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On Some General Inequalities Related to Operator AM-GM Inequality

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On Some General Inequalities Related to Operator AM-GM Inequality Filomat 34:11 (2020), 3601–3607 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL2011601S University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Some General Inequalities Related to Operator AM-GM Inequality Rahim Safshekana, Ali Farokhiniaa aDepartment of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran 1. Introduction Let B ( ) be the C∗–algebra of all bounded linear operators on a Hilbert space . As usual, we reserve m, M for scalarsH and I for the identity operator on . A self adjoint operator A is saidH to be positive (written A 0) if Ax; x 0 for all x , while it is saidH to be strictly positive (written A > 0) if A is positive and invertible.≥ h If Aiand≥ B are self2 adjoint, H we write B A in case B A 0. The Gelfand map f (t) f (A) is an ≥ − ≥ 7! isometrical –isomorphism between the C∗–algebra C (σ (A)) of continuous functions on the spectrum σ (A) ∗ of a self adjoint operator A and the C∗–algebra generated by A and the identity operator I. This is called the functional calculus of A. If f; 1 C (σ (A)), then f (t) 1 (t) (t σ (A)) implies f (A) 1 (A) (see [11, p. 3]). A linear map Φ : ( ) ( 2) is positive if Φ (A) ≥ 0 whenever2 A 0. It’s said to≥ be unital if Φ (I) = I. B H !B K ≥ ≥ For any strictly positive operator A; B B (H ) and 0 v 1, we write 2 ≤ ≤ 1 1 1 1 v 1 1 1 − A!vB = A− vB− ; A]vB = A 2 A− 2 BA− 2 A 2 ; A B = (1 v) A + vB: r r − 1 For the case v = 2 , we write !, ], and , respectively. We use the same notions for scalars. It is well-known that the arithmetic–geometricr mean inequality (in short, AM-GM inequality), with respect to operator order, says that A]vB A vB: (1) ≤ r The Lowner–Heinz¨ theorem [11, Theorem 1.8] says that if A; B B ( ) are positive, then for 0 p 1, 2 H ≤ ≤ A B Ap Bp: (2) ≤ ) ≤ In general (2) is not true for p > 1. Lin [7] nicely reduced the study of squared operator inequalities to that of some norm inequalities. Actually, he found that a reverse of operator AM-GM inequality can be squared 2 2 2 (A B) K(h) A]B ; (3) r ≤ 2010 Mathematics Subject Classification. Primary 47A63; Secondary 47A30, 47B20 Keywords. Operator inequality, Jensen’s inequality, convex functions, self adjoint operators, positive operators Received: 13 November 2019; Revised: 23 December 2019; Accepted: 03 January 2020 Communicated by Fuad Kittaneh Corresponding author: Ali Farokhinia Email addresses: ra [email protected] (Rahim Safshekan), [email protected] (Ali Farokhinia) R. Safshekan, A. Farokhinia / Filomat 34:11 (2020), 3601–3607 3602 whenever A; B B ( ) are two positive operators satisfying mI A; B MI for some scalars 0 < m < M, 2 (h+1)2 H M ≤ ≤ and K (h) = 4h with h = m . It follows from (2) and (3) that p p p (A B) K(h) A]B 0 < p 2 : (4) r ≤ ≤ It is natural to ask whether inequality (4) is true for p 2? Recently, an affirmative answer to this question has been given by Fu and He [2], where it has been proved≥ that 2 !p p (M + m) p (A B) 2 A]B : r ≤ 4 p Mm The problem of squaring operator inequalities has been studied extensively in the literature. We refer the reader to [4, 8–10, 12] as sample of this work. As mentioned above, Lin’s method was based on some observations about the operator norm and an arithmetic-geometric mean inequality of Bhatia and Kittaneh (see [7, Lemma 2.2]). This paper intends to square a reverse of operator AM-GM inequality in a different way. Moreover, we square the operator Polya–Szeg´ o¨ inequality [6, 10]. 