Let A be a positive (semidefinite) bounded linear operator on a complex Hilbert space (H,⟨·,·⟩)\documentclass[12pt]{minimal}
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\begin{document}$$\big ({\mathcal {H}}, \langle \cdot , \cdot \rangle \big )$$\end{document}. The semi-inner product induced by A is defined by ⟨x,y⟩A:=⟨Ax,y⟩\documentclass[12pt]{minimal}
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\begin{document}$${\langle x, y\rangle }_A := \langle Ax, y\rangle $$\end{document} for all x,y∈H\documentclass[12pt]{minimal}
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\begin{document}$$x, y\in {\mathcal {H}}$$\end{document} and defines a seminorm ‖·‖A\documentclass[12pt]{minimal}
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\begin{document}$${\Vert \cdot \Vert }_A$$\end{document} on H\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}$$\end{document}. This makes H\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}$$\end{document} into a semi-Hilbert space. For p∈[1,+∞)\documentclass[12pt]{minimal}
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\begin{document}$$p\in [1,+\infty )$$\end{document}, the generalized A-joint numerical radius of a d-tuple of operators T=(T1,…,Td)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {T}}=(T_1,\ldots ,T_d)$$\end{document} is given by ωA,p(T)=sup‖x‖A=1∑k=1d|〈Tkx,x〉A|p1p.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \omega _{A,p}({\mathbf {T}}) =\sup _{\Vert x\Vert _A=1}\left( \sum _{k=1}^d|\big \langle T_kx, x\big \rangle _A|^p\right) ^{\frac{1}{p}}. \end{aligned}$$\end{document}Our aim in this paper is to establish several bounds involving ωA,p(·)\documentclass[12pt]{minimal}
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\begin{document}$$\omega _{A,p}(\cdot )$$\end{document}. In particular, under suitable conditions on the operators tuple T\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {T}}$$\end{document}, we generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73–80, 2005).
