Matrix Fej\'{e}r and Levin-Ste\v{c}kin Inequalities
Download PDF
Authors: M. SABABHEH, S. SHEYBANI AND H. R. MORADI
DOI: 10.46793/KgJMat2505.807S
Abstract:
Fejér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fejér type for both convex and log-convex functions. For the matrix Levin-Stečkin type, we present more rigorous results involving the partial Löewner ordering for Hermitian matrices. Further related results involving synchronous functions are presented, too.
Keywords:
Levin-Stečkin inequality, Fejér inequality, positive matrices.
References:
[1] R. Bhatia, Matrix Analysis, Springer Verlag, New York, 1997.
[2] P. L. Chebyshev, O približennyh vyraženijah odnih integralov čerez drugie, in: Šžit soobšćenija i protokoly zasedani Matemmatičeskogo občestva pri Imperatorskom Har’kovskom Universitete, No. 2, 93–98; P. L. Chebyshev, Polnoe sobranie sočineni, Moskva, Leningrad 1948a (1882), 128–131, (in Russian).
[3] S. S. Dragomir, Hermite-Hadamard’s type inequalities for operator convex functions, Appl. Math. Comput. 218(3) (2011), 766–772. https://doi.org/10.1016/j.amc.2011.01.056
[4] L. Fejér, Über die fourierreihen, II, Math. Naturwiss Anz. Ungar. Akad. Wiss 24 (1906), 369–390.
[5] J. Hadamard, Étude sur les proprits des fonctions entiéres et en particulier d’une fonction considérée par Riemann, J. Math. Pures Appl. 58 (1893), 171–215, (in French).
[6] V. I. Levin and S. B. Stěckin, Inequalities, Trans. Amer. Math. Soc. 14 (1960), 1–29.
[7] H. R. Moradi, M. E. Omidvar and S. S. Dragomir, An operator extension of Čebyšev inequality, Analele Stiintifice ale Universitatii Ovidius Constanta 25(2) (2017), 135–148. https://doi.org/10.1515/auom-2017-0025
[8] H. R. Moradi, M. Sababheh and S. Furuichi, On the operator Hermite-Hadamard inequality, Complex Anal. Oper. Theory 15(122), (2021). https://doi.org/10.1007/s11785-021-01172-w
[9] H. R. Moradi and M. Sababheh, More accurate numerical radius inequalities (II), Linear Multilinear Algebra 69 (2021), 921–933. https://doi.org/10.1080/03081087.2019.1703886
[10] M. S. Moslehian, Matrix Hermite-Hadamard type inequalities, Houston J. Math. 39(1) (2013), 177–189.
[11] J. Pečarić, T. Furuta, J. Mićić Hot and Y. Seo, Mond-Pečarić Method in Operator Inequalities, Element, Zagreb, 2005.
[12] M. Sababheh and H. R. Moradi, More accurate numerical radius inequalities (I), Linear Multilinear Algebra 69 (2021), 1964–1973. https://doi.org/10.1080/03081087.2019.1651815
Login:
