Hindawi Publishing Corporation Journal of Operators Volume 2014, Article ID 382364, 5 pages http://dx.doi.org/10.1155/2014/382364
Research Article Inequalities of Convex Functions and Self-Adjoint Operators Zlatko PaviT Mechanical Engineering Faculty in Slavonski Brod, University of Osijek, Trg Ivane Brli´c Maˇzurani´c 2, 35000 Slavonski Brod, Croatia Correspondence should be addressed to Zlatko Pavi´c;
[email protected] Received 28 November 2013; Accepted 4 January 2014; Published 9 February 2014 Academic Editor: Palle E. Jorgensen Copyright © 2014 Zlatko Pavi´c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The paper offers generalizations of the Jensen-Mercer inequality for self-adjoint operators and generally convex functions. The obtained results are applied to define the quasi-arithmetic operator means without using operator convexity. The version of the harmonic-geometric-arithmetic operator mean inequality is derived as an example.
1. Introduction Throughout the paper we will use a real interval I with the nonempty interior and real segments [𝑎, 𝑏] and (𝑎, 𝑏) with 𝑎 < 𝑏. We briefly summarize a development path of the operator form of Jensen’s inequality. Let H and K be Hilbert spaces, let B(H) and B(K) be associated C∗ -algebras of bounded linear operators, and let 1𝐻 and 1𝐾 be their identity operators. Combining the results from [1, 2], it follows that every operator convex function 𝑓 : I → R satisfies the Schwarz inequality 𝑓 (Φ (𝐴)) ≤ Φ (𝑓 (𝐴)) ,
(1)
where Φ : B(H) → B(K) is a positive linear mapping such that Φ(1𝐻) = 1𝐾 and 𝐴 ∈ B(H) is a self-adjoint operator with the spectrum Sp(𝐴) ⊆ I. The above inequality was extended in [3] to the inequality 𝑛
𝑛
𝑓 (∑Φ𝑖 (𝐴 𝑖 )) ≤ ∑Φ𝑖 (𝑓 (𝐴 𝑖 )) , 𝑖=1
(2)
𝑖=1
where Φ𝑖 : B(H) → B(K) are positive linear mappings such that ∑𝑛𝑖=1 Φ𝑖 (1𝐻) = 1𝐾 and 𝐴 𝑖 ∈ B(H) are self-adjoint operators with spectra Sp(𝐴 𝑖 ) ⊆ I. The operator inequality of (2) was formulated for convex (without operator) continuous functions in [4] assuming the spectral conditions: Sp(𝐴) ⊆ [𝑎, 𝑏] and Sp(𝐴 𝑖 ) ∩ (𝑎, 𝑏) = 0 for all 𝐴 𝑖 , where 𝐴 = ∑𝑛𝑖=1 Φ𝑖 (𝐴 𝑖 ).
Including positive operators 𝑃𝑖 ∈ B(H) satisfying ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 , we have that every convex continuous function 𝑓 : I → R satisfies the inequality 𝑛
𝑛
𝑖=1
𝑖=1
𝑓 (∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )) ≤ ∑Φ𝑖 (𝑃𝑖1/2 𝑓 (𝐴 𝑖 ) 𝑃𝑖1/2 )
(3)
if provided the spectral conditions: Sp(𝐴) ⊆ [𝑎, 𝑏] and Sp(𝐴 𝑖 ) ∩ (𝑎, 𝑏) = 0 for all self-adjoint operators 𝐴 𝑖 , and the operator sum 𝐴 = ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ). The inequality in (3) is possible because the operators 𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 and 𝑃𝑖1/2 𝑓(𝐴 𝑖 )𝑃𝑖1/2 are self-adjoint.
