Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces
-
Mohammad W. Alomari
Abstract
In this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.
Acknowledgements
The author wish to thank the referee(s) for their fruitful comments and careful reading of the original manuscript of this work that have implemented the final version of this work.
References
[1] S. Abramovich, S. Ivelić and J. E. Pečarić, Improvement of Jensen–Steffensen’s inequality for superquadratic functions, Banach J. Math. Anal. 4 (2010), no. 1, 159–169. 10.15352/bjma/1272374678Search in Google Scholar
[2] S. Abramovich, G. Jameson and G. Sinnamon, Refining Jensen’s inequality, Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 47(95) (2004), no. 1–2, 3–14. Search in Google Scholar
[3] S. Banić, J. Pečarić and S. Varošanec, Superquadratic functions and refinements of some classical inequalities, J. Korean Math. Soc. 45 (2008), no. 2, 513–525. 10.4134/JKMS.2008.45.2.513Search in Google Scholar
[4] M. Bencze, C. P. Niculescu and F. Popovici, Popoviciu’s inequality for functions of several variables, J. Math. Anal. Appl. 365 (2010), no. 1, 399–409. 10.1016/j.jmaa.2009.10.069Search in Google Scholar
[5] S. S. Dragomir, Operator Inequalities of the Jensen, Čebyšev and Grüss Type, Springer Briefs Math., Springer, New York, 2012. 10.1007/978-1-4614-1521-3Search in Google Scholar
[6] W. O. Fechner, Hlawka’s functional inequality on topological groups, Banach J. Math. Anal. 11 (2017), no. 1, 130–142. 10.1215/17358787-3764339Search in Google Scholar
[7] M. Fujii, M. S. Moslehian and J. Mićić, Bohr’s inequality revisited, Nonlinear Analysis, Springer Optim. Appl. 68, Springer, New York, (2012), 279–290. 10.1007/978-1-4614-3498-6_15Search in Google Scholar
[8] O. Hirzallah, Non-commutative operator Bohr inequality, J. Math. Anal. Appl. 282 (2003), no. 2, 578–583. 10.1016/S0022-247X(03)00185-9Search in Google Scholar
[9] J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math. 30 (1906), no. 1, 175–193. 10.1007/BF02418571Search in Google Scholar
[10] M. Kian, Operator Jensen inequality for superquadratic functions, Linear Algebra Appl. 456 (2014), 82–87. 10.1016/j.laa.2012.12.011Search in Google Scholar
[11] M. Kian and S. S. Dragomir, Inequalities involving superquadratic functions and operators, Mediterr. J. Math. 11 (2014), no. 4, 1205–1214. 10.1007/s00009-013-0357-ySearch in Google Scholar
[12] M. Krnić, N. Lovričević, J. Pečarić and J. Perić, Superadditivity and Monotonicity of the Jensen-type Functionals, Monogr. Inequal. 11, Element, Zagreb, 2015. Search in Google Scholar
[13] J. Mićić, J. Pečarić and J. Perić, Extension of the refined Jensen’s operator inequality with condition on spectra, Ann. Funct. Anal. 3 (2012), no. 1, 67–85. 10.15352/afa/1399900024Search in Google Scholar
[14] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and new Inequalities in Analysis, Math. Appl. (East European Series) 61, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-017-1043-5Search in Google Scholar
[15] B. Mond and J. E. Pečarić, Convex inequalities in Hilbert space, Houston J. Math. 19 (1993), no. 3, 405–420. Search in Google Scholar
[16] C. P. Niculescu, Convexity according to the geometric mean, Math. Inequal. Appl. 3 (2000), no. 2, 155–167. 10.7153/mia-03-19Search in Google Scholar
[17] C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications, CMS Books Math./ Ouvrages Math. SMC 23, Springer, New York, 2006. 10.1007/0-387-31077-0Search in Google Scholar
[18] C. P. Niculescu and F. Popovici, A refinement of Popoviciu’s inequality, Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 49(97) (2006), no. 3, 285–290. Search in Google Scholar
[19] J. Pečarić, T. Furuta, J. Mićić Hot and Y. Seo, Mond–Pečarić Method in Operator Inequalities, Monogr. Inequal. 1, Element, Zagreb, 2005. Search in Google Scholar
[20] T. Popoviciu, Sur certaines inégalités qui caractérisent les fonctions convexes, An. Ştinnţ. Univ. Al. I. Cuza Iaşi Mat. (N. S.) 11B (1965), 155–164. Search in Google Scholar
[21] D. M. Smiley and M. F. Smiley, The polygonal inequalities, Amer. Math. Monthly 71 (1964), 755–760. 10.1080/00029890.1964.11992316Search in Google Scholar
[22] S.-E. Takahasi, Y. Takahashi and A. Honda, A new interpretation of Djoković’s inequality, J. Nonlinear Convex Anal. 1 (2000), no. 3, 343–350. Search in Google Scholar
[23] S.-E. Takahasi, Y. Takahashi, S. Miyajima and H. Takagi, Convex sets and inequalities, J. Inequal. Appl. (2005), no. 2, 107–117. 10.1155/JIA.2005.107Search in Google Scholar
[24] P. M. Vasić and L. R. Stanković, Some inequalities for convex functions, Math. Balkanica 6 (1976), 281–288. Search in Google Scholar
[25] F. Zhang, On the Bohr inequality of operators, J. Math. Anal. Appl. 333 (2007), no. 2, 1264–1271. 10.1016/j.jmaa.2006.12.024Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- 23rd Meeting of the Tunisian Mathematical Society (March 2018, Tabarka, Tunisia)
- Odd-quadratic Leibniz superalgebras
- Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems
- Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces
- A virtual element method for a biharmonic Steklov eigenvalue problem
- Convergence analysis of an inertial accelerated iterative algorithm for solving split variational inequality problem
- Dynamics of an ecological system
- Hilbert space valued Gabor frames in weighted amalgam spaces
- Laguerre–Freud equations associated with the D-Laguerre–Hahn forms of class one
- A weighted inequality for potential type operators
- W-semisymmetric generalized Sasakian-space-forms
- Proximal point algorithm involving fixed point of nonexpansive mapping in 𝑝-uniformly convex metric space
- Existence of positive solutions for a Neumann boundary value problem on the half-line via coincidence degree
- Multi-norm structure based on enveloping 𝐶∗-algebras
- Multiplicative convolution of real asymmetric and real anti-symmetric matrices
Articles in the same Issue
- Frontmatter
- 23rd Meeting of the Tunisian Mathematical Society (March 2018, Tabarka, Tunisia)
- Odd-quadratic Leibniz superalgebras
- Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems
- Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces
- A virtual element method for a biharmonic Steklov eigenvalue problem
- Convergence analysis of an inertial accelerated iterative algorithm for solving split variational inequality problem
- Dynamics of an ecological system
- Hilbert space valued Gabor frames in weighted amalgam spaces
- Laguerre–Freud equations associated with the D-Laguerre–Hahn forms of class one
- A weighted inequality for potential type operators
- W-semisymmetric generalized Sasakian-space-forms
- Proximal point algorithm involving fixed point of nonexpansive mapping in 𝑝-uniformly convex metric space
- Existence of positive solutions for a Neumann boundary value problem on the half-line via coincidence degree
- Multi-norm structure based on enveloping 𝐶∗-algebras
- Multiplicative convolution of real asymmetric and real anti-symmetric matrices
