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In this article, we developed a series of new inequalities involving the $q$-numerical radius for operators and $2\times 2$ operator matrices. These inequalities serve to establish both lower and upper bounds for the $q$-numerical radius of operators. Additionally, we established $q$-numerical radius inequalities for operators via Buzano inequality., Comment: 23 pages

In this article, a series of new inequalities involving the $q$-numerical radius for $n\times n$ tridiagonal, and anti-tridiagonal operator matrices has been established. These inequalities serve to establish both lower and upper bounds for the $q$-numerical radius of operator matrices. Additionally, we developed $q$-numerical radius inequalities for $n\times n$ circulant, skew circulant, imaginary circulant, imaginary skew circulant operator matrices. Important examples have been used to illustrate the developed inequalities. In this regard, analytical expressions and a numerical algorithm have also been employed to obtain the $q$-numerical radii. We also provide a concluding section, which may lead to several new problems in this area., Comment: 23 pages, 9 figures

We develop a new refinement of the Kato's inequality and using this refinement we obtain several upper bounds for the numerical radius of a bounded linear operator as well as the product of operators, which improve the well known existing bounds. Further, we obtain a necessary and sufficient condition for the positivity of $2\times 2$ certain block matrices and using this condition we deduce an upper bound for the numerical radius involving a contraction operator. Furthermore, we study the Schatten $p$-norm inequalities for the sum of two $n\times n$ complex matrices via singular values and from the inequalities we obtain the $p$-numerical radius and the classical numerical radius bounds. We show that for every $p>0$, the $p$-numerical radius $w_p(\cdot): \mathcal{M}_n(\mathbb C)\to \mathbb R$ satisfies $ w_p(T) \leq \frac12 \sqrt{\left\| |T|^{2(1-t)}+|T^*|^{2(1-t)} \right\|^{} \, \big \||T|^{2t}+|T^*|^{2t} \big\|_{p/2}^{} } $ for all $t\in [0,1]$. Considering $p\to \infty$, we get a nice refinement of the well known classical numerical radius bound $w(T) \leq \sqrt{\frac12 \left\| T^*T+TT^* \right \|}.$ As an application of the Schatten $p$-norm inequalities we develop a bound for the energy of graph. We show that $ \mathcal{E}(G) \geq \frac{2m}{ \sqrt{ \max_{1\leq i \leq n} \left\{ \sum_{j, v_i \sim v_j}d_j\right\}} },$ where $\mathcal{E}(G)$ is the energy of a simple graph $G$ with $m$ edges and $n$ vertices $v_1,v_2,\ldots,v_n$ such that degree of $v_i$ is $d_i$ for each $i=1,2,\ldots,n.$, Comment: 19 pages. There is a typo in the abstract in V2, in this version we have corrected this

The main goal of this article is to establish several new $\mathbb{A}$-numerical radius equalities and inequalities for $n\times n$ cross-diagonal, left circulant, skew left circulant operator matrices, where $\mathbb{A}$ is the $n\times n$ diagonal operator matrix whose diagonal entries are positive bounded operator $A$. Also, we introduce two new matrices called left imaginary circulant operator matrix and left imaginary skew circulant operator matrix and present their $\mathbb{A}$-numerical radii. Certain $\mathbb{A}$-numerical radii of general $n\times n$ operator matrices are obtained. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Applications of our results are established through some interesting examples. We also provide a concluding section by posing a problem for future research., Comment: 35 pages

In this paper, we establish some upper bounds for Berezin number inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{cc} 0&X, Y&0 \end{array}\right]$, then \begin{align*} \textbf{ber}^{r}(T)\leq 2^{r-2}\left(\textbf{ber}(f^{2r}(|X|)+g^{2r}(|Y^*|))+\textbf{ber}(f^{2r}(|Y|)+g^{2r}(|X^*|))\right)\\ -2^{r-2} \inf_{\|(k_{\lambda_{1}},k_{\lambda_{2}})\|=1} \eta(k_{\lambda_{1}},k_{\lambda_{2}}), \end{align*} where $\eta (k_{\lambda_{1}}, k_{\lambda_{2}}) = \left(\left\langle(f^{2r}(|X|)+g^{2r}(|Y^*|)\right)k_{\lambda_{2}},k_{\lambda_{2}}\right\rangle^\frac{1}{2}-\left\langle \left(f^{2r}(|Y|)+g^{2r}(|X^*|)\right)k_{\lambda_{1}},k_{\lambda_{1}}\right\rangle^\frac{1}{2})^2$, $X, Y$ are bounded linear operators on a Hilbert space $\mathcal H=\mathcal H(\Omega)$, $r\geq 1$ and $f$, $g$ are nonnegative continuous functions on $[0, \infty)$ satisfying the relation $f(t)g(t)=t\,(t\in[0, \infty))$., Comment: ROCKY MOUNTAIN JOURNAL OF MATHEMATICS

An omitted value of a transcendental meromorphic function $f$ is called a Baker omitted value, in short \textit{bov} if there is a disk $D$ centered at the bov such that each component of the boundary of $f^{-1}(D)$ is bounded. Assuming that the bov is in the Fatou set of $f$, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If $U$ is the Fatou component containing the bov then it is proved that a Fatou component $U'$ is infinitely connected if and only if it lands on $U$, i.e. $f^{k}(U') \subset U$ for some $k \geq 1$. Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbits do not intersect $U$ is finite, we have shown that the connectivity of each Fatou component belongs to a finite set. This set is independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and compactly contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent whenever the bov is in the Fatou set. It is also proved that, if there is a $2$-periodic Baker domain (these are not ruled out when the bov is in the Julia set), or a $2$-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components., Comment: 21 pages, 7 figures

The main goal of this article is to establish several new upper and lower bounds for the $\mathbb{A}$-numerical radius of $2\times 2$ operator matrices, where $\mathbb{A}$ be the $2\times 2$ diagonal operator matrix whose diagonal entries are positive bounded operator $A$., Comment: 18 pages

Let $\mathbb{A}= \begin{pmatrix} A & 0 \\ 0 & A \\ \end{pmatrix} $ be a $2\times2$ diagonal operator matrix whose each diagonal entry is a bounded positive (semidefinite) linear operator $A$ acting on a complex Hilbert space $\mathcal{H}$. In this paper, we derive several $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator $A$ on $\mathcal{H}$. Some applications of our inequalities are also given.

Let ($\mathcal{H}, \langle . , .\rangle )$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $\|T\|_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(\mathcal{H}, \langle .,.\rangle_A),$ where $\langle x, y\rangle_A:=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2\times 2$ operator matrices, where $B=\begin{bmatrix} A & 0 0 & A \end{bmatrix}$. Further, we prove some refinements of earlier $A$-numerical radius inequalities for operators., Comment: 16 pages