2. Discrete and Operator Inequalities for Convex Functions and Trinomial Affine Combinations 2.1. Discrete Variants. Every number 𝑥 ∈ R can be uniquely presented as the binomial affine combination 𝑏−𝑥 𝑥−𝑎 (4) 𝑎+ 𝑏 𝑏−𝑎 𝑏−𝑎 which is convex if and only if the number 𝑥 belongs to the line : interval [𝑎, 𝑏]. Given the function 𝑓 : R → R, let 𝑓{𝑎,𝑏} R → R be the function of the chord line passing through the points 𝐴(𝑎, 𝑓(𝑎)) and 𝐵(𝑏, 𝑓(𝑏)) of the graph of 𝑓. Applying line to the combination in (4), we get the affinity of 𝑓{𝑎,𝑏} 𝑥=
line 𝑓{𝑎,𝑏} (𝑥) =
𝑏−𝑥 𝑥−𝑎 𝑓 (𝑎) + 𝑓 (𝑏) . 𝑏−𝑎 𝑏−𝑎
(5)
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If the function 𝑓 is convex, then we have the inequality line 𝑓 (𝑥) ≤ 𝑓{𝑎,𝑏} (𝑥)
if 𝑥 ∈ [𝑎, 𝑏] ,
(6)
if 𝑥 ∉ (𝑎, 𝑏) .
(7)
and the reverse inequality line 𝑓 (𝑥) ≥ 𝑓{𝑎,𝑏} (𝑥)
Let 𝛼, 𝛽, 𝛾 ∈ R be coefficients such that 𝛼 + 𝛽 − 𝛾 = 1. Let 𝑎, 𝑏, 𝑐 ∈ R be points where 𝑎 < 𝑏. We consider the affine combination 𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐. Inserting the affine combination 𝑐 = 𝜆𝑎+𝜇𝑏 assuming that 𝜆+𝜇 = 1, we get the binomial form 𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐 = (𝛼 − 𝛾𝜆) 𝑎 + (𝛽 − 𝛾𝜇) 𝑏.
(8)
Lemma 1. Let 𝛼, 𝛽, 𝛾 ∈ [0, 1] be coefficients such that 𝛼 + 𝛽 − 𝛾 = 1. Let 𝑎, 𝑏, 𝑐 ∈ R be points such that 𝑎 < 𝑏 and 𝑐 ∈ [𝑎, 𝑏]. Then the affine combination 𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐 ∈ [𝑎, 𝑏] ,
(9)
and every convex function 𝑓 : [𝑎, 𝑏] → R satisfies the inequality 𝑓 (𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐) ≤ 𝛼𝑓 (𝑎) + 𝛽𝑓 (𝑏) − 𝛾𝑓 (𝑐) .
(10)
Proof. The condition 𝑐 = 𝜆𝑎 + 𝜇𝑏 ∈ [𝑎, 𝑏] involves 𝜆, 𝜇 ∈ [0, 1]. Then the binomial combination of the right-hand side in (8) is convex since its coefficients 𝛼−𝛾𝜆 ≥ 𝛼−𝛾 = 1−𝛽 ≥ 0 and also 𝛽−𝛾𝜇 ≥ 0. So, the combination 𝛼𝑎+𝛽𝑏−𝛾 belongs to line , [𝑎, 𝑏]. Applying the inequality in (6) and the affinity of 𝑓{𝑎,𝑏} we get line (𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐) 𝑓 (𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐) ≤ 𝑓{𝑎,𝑏} line = 𝛼𝑓 (𝑎) + 𝛽𝑓 (𝑏) − 𝛾𝑓{𝑎,𝑏} (𝑐)
(11)
≤ 𝛼𝑓 (𝑎) + 𝛽𝑓 (𝑏) − 𝛾𝑓 (𝑐)
Lemma 1 is trivially true if 𝑎 = 𝑏. It is also valid for 𝛾 ∈ [−1, 1] because then the observed affine combinations with 𝛾 ≤ 0 become convex, and associated inequalities follow from Jensen’s inequality. The similar combinations including 𝛾 ∈ [−1, 1] were observed in [5, Corollary 11 and Theorem 12] additionally using a monotone function 𝑔. If 𝛼 = 𝛽 = 𝛾 = 1, then the inequality in (10) is reduced to simple Mercer’s variant of Jensen’s inequality obtained in [6]. Lemma 2. Let 𝛼, 𝛽, 𝛾 ∈ [1, ∞) be coefficients such that 𝛼 + 𝛽 − 𝛾 = 1. Let 𝑎, 𝑏, 𝑐 ∈ R be points such that 𝑎 < 𝑏 and 𝑐 ∉ (𝑎, 𝑏). Then the affine combination (12)
and every convex function 𝑓 : I → R, where I = conv{𝑎, 𝑏, 𝑐} satisfies the inequality 𝑓 (𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐) ≥ 𝛼𝑓 (𝑎) + 𝛽𝑓 (𝑏) − 𝛾𝑓 (𝑐) .
