Extensions of interpolation between the arithmetic-geometric mean inequality for matrices
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, X are n × n $n\times n$ matrices, then ∥ A X B ∗ ∥ 2 ≤ ∥ f 1 ( A ∗ A ) X g 1 ( B ∗ B ) ∥ ∥ f 2 ( A ∗ A ) X g 2 ( B ∗ B ) ∥ , $$\b...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-09-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | http://link.springer.com/article/10.1186/s13660-017-1485-x |
| _version_ | 1828390245591154688 |
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| author | Mojtaba Bakherad Rahmatollah Lashkaripour Monire Hajmohamadi |
| author_facet | Mojtaba Bakherad Rahmatollah Lashkaripour Monire Hajmohamadi |
| author_sort | Mojtaba Bakherad |
| collection | DOAJ |
| description | Abstract In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are n × n $n\times n$ matrices, then ∥ A X B ∗ ∥ 2 ≤ ∥ f 1 ( A ∗ A ) X g 1 ( B ∗ B ) ∥ ∥ f 2 ( A ∗ A ) X g 2 ( B ∗ B ) ∥ , $$\begin{aligned} \bigl\Vert AXB^{*} \bigr\Vert ^{2}\leq \bigl\Vert f_{1} \bigl(A^{*}A\bigr)Xg_{1}\bigl(B^{*}B\bigr) \bigr\Vert \bigl\Vert f_{2}\bigl(A^{*}A\bigr)Xg_{2}\bigl(B^{*}B\bigr) \bigr\Vert , \end{aligned}$$ where f 1 $f_{1}$ , f 2 $f_{2}$ , g 1 $g_{1}$ , g 2 $g_{2}$ are non-negative continuous functions such that f 1 ( t ) f 2 ( t ) = t $f_{1}(t)f_{2}(t)=t$ and g 1 ( t ) g 2 ( t ) = t $g_{1}(t)g_{2}(t)=t$ ( t ≥ 0 $t\geq0$ ). We also obtain the inequality | | | A B ∗ | | | 2 ≤ | | | p ( A ∗ A ) m p + ( 1 − p ) ( B ∗ B ) s 1 − p | | | | | | ( 1 − p ) ( A ∗ A ) n 1 − p + p ( B ∗ B ) t p | | | , $$\begin{aligned} \bigl\vert \!\bigl\vert \!\bigl\vert AB^{*} \bigr\vert \!\bigr\vert \!\bigr\vert ^{2} &\leq \bigl\vert \!\bigl\vert \!\bigl\vert p \bigl(A^{*}A\bigr)^{\frac{m}{p}}+ (1-p) \bigl(B^{*}B\bigr)^{\frac {s}{1-p}} \bigr\vert \!\bigr\vert \!\bigr\vert \bigl\vert \!\bigl\vert \!\bigl\vert (1-p) \bigl(A^{*}A\bigr)^{\frac{n}{1-p}}+ p\bigl(B^{*}B \bigr)^{\frac{t}{p}} \bigr\vert \!\bigr\vert \!\bigr\vert , \end{aligned}$$ in which m, n, s, t are real numbers such that m + n = s + t = 1 $m+n=s+t=1$ , | | | ⋅ | | | $\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert $ is an arbitrary unitarily invariant norm and p ∈ [ 0 , 1 ] $p\in[0,1]$ . |
| first_indexed | 2024-12-10T06:40:47Z |
| format | Article |
| id | doaj.art-ec8c775a43c04023b1c30d7c86e2abb3 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-10T06:40:47Z |
| publishDate | 2017-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-ec8c775a43c04023b1c30d7c86e2abb32022-12-22T01:58:47ZengSpringerOpenJournal of Inequalities and Applications1029-242X2017-09-012017111010.1186/s13660-017-1485-xExtensions of interpolation between the arithmetic-geometric mean inequality for matricesMojtaba Bakherad0Rahmatollah Lashkaripour1Monire Hajmohamadi2Department of Mathematics, Faculty of Mathematics, University of Sistan and BaluchestanDepartment of Mathematics, Faculty of Mathematics, University