This present work aims to ameliorate the celebrated Cauchy-Schwarz inequality and provide several new consequences associated with the numerical radius upper bounds of Hilbert space operators. More precisely, for arbitrary a, b ? H and ? ? 0, we show that |?a,b?|2 ? 1 ? + 1 ?a??b?|?a, b?| + ?/?+1 ?a?2?b?2 ? ?a?2?b?2. As a consequence, we provide several new upper bounds for the numerical radius that refine and generalize some of Kittaneh?s results in [A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math. 2003;158:11-17] and [Cauchy-Schwarz type inequalities and applications to numerical radius inequalities. Math. Inequal. Appl. 2020;23:1117-1125], respectively. In particular, for arbitrary A, B ? B(H) and ? ? 0, we show the following sharp upper bound w2 (B*A) ? 1/2?+2 ?|A|2 + B|2?w(B*A)+ ?/2?+2 ?|A|4 + |B?4, with equality holds when A=B= (0100). It is also worth mentioning here that some specific values of ? ? 0 provide more accurate estimates for the numerical radius. Finally, some related upper bounds are also provided.

Let A_i ∈ B(H), (i = 1, 2, ..., n), and T = [ 0 ⋯ 0 A 1 ⋮ ⋰ A 2 0 0 ⋰ ⋰ ⋮ A n 0 ⋯ 0 ] $ T = \left[ {\matrix{ 0 & \cdots & 0 & {A_1 } \cr \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {A_2 } & 0 \cr 0 & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots \cr {A_n } & 0 & \cdots & 0 \cr } } \right] $ . In this paper, we present some upper bounds and lower bounds for w(T). At the end of this paper we drive a new bound for the zeros of polynomials.

In this paper, we establish some numerical radius inequalities for bounded linear operator defined on a complex Hilbert space. As a natural application, the existence of the all polynomial zeros is identified in a specific small disk. Moreover, we provide a refinement of an earlier numerical radius inequality due to Herzallah et al. [Numerical radius inequalities for certain operator matrices. Integr Equ Oper Theory. 2011;71:129-147] and a generalization of Shebrawi's inequality [Numerical radius inequalities for certain operator matrices II. Linear Algebra Appl. 2017;523:1-12].

Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce the concept of graded classical primary submodules. Various properties of graded classical primary submodules are considered.

In this paper, we will present several upper bounds for the numerical radii of a operator matrices. We use these bounds to generalize and improve some well-known numerical radius inequalities. We provide a refinement of an earlier numerical radius inequality due to (Bani-Domi & Kittaneh, 2021) [Norm and numerical radius inequalities for Hilbert space operators], (Bani-Domi & Kittaneh, 2021) [Refined and generalized numerical radius inequalities for operator matrices] and (Al-Dolat & Kittaneh, 2023) [Upper bounds for the numerical radii of powers of Hilbert space operators].