It is not necessary to require 𝛾 ∈ [1, ∞) in Lemma 2, because it follows from the other coefficient conditions. 2.2. Operator Variants. We write 𝐴 ≤ 𝐵 for self-adjoint operators 𝐴, 𝐵 ∈ B(H) if the inner product inequality ⟨𝐴𝑥, 𝑥⟩ ≤ ⟨𝐵𝑥, 𝑥⟩ holds for every vector 𝑥 ∈ H. A selfadjoint operator 𝐴 is positive (nonnegative) if it is greater than or equal to null operator (𝐴 ≥ 0). If Sp(𝐴) ⊆ I and 𝑓, 𝑔 : I → R are continuous functions such that 𝑓(𝑥) ≤ 𝑔(𝑥) for every 𝑥 ∈ Sp(𝐴), then the operator inequality 𝑓(𝐴) ≤ 𝑔(𝐴) is valid. The bounds of a self-adjoint operator 𝐴 are defined with 𝑎𝐴 = inf ⟨𝐴𝑥, 𝑥⟩ , ‖𝑥‖=1
𝑏𝐴 = sup ⟨𝐴𝑥, 𝑥⟩ , ‖𝑥‖=1
(14)
and its spectrum Sp(𝐴) is contained in [𝑎𝐴, 𝑏𝐴 ] wherein we have the operator inequality 𝑎𝐴 1𝐻 ≤ 𝐴 ≤ 𝑏𝐴1𝐻.
(15)
More details on the theory of bounded operators and their inequalities can be found in [7]. The operator versions of Lemmas 1 and 2 follow. Corollary 3. Let 𝛼, 𝛽, 𝛾 ∈ [0, 1] be coefficients such that 𝛼 + 𝛽 − 𝛾 = 1. Let 𝐴 ∈ B(H) be a self-adjoint operator such that Sp(𝐴) ⊆ [𝑎, 𝑏]. Then Sp (𝛼𝑎1𝐻 + 𝛽𝑏1𝐻 − 𝛾𝐴) ⊆ [𝑎, 𝑏] ,
(16)
and every convex continuous function 𝑓 : [𝑎, 𝑏] → R satisfies the inequality
line because 𝑓{𝑎,𝑏} (𝑐) ≥ 𝑓(𝑐).
𝛼𝑎 + 𝛽𝑏 − 𝛾𝑐 ∉ (𝑎, 𝑏) ,
Proof. The condition 𝑐 = 𝜆𝑎 + 𝜇𝑏 ∉ (𝑎, 𝑏) entails 𝜆 ≤ 0 or 𝜆 ≥ 1, and the coefficients of the binomial form of (8) satisfy 𝛼 − 𝛾𝜆 ≥ 𝛼 ≥ 1 if 𝜆 ≤ 0, or 𝛼 − 𝛾𝜆 ≤ 𝛼 − 𝛾 = 1 − 𝛽 ≤ 0 if 𝜆 ≥ 1. So, the combination 𝛼𝑎+𝛽𝑏−𝛾 does not belong to (𝑎, 𝑏). Applying the inequality in (7), we get the series of inequalities as in (11) but with the reverse inequality signs.
