Extending the Hölder type inequality of Blakley and Roy to non-symmetric non-square matrices

Author:
Thomas H. Pate

Journal:
Trans. Amer. Math. Soc. **364** (2012), 4267-4281

MSC (2010):
Primary 15A63, 15A42, 15A60, 15A15, 05C50

Published electronically:
March 20, 2012

MathSciNet review:
2912454

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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose , , and are positive integers, and let denote standard inner product on the spaces , . We show that if is an non-negative real matrix, and and are non-negative unit vectors in and , respectively, then

with equality if and only if , or there exists such that and . This inequality extends to non-symmetric non-square matrices a 1965 result of Blakley and Roy which asserts that if is a non-negative symmetric matrix, and is a non-negative unit vector, then

with equality, when , if and only if , or there exists such that . The generality of the inequality (1) derives not only from the fact that is not assumed to be symmetric or square, but from the fact that we admit two unit vectors and instead of the single unit vector appearing in inequality (2) of Blakley and Roy. We apply our result to verify the conjecture of A. Sidorenko (1993) in the non-symmetric case provided that the underlying graph is a path.

**1.**G. R. Blakley and Prabir Roy,*A Hölder type inequality for symmetric matrices with nonnegative entries*, Proc. Amer. Math. Soc.**16**(1965), 1244–1245. MR**0184950**, 10.1090/S0002-9939-1965-0184950-9**2.**G. Hardy, J.E. Littlewood, and G. Pólya,*Inequalities*, Cambridge University Press, Cambridge, 1999.**3.**Roger A. Horn and Charles R. Johnson,*Matrix analysis*, Cambridge University Press, Cambridge, 1985. MR**832183****4.**Alexander Sidorenko,*A correlation inequality for bipartite graphs*, Graphs Combin.**9**(1993), no. 2, 201–204. MR**1225933**, 10.1007/BF02988307**5.**A. F. Sidorenko,*Inequalities for functionals generated by bipartite graphs*, Diskret. Mat.**3**(1991), no. 3, 50–65 (Russian); English transl., Discrete Math. Appl.**2**(1992), no. 5, 489–504. MR**1138091**, 10.1515/dma.1992.2.5.489

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Additional Information

**Thomas H. Pate**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

DOI:
http://dx.doi.org/10.1090/S0002-9947-2012-05501-2

Keywords:
Positive matrices,
inequalities,
Hölder’s inequality,
bipartite graphs

Received by editor(s):
August 20, 2010

Received by editor(s) in revised form:
November 5, 2010

Published electronically:
March 20, 2012

Article copyright:
© Copyright 2012
American Mathematical Society