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Let $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$ be a complexHilbert space and $A$ be a positive (semidefinite) bounded linear operator on$\mathcal{H}$. The semi-inner product induced by $A$ is given by ${\langlex\mid y\rangle}_A := \langle Ax\mid y\rangle$, $x, y\in\mathcal{H}$ and definesa seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. This makes $\mathcal{H}$ into asemi-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|{\langleSx\mid x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove severalbounds involving $\omega_{A,\text{e}}(T,S)$. Moreover, several inequalitiesrelated to the $A$-Davis-Wielandt radius of semi-Hilbert space operators isestablished. Some of the obtained bounds generalize and refine some earlierresults of Zamani and Shebrawi [Mediterr. J. Math. 17, 25 (2020)].

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