(13)
𝑓 (𝛼𝑎1𝐻 + 𝛽𝑏1𝐻 − 𝛾𝐴) ≤ 𝛼𝑓 (𝑎) 1𝐻 + 𝛽𝑓 (𝑏) 1𝐻 − 𝛾𝑓 (𝐴) . (17) Proof. The spectral inclusion in (16) follows from the incluline sion in (9). Using the affinity of the function 𝑓{𝑎,𝑏} and the line operator inequalities 𝑓{𝑎,𝑏} (⋅) ≥ 𝑓(⋅), we can replace the discrete inequalities in (11) with the operator inequalities. Corollary 4. Let 𝛼, 𝛽, 𝛾 ∈ [1, ∞) be coefficients such that 𝛼 + 𝛽 − 𝛾 = 1. Let 𝐴 ∈ B(H) be a self-adjoint operator such that Sp(𝐴) ∩ (𝑎, 𝑏) = 0. Then Sp (𝛼𝑎1𝐻 + 𝛽𝑏1𝐻 − 𝛾𝐴) ∩ (𝑎, 𝑏) = 0,
(18)
and every convex continuous function 𝑓 : I → R, where I contains Sp(𝐴) and [𝑎, 𝑏], satisfies the inequality 𝑓 (𝛼𝑎1𝐻 + 𝛽𝑏1𝐻 − 𝛾𝐴) ≥ 𝛼𝑓 (𝑎) 1𝐻 + 𝛽𝑓 (𝑏) 1𝐻 − 𝛾𝑓 (𝐴) . (19)
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3
3. Main Results We want to extend and generalize the inequalities in (17) and (19) including positive operators and positive linear mappings. The main results are Theorems 8 and 9. Lemma 5. Let Φ𝑖 : B(H) → B(K) be linear mappings and let 𝑃𝑖 ∈ B(H) be positive linear operators so that ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 . Let 𝐴 𝑖 ∈ B(H) be self-adjoint operators. Then every affine function 𝑔(𝑥) = 𝑢𝑥 + V, where 𝑢 and V are real constants, satisfies the operator equality 𝑛
𝑔 (∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )) 𝑖=1
=
∑Φ𝑖 (𝑃𝑖1/2 𝑔 (𝐴 𝑖 ) 𝑃𝑖1/2 ) . 𝑖=1
(20)
𝑖=1
𝑖=1
𝑔 (∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )) = 𝑢∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ) + V1𝐾 𝑛
𝑛
𝑖=1
𝑖=1
=
∑Φ𝑖 (𝑃𝑖1/2 𝑖=1
𝑖=1
𝑖=1
𝑛
line ≤ 𝑓{𝑎,𝑏} (∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ))
≤ max {𝑓 (𝑎) 1𝐾 , 𝑓 (𝑏) 1𝐾 } .
