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Inequalities for Sector Matrices with Negative Power

In this paper, we present some inequalities for sector matrices with negative power. Among other results, we prove that if A,B ∈Mn(C) with W(A),W(B) ⊆ Sα, then for any positive unital linear map Φ, it holds <((1 − v)Φ(A) + vΦ(B))r ≤ cos2r(α)<Φ((1 − v)Ar + vBr), where v ∈ [0, 1] and r ∈ [−1, 0]. This improves Tan and Xie’s Theorem 2.4 in [22] if setting Φ(X) = X for every X ∈Mn(C) and replacing A by A−1, B by B−1, respectively, and r = −1, which is also a special result of Bedrani, Kittaneh and Sababheh’s Theorem 4.1 in [4].


Introduction
Let M n (C) denote the set of n × n complex matrices. For A ∈ M n (C), the conjugate transpose of A is denoted by A * , and the matrices A = 1 2 (A + A * ) and A = 1 2i (A − A * ) are called the real part and imaginary part of A, respectively ([6, p. 6] and [10, p. 7 ]). Recall that a norm · on M n (C) is unitarily invariant if UAV = A for any A ∈ M n (C) and all unitarily matrices U, V ∈ M n (C). A matrix A is called accretive if A is positive definite. For two Hermitian matrices A, B ∈ M n (C), we use A ≥ B to mean that A − B is positive semidifinite. A linear map Φ : M n → M n is called positive if it maps positive definite matrices to positive definite matrices and is said to be unital if it maps identity matrices to identity matrices.
The numerical range of A ∈ M n (C) is defined by For α ∈ [0, π 2 ), S α denotes the sector region in the complex plane as follows: S α = z ∈ C : z > 0, | z| ≤ ( z) tan α .
A matrix A ∈ M n (C) is called a sector matrix when satisfying W(A) ⊂ S α . If W(A) ⊆ S 0 , then A is positive definite, and if W(A), W(B) ⊆ S α for some α ∈ [0, π 2 ), then W(A + B) ⊆ S α . From the definition of S α above we know that if W(A) ⊆ S α , then A is nonsingular and (A) is positive definite. Moreover, W(A) ⊆ S α implies W(X * AX) ⊆ S α for any nonzero n × m matrix X, thus W(A −1 ) ⊆ S α . Recently, Tan and Chen [21] also proved that for any positive linear map Φ, W(A) ⊆ S α implies that W(Φ(A)) ⊆ S α . Recent developments on sector matrices can be found in [9,11,[13][14][15][16][17]21].
For two positive definite matrices A, B ∈ M n (C), the weighted geometric mean of A and B is defined as where 0 ≤ λ ≤ 1. For more information about the weighted geometric mean, we refer the reader to [12]. For two accretive matrices A, B ∈ M n (C), Drury [8] defined the geometric mean of A and B as follows This new geometric mean defined by (1) possesses some similar properties compared to the geometric mean of positive matrices. For instance, Later, Raissouli, Moslehian and Furuichi [20] defined the following weighted geometric mean of two accretive matrices A, B ∈ M n (C), where λ ∈ [0, 1]. If λ = 1 2 , then the formula (2) coincides with the formula (1). Very recently, Bedrani, Kittaneh and Sababheh [4] defined a more general operator mean for two accretive matrices A, B ∈ M n (C), where f : (0, ∞) → (0, ∞) is an operator monotone function with f (1) = 1 and v f is the probability measure characterizing σ f . Moreover, they also characterize the operator monotone function for an accretive matrix: let A ∈ M n (C) be accretive and f : (0, ∞) → (0, ∞) be an operator monotone function with f (1) = 1, then where v f is the probability measure satisfying f (x) Ando [1] proved that if A, B ∈ M n (C) are positive definite, then for any positive linear map Φ, where λ ∈ [0, 1]. Ando's formula (5) is known as a matrix Hölder inequality. The famous Choi's inequality [5, p. 41] says: if Φ is a positive unital linear map and A > 0, then Utilizing the weighted geometric mean defined by Raissouli, Moslehian and Furuichi for two accretive matrices, Tan and Xie [22] gave an AM-GM-HM inequality for sectorial matrices as follows.
Very recently, Bedrani, Kittaneh and Sababheh [4] utilized the newly defined operator mean to obtain a more general one compared to Tan and Xie.
Under Bedrani, Kittaneh and Sababheh's definitions, they generalize many inequalities from positive definite matrices to sector matrices for operator monotone functions. Thus it is interesting to find out whether one can replace operator monotone functions by operator convex functions or more generally, convex functions. In this paper, we will utilize operator convex function namely, f (t) = t r , r ∈ [−1, 0], instead of operator monotone functions to present some inequalities for sector matrices, which complements Bedrani, Kittaneh and Sababheh's results. In particular, we strengthen Theorem 1.1 in a new perspective.

Lemmas
We begin this section with some lemmas which will be necessary to prove our main results. Very recently, Choi, Tam and Zhang gave the following two results in [7], which will be utilized massively in our argument.
A reverse of Lemma 2.1 is as follows.
Very recently, Bedrani, Kittaneh and Sababheh [4] obtained the following inequality for general operator mean of sector matrices.

Main results
The first theorem presents a weighted arithmetic mean type inequality for sectorial matrices involving negative power, which also can be viewed as a convex function, leading to improvements of Theorem 1.1 and Theorem 1.2.
Proof. We have the following chain of inequalities completing the proof.
By setting Φ(X) = X for every X ∈ M n (C) and replacing A by A −1 , B by B −1 , respectively, and r = −1 in Theorem3.1, we obtain the following corollary, which improves Theorem 1.1. As pointed out by a referee, Corollary 3.2 could also be derived by Theorem 1.2 if setting σ f = ∇ v , where ∇ v denotes the weighted arithmetic mean.
The following two corollaries considerably refine Theorem 2.1 and Theorem 2.2 in [19]. Proof. In Cocollary3.2, setting v = 1 2 and replacing A by A −1 and B by B −1 , we have The rest of the proof follows from Theorem 2.1 in [19], thus we omit the details. Proof. First of all, by Lemma 2.1 we have ( A) r ≤ cos 2r (α) A r .
By the operator monotone decreasing of the inverse we have Hence we get By the operator monotone decreasing of the inverse again, we obtain Next we compute which completes the proof.
Proof. We prove inequality (9) first. Compute where the first inequality follows by Lemma 2.4, the second one is due to Theorem 3.1 and the last one is by Lemma 2.4 again. Now we prove inequality (10) as promised.
where the first inequality is by Lemma 2.4, the second one is by Theorem 3.5, the third one is due to Lemma 2.3 and the last one is by Lemma 2.4 again. This completes the proof.
Proof. We estimate where the first inequality follows by Lemma 2.5, the second one follows from Theorem 3.1 and the last one is by Lemma 2.5 again. Next we prove inequality (12).