Inequalities for the $A$-joint numerical radius of two operators and their applications

Let $(\mathcal{H},⟨\cdot \mid \cdot ⟩)$ be a complex Hilbert space and $A$ be a positive (semidefinite) bounded linear operator on $\mathcal{H}$. The semi-inner product induced by $A$ is given by ${\langle x\mid y\rangle}A := \langle Ax\mid y\rangle$,$x, y\in$\mathcal{H}$$ and defines a seminorm ${\mathrm{\parallel }\cdot \mathrm{\parallel }}_A$ on $\mathcal{H}$. This makes $\mathcal{H}$ into a semi-Hilbert space. The $A$-joint numerical radius of two $A$-bounded operators $T$ and $S$ is given by \begin{align*} \omega{A,$\text{e}$}(T,S) = \sup_{|x|_A= 1}\sqrt{\big|{\langle Tx\mid x\rangle}_A\big|^2+\big|{\langle Sx\mid x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove several bounds involving ${\omega }_{A,\text{e}}(T,S)$. Moreover, several inequalities related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators is established. Some of the obtained bounds generalize and refine some earlier results of Zamani and Shebrawi [Mediterr. J. Math. 17, 25 (2020)].