Proof. The inequality in (25) is the consequence of Lemmas 5 and 6, and the discrete inequality 𝑛
𝑛
𝑖=1
𝑖=1
max {𝑓 (∑𝑝𝑖 𝑥𝑖 ) , ∑𝑝𝑖 𝑓 (𝑥𝑖 )}
= 𝑢∑Φ𝑖 (𝑃𝑖 𝐴 𝑖 ) + V∑Φ𝑖 (𝑃𝑖 ) 𝑛
𝑛
(25)
Proof. Applying the affinity of the function 𝑔 and the assumption ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 , it follows that 𝑛
𝑛
max {𝑓 (∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )) , ∑Φ𝑖 (𝑃𝑖1/2 𝑓 (𝐴 𝑖 ) 𝑃𝑖1/2 )}
𝑖=1
𝑛
𝑛
that ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 . Let 𝐴 𝑖 ∈ B(H) be self-adjoint operators such that Sp(𝐴 𝑖 ) ⊆ [𝑎, 𝑏]. Then every convex continuous function 𝑓 : [𝑎, 𝑏] → R satisfies the inequality
𝑛
(𝑢𝐴 𝑖 +
V1𝐻) 𝑃𝑖1/2 )
𝑖=1
≤ max {𝑓 (𝑎) , 𝑓 (𝑏)} ,
𝑛
= ∑Φ𝑖 (𝑃𝑖1/2 𝑔 (𝐴 𝑖 ) 𝑃𝑖1/2 ) 𝑖=1
(21) achieving the equality in (20). Lemma 6. Let Φ𝑖 : B(H) → B(K) be positive linear mappings and let 𝑃𝑖 ∈ B(H) be positive linear operators so that ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 . Let 𝐴 𝑖 ∈ B(H) be self-adjoint operators such that Sp(𝐴 𝑖 ) ⊆ [𝑎, 𝑏]. Then the spectrum of the operator sum 𝐴 = 𝑛 1/2 1/2 ∑𝑖=1 Φ𝑖 (𝑃𝑖 𝐴 𝑖 𝑃𝑖 ) is contained in [𝑎, 𝑏]. Proof. Applying the positive operators 𝑃𝑖 and the positive mappings Φ𝑖 to the assumed spectral inequalities 𝑎1𝐻 ≤ 𝐴 𝑖 ≤ 𝑏1𝐻,
(26)
line ≤ 𝑓{𝑎,𝑏} (∑𝑝𝑖 𝑥𝑖 )
where 𝑥𝑖 ∈ [𝑎, 𝑏] are points and 𝑝𝑖 ∈ [0, 1] are coefficients of the sum equal to 1. Theorem 8. Let 𝛼, 𝛽, 𝛾 ∈ [0, 1] be coefficients such that 𝛼 + 𝛽 − 𝛾 = 1. Let Φ𝑖 : B(H) → B(K) be positive linear mappings and let 𝑃𝑖 ∈ B(H) be positive linear operators so that ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 . Let 𝐴 𝑖 ∈ B(H) be self-adjoint operators such that Sp(𝐴 𝑖 ) ⊆ [𝑎, 𝑏]. Then the spectrum of the operator 𝑛
𝐴 = 𝛼𝑎1𝐾 + 𝛽𝑏1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )
(27)
𝑖=1
(22) is contained in [𝑎, 𝑏], and every convex continuous function 𝑓 : [𝑎, 𝑏] → R satisfies the inequality
we get 𝑎Φ𝑖 (𝑃𝑖 ) ≤ Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ) ≤ 𝑏Φ𝑖 (𝑃𝑖 ) .
(23)
Summing the above inequalities and using the assumption ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 , we have 𝑎1𝐾 ≤
𝑛
∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ) 𝑖=1
≤ 𝑏1𝐾
𝑛
𝑓 (𝐴) ≤ 𝛼𝑓 (𝑎) 1𝐾 + 𝛽𝑓 (𝑏) 1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝑓 (𝐴 𝑖 ) 𝑃𝑖1/2 ) . 𝑖=1
(28)
(24)
which provides that Sp(𝐴) ⊆ [𝑎, 𝑏].
If the function 𝑓 is concave, then the reverse inequality is valid in (28).