2. Main Results To prove our generalized operator AM-GM inequalities, we need several well known lemmas. The first lemma is a simple consequence of the Jensen inequality concerning the convexity of certain power function [11, Theorem 1.4]. Lemma 2.1. (H¨older–McCarthyinequality) Let A B ( ) be a positive operator. Then for any unit vector x , 2 H 2 H p Ax; x Apx; x p 1 : h i ≤ h i ≥ The second lemma is the converses of Holder–McCarthy¨ inequality [11, Theorem 1.29]. Lemma 2.2. Let A B ( ) such that mI A MI for some scalars 0 < m < M. Then for any unit vector x , 2 H ≤ ≤ 2 H D E A2x; x K (h) Ax; x 2; ≤ h i 2 (h+1) M where K (h) = 4h with h = m . The third lemma is a reverse of operator AM–GM inequality, which has been proved in [5, Theorem 1]. Lemma 2.3. Let A; B B ( ) such that mI A; B MI for some scalars 0 < m < M. Then for any 0 v 1, 2 H ≤ ≤ ≤ ≤ m λM A vB r A]vB; r ≤ m]λM where λ = min v; 1 v . In particular, f − g p A B K (h)A]B; r ≤ 2 (h+1) M where K (h) = 4h with h = m . Our first result is a generalization of the inequality (3). Theorem 2.4. Let A; B B ( ) such that mI A; B MI for some scalars 0 < m < M. Then for any 0 v 1, 2 H ≤ ≤ ≤ ≤ !2 2 m λM 2 (A vB) K (h) r A]vB ; (5) r ≤ m]λM 2 where K (h) = (h+1) with h = M , and λ = min v; 1 v . In particular, 4h m f − g 2 2 2 (A B) K(h) A]B : r ≤ R. Safshekan, A. Farokhinia / Filomat 34:11 (2020), 3601–3607 3603 Proof. One can see that Lemma 2.3 implies * + m λM A vBx; x r A]vBx; x (6) h r i ≤ m]λM for any unit vector x . Taking the square in (6), we have 2 H * +2 2 m λM A vBx; x r A]vBx; x h r i ≤ m]λM !2 m λM 2 = r A]vBx; x (7) m]λM !2 m λM D 2 E r A]vB x; x (by Lemma 2.1): ≤ m]λM On the other hand, mI A; B MI implies ≤ ≤ (1 v) mI (1 v) A (1 v) MI; (8) − ≤ − ≤ − and vmI vB vMI: (9) ≤ ≤ It follows from (8) and (9) that mI A vB MI: ≤ r ≤ By applying Lemma 2.2, we get 1 D 2 E 2 (A vB) x; x A vBx; x : (10) K (h) r ≤ h r i Combining (7) and (10) we infer !2 D 2 E m λM D 2 E (A vB) x; x K (h) r A]vB x; x r ≤ m]λM for any unit vector x . This completes the proof. 2 H Another result of this type is the following one: Corollary 2.5. Let A; B B ( ) such that mI A; B MI for some scalars 0 < m < M. Then for any 0 v 1, 2 H ≤ ≤ ≤ ≤ 0 !2 1 2 B m λM C 2 2 (A vB) BK (h) r 1C M I + A]vB ; r ≤ @ m]λM − A where λ = min v; 1 v . f − g Proof. It follows from (5) that 0 !2 1 2 2 B m λM C 2 (A vB) A]vB BK (h) r 1C A]vB r − ≤ @ m]λM − A 0 !2 1 B m λM C 2 BK (h) r 1C M I; ≤ @ m]λM − A where the second inequality follows from the fact that A; B MI A]vB MI]vMI = MI: ≤ ) ≤ R. Safshekan, A. Farokhinia / Filomat 34:11 (2020), 3601–3607 3604 As pointed out by Fujii and Nakamura in their paper [3, Theorem 2], if A B ( ) is a positive operator such that mI A MI for some scalars 0 < m < M, and x is a unit vector,2 thenH for any p 1, ≤ ≤ 2 H ≥ p Apx; x K m; M; p Ax; x ; (11) h i ≤ h i where !p mMp Mmp p 1 Mp mp K m; M; p = − − − : (12) p 1 (M m) p · mMp Mmp − − − (M+m)2 We note that K (m; M; 1) = K (m; M; 2) = 4Mm is the original Kantorovich constant. Now, by employing− (11) and applying a same arguments as in the proof of Theorem 2.4 we reach the following result. Corollary 2.6. Let A; B B ( ) such that mI A; B MI for some scalars 0 < m < M. Then for any p 1 and 0 v 1, 2 H ≤ ≤ ≥ ≤ ≤ !p p m λM p (A vB) K m; M; p r A]vB ; r ≤ m]λM where K m; M; p is defined as in (12), and λ = min v; 1 v . f − g 1 1 1 m λM 1 1 1 1 − Since A− vB− r A− ]vB− and A− ]vB− = A]vB, it follows that r ≤ m]λM m λM A]vB r A!vB: (13) ≤ m]λM The following result concerning (13) may be stated: Corollary 2.7. Let A; B B ( ) such that mI A; B MI for some scalars 0 < m < M. Then for any p 1 and 0 v 1, 2 H ≤ ≤ ≥ ≤ ≤ !p p m λM p A]vB K m; M; p r (A!vB) ; ≤ m]λM where K m; M; p is defined as in (12), and λ = min v; 1 v .
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