. J. 32, 540–542 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Saker, S.H., Saied, A.I. & Krnić, M. Some New Weighted Dynamic Inequalities for Monotone Functions Involving Kernels. Mediterr. J. Math. 17, 39 (2020). https://doi.org/10.1007/s00009-020-1473-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-1473-0