Corollary 7. Let Φ𝑖 : B(H) → B(K) be positive linear mappings and let 𝑃𝑖 ∈ B(H) be positive linear operators so
Proof. Taking the operator sum 𝐴 = ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ), the spectral inclusion Sp(𝐴) ⊆ [𝑎, 𝑏] follows from Lemma 6 and
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the inclusion in (16). Assuming and applying the convexity of line 𝑓 and the affinity of 𝑓{𝑎,𝑏} according to Lemma 5, we get 𝑛
line 𝑓 (𝐴) ≤ 𝑓{𝑎,𝑏} (𝛼𝑎1𝐾 + 𝛽𝑏1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )) 𝑖=1
= 𝛼𝑓 (𝑎) 1𝐾 + 𝛽𝑓 (𝑏) 1𝐾
that 𝜓 is 𝜑-convex. A similar notation is used for concavity. This terminology is taken from [9, Definition 1.19]. A continuous function 𝑓 : I → R is said to be operator increasing on I if 𝐴 ≤ 𝐵 implies 𝑓(𝐴) ≤ 𝑓(𝐵) for every pair of self-adjoint operators 𝐴, 𝐵 ∈ B(H) with spectra in I. A function 𝑓 is said to be operator decreasing if the function −𝑓 is operator increasing. Take an operator affine combination
𝑛
line − 𝛾𝑓{𝑎,𝑏} (∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ))
𝑛
𝐴 = 𝛼𝑎1𝐾 + 𝛽𝑏1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )
𝑖=1
𝑛
line (𝐴 𝑖 ) 𝑃𝑖1/2 ) = 𝛼𝑓 (𝑎) 1𝐾 + 𝛽𝑓 (𝑏) 1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝑓{𝑎,𝑏} 𝑖=1 𝑛
≤ 𝛼𝑓 (𝑎) 1𝐾 + 𝛽𝑓 (𝑏) 1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝑓 (𝐴 𝑖 ) 𝑃𝑖1/2 ) 𝑖=1
(29) line because 𝑓{𝑎,𝑏} (𝐴 𝑖 ) ≥ 𝑓(𝐴 𝑖 ).
as in Theorem 8. If 𝜑 : [𝑎, 𝑏] → R is a strictly monotone continuous function, we define the 𝜑-quasi-arithmetic mean of the combination 𝐴 as the operator 𝑀𝜑 (𝐴) = 𝜑−1 (𝛼𝜑 (𝑎) 1𝐾 + 𝛽𝜑 (𝑏) 1𝐾 (33)
𝑛
The version of Theorem 8 for 𝛼 = 𝛽 = 𝛾 = 1 and all 𝑃𝑖 = 1𝐻 was obtained in [8] as the main result. Theorem 9. Let 𝛼, 𝛽, 𝛾 ∈ [1, ∞) be coefficients such that 𝛼 + 𝛽 − 𝛾 = 1. Let Φ𝑖 : B(H) → B(K) be positive linear mappings and let 𝑃𝑖 ∈ B(H) be positive linear operators so that ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖 ) = 1𝐾 . Let 𝐴 𝑖 ∈ B(H) be self-adjoint operators such that Sp(𝐴 𝑖 ) ∩ (𝑎, 𝑏) = 0, and let 𝐴 = ∑𝑛𝑖=1 Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ) be the operator sum such that Sp(𝐴) ∩ (𝑎, 𝑏) = 0. Then the spectrum of the operator 𝑛
𝐴 = 𝛼𝑎1𝐾 + 𝛽𝑏1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 )
(30)
𝑖=1
satisfies the relation Sp(𝐴) ∩ (𝑎, 𝑏) = 0, and every convex continuous function 𝑓 : I → R, where I contains all spectra and [𝑎, 𝑏], satisfies the inequality 𝑓 (𝐴) ≥ 𝛼𝑓 (𝑎) 1𝐾 + 𝛽𝑓 (𝑏) 1𝐾 𝑛
− 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝑓 (𝐴 𝑖 ) 𝑃𝑖1/2 ) .
