Korean J. Math.  Vol 28, No 4 (2020)  pp.717-737
DOI: https://doi.org/10.11568/kjm.2020.28.4.717

On weighted generalization of opial type inequalities in two variables

Hüseyin Budak, Mehmet Zeki Sarikaya, Artion Kashuri


In this paper, we establish some weighted generalization of Opial type inequalities in two independent variables for two functions. We also obtain weighted Opial type inequalities by using $p$-norms. Special cases of our results reduce to the inequalities in earlier study.


Opial inequality, H ̈older’s inequality.

Subject classification

26D15, 26D10, 26B15


Full Text:



R.P. Agarwal and P. Y. H. Pang, Sharp opial-type inequalities in two variables, Appl Anal. 56 (3) (1996), 227–242. (Google Scholar)

H. Budak and Sarikaya, Refinements of Opial type inequalities in two variables, ResearchGate Article: www.researchgate.net/publication/329091454. (Google Scholar)

H. Budak, Generalizations of Opial type inequalities two variables using p- norms, Transylvanian Journal of Mathematics and Mechanics, 11 (1-2) (2019), 63–75. (Google Scholar)

Z. Changjian, and W. Cheung, On improvements of Opial-type inequalities, Georgian Mathematical Journal, 21 (4) (2014), 415–419. (Google Scholar)

W.S. Cheung, Some new Opial-type inequalities, Mathematika 37 (1990), 136– 142. (Google Scholar)

W.S. Cheung, Some generalized Opial-type inequalities, J. Math. Anal. Appl. 162 (1991), 317– 321. (Google Scholar)

W.S. Cheung, On Opial-type inequalities in two variables, Aequationes Mathematicae 38 (1989), 236–244. (Google Scholar)

W.S. Cheung, Opial-type inequalities with m functions in n variables, Mathematika 39 (2) (1992), 319–326. (Google Scholar)

S. S. Dragomir, Generalizations of Opial ́ıs inequalities for two functions and applications, Preprint RGMIA Res. Rep. Coll. 21 (2018), Art. 64. (Google Scholar)

C. T. Lin and G. S.Yang, A generalized Opial’s inequality in two variables, Tamkang J. Math. 15 (1984), 115–122. (Google Scholar)

Z. Opial, Sur une inegaliti, Ann. Polon. Math. 8 (1960), 29–32. (Google Scholar)

B. G. Pachpatte, On Opial-type integral inequalities, J. Math. Anal. Appl. 120 (1986), 547–556. (Google Scholar)

B. G. Pachpatte, Some inequalities similar to Opial’s inequality, Demonstratio Math. 26 (1993), 643–647. (Google Scholar)

B. G. Pachpatte, A note on some new Opial type integral inequalities, Octogon Math. Mag. 7 (1999), 80–84. (Google Scholar)

B. G. Pachpatte, On some inequalities of the Weyl type, An. Stiint. Univ. “Al.I. Cuza” Iasi 40 (1994), 89–95. (Google Scholar)

B. G. Pachpatte, On Opial type integral inequalities, J. Math. Analy. Appl. 120, 547–556 (1986). (Google Scholar)

B. G. Pachpatte, On two inequalities similar to Opial’s inequality in two independent variables, Periodica Math. Hungarica 18 (1987), 137–141. (Google Scholar)

B. G. Pachpatte, On an inequality of opial type in two variables, Indian J. Pure Appl. Math. 23 (9) (1992), 657–661. (Google Scholar)

B.C. Pachpatte, On two independent variable Opial-type integral inequalities, J. Math. Anal. Appl. 125 (1987), 47–57. (Google Scholar)

B.C. Pachpatte, On Opial type inequalities in two independent variables, Proc. Royal Soc. Edinburgh, 100A (1985), 263–270. (Google Scholar)

B.C. Pachpatte, On certain two dimensional integral inequalities, Chinese J. (Google Scholar)

Math. 17 (4) (1989), 273–279. (Google Scholar)

B.C. Pachpatte, On multidimensional Opial-type inequalities, J. Math. Anal. Appl. 126 (1) (1987), 85–89. (Google Scholar)

B.C. Pachpatte, On some new integral inequalities in ceveral independent variables, Chinese Journal of Mathematics 14 (2) (1986), 69–79. (Google Scholar)

B.C. Pachpatte, Inequalities of Opial type in three independent variables, Tamkang Journal of Mathematics 35 (2) (2004), 145–158. (Google Scholar)

H. M. Srivastava, K.-L. Tseng, S.-J. Tseng and J.-C. Lo, Some weighted Opial-type inequalities on time scales, Taiwanese J. Math. 14 (2010), 107–122. (Google Scholar)

J. Traple, On a boundary value problem for systems of ordinary differential equations of second order, Zeszyty Nauk. Univ. Jagiello. Prace Mat. 15 (1971), 159–168. (Google Scholar)

C.-J. Zhao and W.-S. Cheung, On Opial-type integral inequalities and applications, Math. Inequal. Appl. 17 (1) (2014), 223–232. (Google Scholar)

G. S. Yang. Inequality of Opial-type in two variables, Tamkang J. Math. 13 (1982), 255–259. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr