Refinements and Reverses of Operator Callebaut Inequality Involving Tracy-Singh Products and Khatri-Rao Products

Arnon Ploymukda, Pattrawut Chansangiam


In this paper, we establish certain refinements and reverses of Callebaut-type inequality for bounded continuous fields of Hilbert space operators, parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. These inequalities involve Tracy-Singh products, Khatri-Rao products and weighted geometric means. In addition, we obtain integral Callebauttype inequalities for tensor products and Hadamard products. Our results extend Callebaut-type inequalities for real numbers, matrices and operators.


Callebaut inequality; Tracy-Singh product; Khatri-Rao product; Weighted geometric mean; Continuous field of operators

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M. Bakherad and M.S. Moslehian, Complementary and refined inequalities of Callebaut inequality for operators, Linear Multilinear Algebra 63(8) (2015), 1678 – 1692, DOI: 10.1080/03081087.2014.967234.

M. Bakherad, Some reversed and refined Callebaut inequalities via Kantorovich constant, Bull. Malays. Math. Sci. Soc. 41 (2018), 765 – 777, DOI: 10.1007/s40840-016-0364-9.

W.M. Bogdanowicz, Fubini theorems for generalized Lebesgue-Bochner-Stieltjes integral, Proc. Japan Acad. 41(10) (1966), 979 – 983, DOI: 10.3792/pja/1195526750.

D.K. Callebaut, Generalization of the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 12(3) (1965), 491 – 494, DOI: 10.1016/0022-247X(65)90016-8.

J.I. Fujii, The Marcus-Khan theorem for Hilbert space operators, Mathematica Japonica 41(3) (1995), 531 – 535.

F. Hiai and X. Zhan, Inequalities involving unitarily invariant norms and operator monotone functions, Linear Algebra Appl. 341(1-3) (2002), 151 – 169, DOI: 10.1016/S0024-3795(01)00353-6.

M.S. Moslehian, J.S. Matharu and J.S. Aujla, Non-commutative Callebaut inequality, Linear Algebra Appl. 436(3) (2012), 3347 – 3353, DOI: 10.1016/j.laa.2011.11.024.

A. Ploymukda, P. Chansangiam and W. Lewkeeratiyutkul, Algebraic and order properties of Tracy-Singh products for operator matrices, J. Comput. Anal. Appl. 24(4) (2018), 656 – 664.

A. Ploymukda, P. Chansangiam and W. Lewkeeratiyutkul, Analytic properties of Tracy-Singh products for operator matrices, J. Comput. Anal. Appl. 24(4) (2018), 665 – 674.

A. Ploymukda and P. Chansangiam, Integral inequalities of Chebyshev type for continuous fields of Hermitian operators involving Tracy-Singh products and means, J. Math. Inequal. submitted.

A. Ploymukda and P. Chansangiam, Khatri-Rao products of operator matrices acting on the direct sum of Hilbert spaces, Journal of Mathematics (2016), 7 pages, DOI: 10.1155/2016/8301709.

A. Ploymukda and P. Chansangiam, Monotonicity of certain maps and Callebaut-type integral inequalities for continuous fields of operators, Annals of the Alexandru Ioan Cuza University - Mathematics, submitted.

S. Wada, On some refinement of the Cauchy-Schwarz inequality, Linear Algebra Appl. 420(2-3) (2007), 433–440, DOI: 10.1016/j.laa.2006.07.019.

J. Zhao and J. Wu, Operator inequalities involving improved Young and its reverse inequalities, J. Math. Anal. Appl. 421(2) (2015), 1779–1789, DOI: 10.1016/j.jmaa.2014.08.032.



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