of Sistan and BaluchestanDepartment of Mathematics, Faculty of Mathematics, University of Sistan and BaluchestanAbstract In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are n × n $n\times n$ matrices, then ∥ A X B ∗ ∥ 2 ≤ ∥ f 1 ( A ∗ A ) X g 1 ( B ∗ B ) ∥ ∥ f 2 ( A ∗ A ) X g 2 ( B ∗ B ) ∥ , $$\begin{aligned} \bigl\Vert AXB^{*} \bigr\Vert ^{2}\leq \bigl\Vert f_{1} \bigl(A^{*}A\bigr)Xg_{1}\bigl(B^{*}B\bigr) \bigr\Vert \bigl\Vert f_{2}\bigl(A^{*}A\bigr)Xg_{2}\bigl(B^{*}B\bigr) \bigr\Vert , \end{aligned}$$ where f 1 $f_{1}$ , f 2 $f_{2}$ , g 1 $g_{1}$ , g 2 $g_{2}$ are non-negative continuous functions such that f 1 ( t ) f 2 ( t ) = t $f_{1}(t)f_{2}(t)=t$ and g 1 ( t ) g 2 ( t ) = t $g_{1}(t)g_{2}(t)=t$ ( t ≥ 0 $t\geq0$ ). We also obtain the inequality | | | A B ∗ | | | 2 ≤ | | | p ( A ∗ A ) m p + ( 1 − p ) ( B ∗ B ) s 1 − p | | | | | | ( 1 − p ) ( A ∗ A ) n 1 − p + p ( B ∗ B ) t p | | | , $$\begin{aligned} \bigl\vert \!\bigl\vert \!\bigl\vert AB^{*} \bigr\vert \!\bigr\vert \!\bigr\vert ^{2} &\leq \bigl\vert \!\bigl\vert \!\bigl\vert p \bigl(A^{*}A\bigr)^{\frac{m}{p}}+ (1-p) \bigl(B^{*}B\bigr)^{\frac {s}{1-p}} \bigr\vert \!\bigr\vert \!\bigr\vert \bigl\vert \!\bigl\vert \!\bigl\vert (1-p) \bigl(A^{*}A\bigr)^{\frac{n}{1-p}}+ p\bigl(B^{*}B \bigr)^{\frac{t}{p}} \bigr\vert \!\bigr\vert \!\bigr\vert , \end{aligned}$$ in which m, n, s, t are real numbers such that m + n = s + t = 1 $m+n=s+t=1$ , | | | ⋅ | | | $\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert $ is an arbitrary unitarily invariant norm and p ∈ [ 0 , 1 ] $p\in[0,1]$ .http://link.springer.com/article/10.1186/s13660-017-1485-xarithmetic-geometric meanunitarily invariant normHilbert-Schmidt normCauchy-Schwarz inequality |
| spellingShingle | Mojtaba Bakherad Rahmatollah Lashkaripour Monire Hajmohamadi Extensions of interpolation between the arithmetic-geometric mean inequality for matrices Journal of Inequalities and Applications arithmetic-geometric mean unitarily invariant norm Hilbert-Schmidt norm Cauchy-Schwarz inequality |
| title | Extensions of interpolation between the arithmetic-geometric mean inequality for matrices |
| title_full | Extensions of interpolation between the arithmetic-geometric mean inequality for matrices |
| title_fullStr | Extensions of interpolation between the arithmetic-geometric mean inequality for matrices |
| title_full_unstemmed | Extensions of interpolation between the arithmetic-geometric mean inequality for matrices |
| title_short | Extensions of interpolation between the arithmetic-geometric mean inequality for matrices |
| title_sort | extensions of interpolation between the arithmetic geometric mean inequality for matrices |
| topic | arithmetic-geometric mean unitarily invariant norm Hilbert-Schmidt norm Cauchy-Schwarz inequality |
| url | http://link.springer.com/article/10.1186/s13660-017-1485-x |
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