Bakherad et al. Journal of Inequalities and Applications 017) 017:09 DOI 10.1186/s13660-017-1485-x R E S E A R C H Open Access Extensions of interpolation between the arithmetic-geometric mean inequality for matrices Motaba Bakherad Rahmatollah Lashkaripour * and Monire Hamohamadi * Correspondence: lashkari@hamoon.usb.ac.ir Department of Mathematics Faculty of Mathematics University of Sistan and Baluchestan Zahedan Iran Abstract In this paper we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities it is shown that if A B are n n matrices then AB f1 A A)g 1 B B) f A A)g B B) where f 1 f g 1 g are non-negative continuous functions such that f 1 t)f t)=t and g 1 t)g t)=t t 0). We also obtain the inequality AB pa A) m p +)B B) s 1 p)a A) n + pb B) p t in which m n s t are real numbers such that m + n = s + t =1 is an arbitrary unitarily invariant norm and p [0 1]. MSC: Primary 47A64; secondary 15A60 Keywords: arithmetic-geometric mean; unitarily invariant norm; Hilbert-Schmidt norm; Cauchy-Schwarz inequality 1 Introduction and preliminaries Let M n be the C -algebra of all n n complex matrices and be the standard scalar product in C n with the identity I. TheGelfandmapf t) f A) isanisometrical isomorphism between the C -algebra CspA)) of continuous functions on the spectrum spa) of a Hermitian matrix A and the C -algebra generated by A and I. Anorm on M n is said to be unitarily invariant norm if UAV = A for all unitary matrices U and V.ForA M n lets 1 A) s A) s n A) denote the singular values of A i.e. the eigenvalues of the positive semidefinite matrix A =A A) 1 arranged in a decreasing order with their multiplicities counted. Note that s A) =s A )=s A ) 1 n)and A = s 1 A). The Ky Fan norm of a matrix A is defined as A k) = k s A) 1 k n). The Fan dominance theorem asserts that A k) B k) for k = 1...n if and only if A B for every unitarily invariant normsee [1] p.93). The Hilbert- Schmidt norm is defined by A = n s A))1/ which is unitarily invariant. The Authors) 017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors) and the source provide a link to the Creative Commons license and indicate if changes were made.
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page of 10 The classical Cauchy-Schwarz inequality for a 0 b 01 n)statesthat n ) n a b ) n ) a b with equality if and only if a 1...a n )andb 1...b n ) are proportional []. Bhatia and Davis gaveamatrixcauchy-schwarzinequalityasfollows: AB A A B B 1) where A B M n. For further information as regards the Cauchy-Schwarz inequality see [3 5] and the references therein. Recently Kittaneh et al. [6] extended inequality 1) to the form AB A A ) p B B ) A A ) B B ) p ) where A B M n and p [0 1]. Audenaert [7] provedthatforalla B M n and all p [0 1] we have AB pa A +)B B 1 p)a A + pb B. 3) In [8] the authors generalized inequality 3) for all A B M n and all p [0 1] to the form AB pa A +)B B 1 p)a A + pb B. 4) Inequality 4) interpolates between the arithmetic-geometric mean inequality. In [6] the authors showed a refinement of inequality 4) for the Hilbert-Schmidt norm as follows: AB pa A +)B B r A A B B ) 1 p)a A + pb B r A A B B ) 5) in which A B M n p [0 1] and r = min{p1 p}. The Young inequality for every unitarily invariant norm states that A p B pa +)B wherea B are positive definite matrices and p [0 1] see [9] andalso[10 11]). Kosaki [1] extendedthelast inequality for the Hilbert-Schmidt norm as follows: A p B pa +)B 6) where A B are positive definite matrices is any matrix and p [0 1]. In [13] the authors considered as a refined matrix Young inequality for the Hilbert-Schmidt norm A p B + r A B pa +)B 7) in which A B are positive semidefinite matrices M n p [0 1] and r = min{p}.
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page 3 of 10 Based on the refined Young inequality 7) Zhao and Wu [14]provedthat A p B + r 0A 1 B 1 A +) A B pa +)B 8) for 0 < p 1 and A p B + r 1 0 A B 1 B + p A B pa +)B for 1 < p <1suchthatr = min{p} and r 0 = min{r1 r}. In this paper we obtain some operator and unitarily invariant norms inequalities. Among other results we obtain a refinement of inequality 5) and we also extend inequalities ) 3)and5) to the function f t)=t p p R). Main results In this section using some ideas of [6 15] we extend the Audenaert results for the operator norm. Theorem 1 Let A B M n and f 1 f g 1 g be non-negative continuous functions such that f 1 t)f t)=tandg 1 t)g t)=t t 0). Then AB f 1 A A ) g 1 B B ) f A A ) g B B ). 9) Proof It follows from AB = B A AB = s 1 B A AB ) = λ max B A AB ) since B A AB is positive semidefinite ) = λ max A AB B ) = λ max f 1 A A ) f A A ) g B B ) g 1 B B )) = λ max g1 B B ) f 1 A A ) f A A ) g B B )) g 1 B B ) f 1 A A ) f A A ) g B B ) g1 B B ) f 1 A A ) f A A ) g B B ) = f1 A A ) g 1 B B ) f A A ) g B B ) that we get the desired result. Corollary If A B M n and m n s t are real numbers such that m + n = s + t =1 then AB A A ) m B B ) s B B ) t. 10) In the next resultswe show some generalizationsof inequality3) for the operator norm.
