More inequalities on numerical radii of sectorial matrices

Let Mn(C) denote the set of n × n complex matrices. For A ∈ Mn(C), the conjugate transpose of A is denoted by A∗, and the matrices <A = 2 (A + A∗) and =A = 1 2i (A − A∗) are called the real part and imaginary part of A, respectively ( [6, p. 6] and [12, p. 7 ]), Moreover, A is called accretive if<A > 0. For two Hermitian matrices A, B ∈ Mn(C), we write A ≥ B (or B ≤ A) if A − B is positive semidifinite. A linear map Φ : Mn(C) → Mk(C) 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 ∈ Mn(C) is defined by


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 [12, p. 7 ]), Moreover, A is called accretive if A > 0. For two Hermitian matrices A, B ∈ M n (C), we write A ≥ B (or B ≤ A) if A − B is positive semidifinite. A linear map Φ : M n (C) → M k (C) 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 while the operator norm of A is defined by A = max {| Ax, y | : x, y ∈ C n , x = y = 1} .
For two positive definite matrices A, B ∈ M n (C) and 0 ≤ λ ≤ 1, the weighted geometric mean is defined by , and the weighted harmonic mean is defined by In particular, when λ = 1 2 , we denote the geometric mean, harmonic mean and arithmetic mean by A B, A!B and A∇B, respectively. When λ [0, 1], we still define A λ B as above, which is then not needed to be a matrix mean.
For two accretive matrices A, B ∈ M n (C), Drury [9] defined the geometric mean of A and B as follows This new geometric mean defined by (1.2) 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 (1.3) coincides with the formula (1.2). Very recently, Bedrani, Kittaneh and Sababheh [2] 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 . For more information about operator mean, more generally, operator monotone functions that preserve the ordering of real parts of operators, we refer the readers to the recent work of Gaál and Pálfia [11].
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, where v f is the probability measure satisfying f (x) Recently, Mao et al. [19] defined the Heinz mean for two sector matrices A, B ∈ M n (C) with Ando [1] proved that if A, B ∈ M n (C) are positive definite, then for any positive linear map Φ, Ando's formula (1.6) is known as a matrix Hölder inequality.
To reduce the brackets, we denote (Φ(A)) t by Φ t (A) throughout this paper. The famous Choi's inequality [5, p. 41] says: if Φ is a positive unital linear map and A > 0, then For the sake of convenience, we shall need the following notation.
In a recent paper [3], Bedrani, Kittaneh and Sababheh studied the numerical radius inequalities of sectorial matrices. They [3] obtained relation between ω −t (A) and ω(A −t ) as follows.
They also [3] gave the Heinz-type inequality for the numerical radii of sectorial matrices below.
In this paper, we intend to improve upon the bounds of Theorem 1.1 and 1.2. Furthermore, we shall present some numerical radius inequalities of sectorial matrices involving positive linear maps.

Main results
We begin this section with some lemmas which will be necessary for proving our main results. (2.1) The following lemma gives a better bound of (2.1).
The famous Löwner-Heinz inequality says that if A, B ∈ M n (C) are such that A ≥ B ≥ 0 and t ∈ [0, 1], then A t ≥ B t . Inspired by Lemma 2.2, a sectorial matrix version is as follows: Next we present a reverse of Lemma 2.2.
Lately, Bedrani, Kittaneh and Sababheh [2] obtained the following inequality for general operator mean of sectorial matrices.
Now we are ready to give our first main result.
Theorem 2.9. Let A i ∈ M n (C) be such that W(A i ) ⊆ S α , i = 1, 2 · · · , k, and a 1 , · · · , a k be positive real numbers with k j=1 a j = 1. Then for t ∈ [−1, 0], which completes the proof. Proof. The result directly derived from Theorem 2.9 by substituting k = 1. We remark that Corollary 2.10 is a refinement of Theorem 1.1.
Corollary 2.11. Let A ∈ M n (C) be such that W(A) ⊆ S α . Then Proof. The result is directly derived from Corollary 2.10 by substituting t = −1.
Thanks to Corollary 2.11, considerable refinements of Theorem 3.6 and 3.12 in [4] are given below.
Corollary 2.12. Let A ∈ M n (C) be such that W(A) ⊆ S α and B > 0. Then for t ∈ (1, 2), Proof. The result directly derived from Theorem 3.6 in [4] and Corollary 2.11.
Corollary 2.13. Let B ∈ M n (C) be such that W(B) ⊆ S α and A > 0. Then for t ∈ (−1, 0), Proof. The result directly derived from Theorem 3.12 in [4] and Corollary 2.11. Next we give a complement of Theorem 2.9.
Theorem 2.14. Let A i ∈ M n (C) be such that W(A i ) ⊆ S α , i = 1, 2 · · · , k, and a 1 , · · · , a k be positive real numbers with k j=1 a j = 1.
completing the proof.
Proof. We have the following chain of inequalities which completes the proof.

Proof.
Compute This completes the proof.
The following result presents a reverse of Theorem 2.16.
Proof. We estimate which proves the first inequality. To prove the second inequality, compute cos 2t (α) cos 3 where the first inequality is obtained by the preceding proof, the second one is by Lemma 2.6 and the last one is due to Lemma 2.5. This completes the proof.
where the first inequality is obtained by the preceding proof, the second one is by Lemma 2.6 and the last one is due to Lemma 2.5. This completes the proof. We remark that (2.4) coincides with Theorem 3.7 in [3] when setting Φ(X) = X for every X ∈ M n (C). which completes the proof. We remark that Theorem 2.19 is an improvement of Theorem 1.2. Consider a partitioned matirx A ∈ M n (C) in the form where A 11 and A 22 are square matrices. If A 11 is invertible, we denote the Schur complement of Whenever we mention S (B), we assume B ∈ M n (C) has the partition mentioned above and the relevant inverse exists.
The following corollary is a complement of Proposition 3.3 in [3]. Proof. Let Φ(X) = X for every X ∈ M n (C) in Theorem 2.22, we get the desired result.