(32)
𝑖=1
− 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝜑 (𝐴 𝑖 ) 𝑃𝑖1/2 )) . 𝑖=1 The spectrum of the operator 𝑀𝜑 (𝐴) is contained in [𝑎, 𝑏] because the spectrum of the operator 𝑛
𝐴𝜑 = 𝛼𝜑 (𝑎) 1𝐾 + 𝛽𝜑 (𝑏) 1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝜑 (𝐴 𝑖 ) 𝑃𝑖1/2 ) 𝑖=1
(34) is contained in 𝜑([𝑎, 𝑏]). The quasi-arithmetic means defined in (33) are invariant with respect to the affinity; that is, the equality 𝑀𝑢𝜑+V (𝐴) = 𝑀𝜑 (𝐴)
(35)
holds for all pairs of real numbers 𝑢 ≠ 0 and V. Indeed, if 𝜓(𝑥) = 𝑢𝜑(𝑥) + V, then 𝐴𝜓 = 𝑢𝐴𝜑 + V1𝐾 ,
(31)
𝑖=1
If the function 𝑓 is concave, then the reverse inequality is valid in (31). Proof. The relation Sp(𝐴) ∩ (𝑎, 𝑏) = 0 is the consequence of the relation in (18). Assuming and using the convexity of 𝑓 line line and the affinity of 𝑓{𝑎,𝑏} , as well as the inequalities 𝑓{𝑎,𝑏} (𝐴 𝑖 ) ≤ 𝑓(𝐴 𝑖 ), we get the series of inequalities as in (29) but with the reverse inequality signs.
1 𝜓−1 (𝑥) = 𝜑−1 ( (𝑥 − V)) , 𝑢
(36)
and therefore, it follows that 1 𝑀𝜓 (𝐴) = 𝜓−1 (𝐴𝜓 ) = 𝜑−1 ( (𝐴𝜓 − V1𝐾 )) 𝑢
(37)
= 𝜑−1 (𝐴𝜑 ) = 𝑀𝜑 (𝐴) .
4. Application to Quasi-Arithmetic Means
The order of the pair of quasi-arithmetic means 𝑀𝜑 and 𝑀𝜓 depends on convexity of the function 𝜓 ∘ 𝜑−1 and monotonicity of the function 𝜓. Theorem 8 can be applied to operator means as follows.
In applications of convexity to quasi-arithmetic means, we use strictly monotone continuous functions 𝜑, 𝜓 : I → R such that the function 𝜓 ∘ 𝜑−1 is convex, in which case we say
Corollary 10. Let 𝐴 be an affine combination as in (32) satisfying the assumptions of Theorem 8. Let 𝜑, 𝜓 : [𝑎, 𝑏] → R be strictly monotone continuous functions.
Journal of Operators
5
If 𝜓 is either 𝜑-convex with operator increasing 𝜓−1 or 𝜑-concave with operator decreasing 𝜓−1 , then one has the inequality 𝑀𝜑 (𝐴) ≤ 𝑀𝜓 (𝐴) .
Proof. Let us prove the case in which 𝜓 is 𝜑-convex with operator increasing 𝜓−1 . Put [𝑐, 𝑑] = 𝜑([𝑎, 𝑏]). Applying the inequality in (28) of Theorem 8 to the affine combination 𝐴𝜑 of (34) with Sp(𝐴𝜑 ) ⊆ [𝑐, 𝑑] and the convex function 𝑓 = 𝜓 ∘ 𝜑−1 : [𝑐, 𝑑] → R, we get 𝜓 ∘ 𝜑−1 (𝐴𝜑 ) ≤ 𝐴𝜓 .
