New inequalities for the $\mathbf{A}$-joint weighted numerical radius with applications

Let $(\mathcal{K},⟨\cdot ,\cdot ⟩)$ be a complex Hilbert space. Let $\mathbf{A}$ be a positive (semidefinite) bounded linear operator on $\mathcal{K}$. We introduce the Euclidean-arithmetic mean $\mathbf{A}$-numerical radius for a pair of $\mathbf{A}$-bounded operators $(\mathbf{U},\mathbf{V})$, defined by ${\omega }_{\mathbf{A},e,\epsilon }(\mathbf{U},\mathbf{V}):=\underset{\eta \in {\mathrm{\Sigma }}_1^{\mathbf{A}}}{sup}{\left((1-\epsilon )|⟨\mathbf{U}\eta ,\eta {⟩}_{\mathbf{A}}{|}^2+\epsilon |⟨\mathbf{V}\eta ,\eta {⟩}_{\mathbf{A}}{|}^2\right)}^{\frac{1}{2}},$ where $\epsilon \in [0,1]$, $⟨{\eta }_1,{\eta }_2{⟩}_{\mathbf{A}}=⟨\mathbf{A}{\eta }_1,{\eta }_2⟩$ for all ${\eta }_1,{\eta }_2\in \mathcal{K}$, and ${\mathrm{\Sigma }}_1^{\mathbf{A}}$ denotes the unit $\mathbf{A}$-sphere. Furthermore, we define the integral Euclidean numerical radius and present several Hermite-Hadamard-type inequalities. Finally, we provide significant applications of our framework by deriving bounds for the Cartesian decomposition of operators and generalizing the Davis-Wielandt radius to a $p$-arithmetic mean context with $p\ge 1$.