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page 4 of 10 Corollary 3 Let A B M n and let f 1 f g 1 g be non-negative continuous functions such that f 1 t)f t)=tandg 1 t)g t)=t t 0). Then AB pf 1 A A ) 1 p +)g 1 B B ) 1 1 p)f A A ) 1 + pg B B ) 1 p where p [0 1]. Proof Applying Theorem 1 for = Iwehave AB f1 A A ) g 1 B B ) f A A ) g B B ) = f 1 A A ) 1 p ) p g1 B B ) 1 ) f A A ) 1 ) g B B ) 1 p ) p by Theorem 1) pf1 A A ) 1 p +)g 1 B B ) 1 1 p)f A A ) 1 + pg B B ) 1 p by the Young inequality). Corollary 4 Let A B M n and let f g be non-negative continuous functions such that f t)gt)=t t 0). Then AB pf A A ) +)g B B ) 1 1 p)f A A ) + pg B B ) 1 pg A A ) +)f B B ) 1 1 p)g A A ) + pf B B ) 1 where p [0 1]. Proof Applying Theorem 1 andtheyounginequalityweget AB 4 f A A ) 1 g B B ) 1 g A A ) 1 f B B ) 1 by Theorem 1 for f and g) f A A ) p g B B ) f A A ) g B B ) p g A A ) p f B B ) g A A ) f B B ) p by inequality 10)) pf A A ) +)g B B ) 1 p)f A A ) + pg B B ) pg A A ) +)f B B ) 1 p)g A A ) + pf B B ) by the Young inequality). 3 Some interpolations for unitarily invariant norms In this section applying some ideas of [6] we generalize some interpolations for an arbitrary unitarily invariant norm. Let Q kn denote the set of all strictly increasing k-tuples chosen from 1...n i.e. I Q kn if I =i 1 i...i k ) where 1 i 1 < i < < i k n. The following lemma gives some properties of the kth antisymmetric tensor powers of matrices in M n ;see[1] p.18.
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page 5 of 10 Lemma 5 Let A B M n. Then a) k A) k B)= k AB) for k =1...n. b) k A) = k A for k =1...n. c) k A) 1 = k A 1 for k =1...n. d) If s 1 s...s n are the singular values of A then the singular values of k A are s i1 s i...s ik where i 1 i...i k ) Q kn. Now we show inequality 10)foran arbitrary unitarily invariant norm. Theorem 6 Let A B M n and be an arbitrary unitarily invariant norm. Then AB A A ) m B B ) s B B ) t 11) where m n s t are real numbers such that m + n = s + t =1.In particular if A Barepositive definite then A 1 B 1 A p B A B p 1) where p [0 1]. Proof If we replace A B and by k A k B and k theirkth antisymmetric tensor powers in inequality 9) and applylemma5thenwehave k AB k A A ) m B B ) s k B B ) t which is equivalent to s 1 k k AB ) s 1 A A ) m B B ) ) k s s 1 B B ) ) t. Applying Lemma 5d) we have k s AB ) k k s 1 A A ) m B B ) s) k s 1 B B ) t) s 1 A A ) m B B ) s) s 1 B B ) t) 13) where k =1...n.Inequality13)impliesthat k s AB ) k s 1 A A ) m B B ) s) s 1 B B ) t) k s A A ) m B B ) s) ) 1 k s B B ) t) ) 1 by the Cauchy-Schwarz inequality)
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page 6 of 10 where k =1...n.Hence AB k) A A ) m B B ) s k) B B ) t k). Now using the Fan dominance theorem [1]p.98we get the desired result. Now using inequality 1) Theorem 6 and the same argument in the proof of Corollaries 3 and 4 we get the following results; these inequalities are generalizations of the Audenaert inequality 3). Corollary 7 Let A B M n m n s t be real numbers such that m + n = s + t =1and let be an arbitrary unitarily invariant norm. Then AB p A A ) m p +) B B ) s 1 p) A A ) n + p B B ) p t where p [0 1]. Corollary 8 Let A B M n m n s t be real numbers such that m + n = s + t =and let be an arbitrary unitarily invariant norm. Then AB p A A ) m +) B B ) s 1 1 p) A A ) m + p B B ) s 1 p +) B B ) t 1 1 p) + p B B ) t 1 14) in which p [0 1]. Remark 9 If we put n = m = s = t = 1 in inequality 14) then we obtain the Audenaert inequality 3). Also if we use inequality 6) Corollaries7 and 8 then similar to Corollaries 3 and 4 we get the following inequalities: AB p A A ) m p +) B B ) s 1 p) + p B B ) t p 15) where A B M n m n s t are real numbers such that m + n = s + t =1p [0 1] and AB p A A ) m +) B B ) s 1 1 p) A A ) m + p B B ) s 1 p +) B B ) t 1 1 p) + p B B ) t 1 for A B M n realnumbersm n s t inwhichm + n = s + t =andp [0 1]. These inequalities are generalizations of 4) for the Hilbert-Schmidt norms. In the following theorem we show a refinement of inequality 15) forthehilbert- Schmidt norm.