(39)
−1
Assigning the increasing function 𝜓 to the above inequality, we attain 𝑀𝜑 (𝐴) = 𝜑−1 (𝐴𝜑 ) ≤ 𝜓−1 (𝐴𝜓 ) = 𝑀𝜓 (𝐴)
(40)
which finishes the proof. Using Corollary 10 we get the following version of the harmonic-geometric-arithmetic mean inequality for operators. Corollary 11. If 𝐴 is an affine operator combination as in (32) satisfying the assumptions of Theorem 8 with the addition that [𝑎, 𝑏] ⊂ (0, ∞), then one has the harmonic-geometricarithmetic operator inequality −1
𝑛 𝛼 𝛽 1/2 (( + ) 1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝐴−1 𝑖 𝑃𝑖 )) 𝑎 𝑏 𝑖=1 𝑛
≤ ln 𝑎𝛼 𝑏𝛽 1𝐾 − 𝛾 exp (∑Φ𝑖 (𝑃𝑖1/2 (ln 𝐴 𝑖 ) 𝑃𝑖1/2 ))
(41)
𝑖=1
𝑛
≤ (𝛼𝑎 + 𝛽𝑏) 1𝐾 − 𝛾∑Φ𝑖 (𝑃𝑖1/2 𝐴 𝑖 𝑃𝑖1/2 ) . 𝑖=1
Proof. To prove the left-hand side of the inequality in (41) we use the functions 𝜑(𝑥) = ln 𝑥 and 𝜓(𝑥) = 𝑥−1 . Then 𝜓 ∘ 𝜑−1 (𝑥) = exp(−𝑥) and 𝜓−1 (𝑥) = 𝑥−1 , so 𝜓 is 𝜑-convex and 𝜓−1 (𝑥) = 𝑥−1 is operator decreasing. Applying Corollary 10 to this case, we have (42)
To prove the right-hand side we use the functions 𝜑(𝑥) = ln 𝑥 and 𝜓(𝑥) = 𝑥. Then 𝜓 ∘ 𝜑−1 (𝑥) = exp 𝑥 and 𝜓−1 (𝑥) = 𝑥, so 𝜓 is 𝜑-convex and 𝜓−1 (𝑥) = 𝑥−1 is operator increasing. Applying the inequality in (38), we get 𝑀ln 𝑥 (𝐴) ≤ 𝑀𝑥 (𝐴) .
The author declares that there is no conflict of interests regarding the publication of this paper.
(38)
If 𝜓 is either 𝜑-convex with operator decreasing 𝜓−1 or 𝜑concave with operator increasing 𝜓−1 , then one has the reverse inequality in (38).
𝑀ln 𝑥 (𝐴) ≥ 𝑀𝑥−1 (𝐴) .
Conflict of Interests
(43)
The double inequality in (41) follows by connecting the inequalities in (42) and (43). Quasi-arithmetic operator means without applying operator convexity were also investigated in [4, 10].
References [1] C. Davis, “A Schwarz inequality for convex operator functions,” Proceedings of the American Mathematical Society, vol. 8, pp. 42– 44, 1957. [2] M. D. Choi, “A Schwarz inequality for positive linear maps on C∗ -algebras,” Illinois Journal of Mathematics, vol. 18, pp. 565– 574, 1974. [3] F. Hansen, J. Peˇcari´c, and I. Peri´c, “Jensen’s operator inequality and its converses,” Mathematica Scandinavica, vol. 100, no. 1, pp. 61–73, 2007. [4] J. Mi´ci´c, Z. Pavi´c, and J. Peˇcari´c, “Jensen’s inequality for operators without operator convexity,” Linear Algebra and its Applications, vol. 434, no. 5, pp. 1228–1237, 2011. [5] Z. Pavi´c, “The applications of functional variants of Jensen’s inequality,” Journal of Function Spaces and Applications, vol. 2013, Article ID 194830, 5 pages, 2013. [6] A. M. Mercer, “A variant of Jensen’s inequality,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, article 73, 2 pages, 2003. [7] T. Furuta, J. Mi´ci´c Hot, J. Peˇcari´c, and Y. Seo, Mond-Peˇcari´c Method in Operator Inequalities, vol. 1, Element, Zagreb, Croatia, 2005. [8] A. Matkovi´c, J. Peˇcari´c, and I. Peri´c, “Refinements of Jensen’s inequality of Mercer’s type for operator convex functions,” Mathematical Inequalities & Applications, vol. 11, no. 1, pp. 113– 126, 2008. [9] J. E. Peˇcari´c, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol. 187, Academic Press, Boston, Mass, USA, 1992. [10] J. Mi´ci´c, Z. Pavi´c, and J. Peˇcari´c, “The inequalities for quasiarithmetic means,” Abstract and Applied Analysis, vol. 2012, Article ID 203145, 25 pages, 2012.
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