Geodesic
Several refinements and counterparts of Radon's inequality
Mathematica Bohemica, Tome 140 (2015) no. 1, pp. 71-80

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We establish that the inequality of Radon is a particular case of Jensen's inequality. Starting from several refinements and counterparts of Jensen's inequality by Dragomir and Ionescu, we obtain a counterpart of Radon's inequality. In this way, using a result of Simić we find another counterpart of Radon's inequality. We obtain several applications using Mortici's inequality to improve Hölder's inequality and Liapunov's inequality. To determine the best bounds for some inequalities, we used Matlab program for different cases.
We establish that the inequality of Radon is a particular case of Jensen's inequality. Starting from several refinements and counterparts of Jensen's inequality by Dragomir and Ionescu, we obtain a counterpart of Radon's inequality. In this way, using a result of Simić we find another counterpart of Radon's inequality. We obtain several applications using Mortici's inequality to improve Hölder's inequality and Liapunov's inequality. To determine the best bounds for some inequalities, we used Matlab program for different cases.
DOI : 10.21136/MB.2015.144180
Classification : 26D15
Keywords: Radon's inequality; Jensen's inequality; Hölder's inequality; Liapunov's inequality 
Raţiu, Augusta; Minculete, Nicuşor. Several refinements and counterparts of Radon's inequality. Mathematica Bohemica, Tome 140 (2015) no. 1, pp. 71-80. doi: 10.21136/MB.2015.144180
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[1] Agarwal, R. P., Dragomir, S. S.: A survey of Jensen type inequalities for functions of selfadjoint operators in Hilbert spaces. Comput. Math. Appl. 59 (2010), 3785-3812 Corrigendum Comput. Math. Appl. 61 (2011), 2931. | DOI | MR | Zbl

[2] Bergström, H.: A triangle-inequality for matrices. Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949 Johan Grundt Tanums Forlag, Oslo (1952), 264-267. | MR | Zbl

[3] Bullen, P. S.: Handbook of Means and Their Inequalities. Mathematics and Its Applications 560 Kluwer Academic Publishers, Dordrecht (2003). | MR | Zbl

[4] Ciurdariu, L.: On Bergström inequality for commuting Gramian normal operators. J. Math. Inequal. 4 (2010), 505-516. | DOI | MR | Zbl

[5] Dragomir, S. S.: A converse result for Jensen's discrete inequality via Grüss' inequality and applications in information theory. An. Univ. Oradea, Fasc. Mat. 7 (1999/2000), 178-189. | MR

[6] Dragomir, S. S., Ionescu, N. M.: Some converse of Jensen's inequality and applications. Rev. Anal. Numér. Théor. Approx. 23 (1994), 71-78. | MR | Zbl

[7] Furuichi, S., Minculete, N., Mitroi, F.-C.: Some inequalities on generalized entropies. J. Inequal. Appl. (electronic only) 2012 (2012), Article No. 2012:226, 16 pages. | MR | Zbl

[8] Gavrea, B.: On an inequality for convex functions. Gen. Math. 19 (2011), 37-40. | MR | Zbl

[9] Jiang, S.-J., Pang, L.-P., Shen, J.: Existence of solutions of generalized vector variational-type inequalities with set-valued mappings. Comput. Math. Appl. 59 (2010), 1453-1461. | DOI | MR | Zbl

[10] Mărghidanu, D.: Generalizations and refinements for Bergström and Radon's inequalities. J. Sci. Arts 8 (2008), 57-62. | Zbl

[11] Mărghidanu, D., Díaz-Barrero, J. L., Rădelescu, S.: New refinements of some classical inequalities. Math. Inequal. Appl. 12 (2009), 513-518. | MR | Zbl

[12] Mortici, C.: A new refinement of the Radon inequality. Math. Commun. 16 (2011), 319-324. | MR | Zbl

[13] Nhan, N. D. V., Duc, D. T., Tuan, V. K.: Weighted norm inequalities for a nonlinear transform. Comput. Math. Appl. 61 (2011), 832-839. | DOI | MR | Zbl

[14] Niculescu, C., Persson, L.-E.: Convex Functions and Their Applications. A Contemporary Approach. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 23 Springer, New York (2006). | MR | Zbl

[15] Pečarić, J., Perić, J.: Remarks on the paper ``Jensen's inequality and new entropy bounds'' of S. Simić. J. Math. Inequal. 6 (2012), 631-636. | DOI | MR | Zbl

[16] Pop, O. T.: About Bergström's inequality. J. Math. Inequal. 3 (2009), 237-242. | DOI | MR | Zbl

[17] Qiang, H., Hu, Z.: Generalizations of Hölder's and some related inequalities. Comput. Math. Appl. 61 (2011), 392-396. | DOI | MR | Zbl

[18] Radon, J.: Theorie und Anwendungen der absolut additiven Mengenfunktionen. Wien. Ber. 122 German (1913), 1295-1438.

[19] Simic, S.: Jensen's inequality and new entropy bounds. Appl. Math. Lett. 22 (2009), 1262-1265. | DOI | MR | Zbl

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