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page 7 of 10 Theorem 10 Let A B M n. Then AB p A A ) m p +) B B ) s r A A ) m p B B ) s ) 1 p) + p B B ) t p r B B ) t p ) in which m n s t are real numbers such that m+n = s+t =1p [0 1] and r = min{p}. Proof Applying inequality 11) we deduce that AB A A ) m B B ) s B B ) t = A A ) m ) p p B B ) s ) A A ) n ) B B ) p t ) p p A A ) m p +) B B ) s r A A ) m p B B ) s ) 1 p) + p B B ) t p r B B ) t p ) where p [0 1] and r = min{p} and the proof is complete. Theorem 10 includes a special case as follows. Corollary 11 [6] Theorem.5) Let A B M n. Then AB pa A +)B B r A A B B ) 1 p) A A ) + pb B r A A B B ) where p [0 1] and r = min{p}. Proof For p [0 1] if we put m = t = p and n = s = in Theorem 10 thenwegetthe desired result. Thenextresultisarefinementofinequality5). Theorem 1 Let A B M n C) and let p 0 1). Then i) For 0<p 1 AB pa A +)B B r 0 A A ) 1 B B ) 1 A A ) A A B B 1 ) 1 p)a A + pb B r 0 A A ) 1 B B ) 1 A A p A A B B ) 1. 16) ii) For 1 < p <1 AB pa A +)B B r 0 A A ) 1 B B ) 1 B B ) ) A A B B 1 )
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page 8 of 10 1 p)a A + pb B r 0 A A ) 1 B B ) 1 B B ) p A A B B ) 1 17) where r = min{p} and r 0 = min{r1 r}. Proof The proof of inequality 17) is similar to that of inequality 16). Thus we only need to prove the inequality 16). If 0 < p 1 replacinga and B by A A and B B in inequality 8) respectively we have A A ) p B B ) pa A +)B B r 0 A A ) 1 B B ) 1 A A ) A A B B ) 1. 18) Interchanging the roles of p and 1 p in the inequality 18) we get A A ) B B ) p 1 p)a A + pb B r 0 A A ) 1 B B ) 1 A A p A A B B 1 ). 19) Applying inequalities 11) 18)and19) we get the desired result. Corollary 13 Let A B M n C) and p 0 1). Then i) For 0<p 1 AB pa A +)B B r 0 A A ) 1 B B ) 1 A A ) A A B B 1 ) 1 p)a A + pb B r 0 A A ) 1 B B ) 1 A A p A A B B ) 1. ii) For 1 < p <1 AB pa A +)B B r 0 A A ) 1 B B ) 1 B B ) ) A A B B 1 ) 1 p)a A + pb B r 0 A A ) 1 B B ) 1 B B ) p A A B B ) 1 where r = min{p} and r 0 = min{r1 r}. Through the following we would like to obtain upper bound for AB forevery unitary invariant norm. The following lemma has been shown in [16] and it is considered as a refined matrix Young inequality for every unitary invariant norm.
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 Page 9 of 10 Lemma 14 Let A B M n such that A Barepositivesemidefinite. Then for 0 p 1 we have A p B + r 0 A B ) p A +) B ) 0) where r 0 = min{p}. Proposition 15 Let A B M n. Then AB p A A +) B B ) r 0 A A B B ) ) 1 1 p) A A + p B B ) r 0 A A B B ) ) 1 where p [0 1] and r 0 = min{p}. Proof In inequality 0) if we replace A by A A and B by B Bthenwehave A A ) p B B ) p A A +) B B ) r 0 A A B B ) ) 1. 1) Interchanging p with 1 p in inequality 1) we get A A ) B B ) p 1 p) A A + p B B ) r 0 A A B B ) ) 1. ) Now applying inequalities 11) 1)and)we get the desired inequality. 4 Conclusions Our application of the methods based on the Audenaert results is presented in this paper to the operator norm and so are some interpolations for an arbitrary unitarily invariant norm. Moreover we refine some previous inequalities as regards the Cauchy-Schwarz inequality for the operator and Hilbert-Schmidt norms. Acknowledgements The first author would like to thank the Tusi Mathematical Research Group TMRG). Funding The authors declare that there is no source of funding for this research. Competing interests The authors declare that they have no competing interests. Authors contributions The authors contributed equally to the manuscript and read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to urisdictional claims in published maps and institutional affiliations. Received: 4 May 017 Accepted: 5 August 017
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