Let A be a positive (semi-definite) bounded linear operator on a complex Hilbert space (H,⟨·,·⟩)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathcal {H}},\langle \cdot ,\cdot \rangle )$$\end{document}. Let ωA(T)\documentclass[12pt]{minimal}
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\begin{document}$$\omega _A(T)$$\end{document} and ‖T‖A\documentclass[12pt]{minimal}
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\begin{document}$$\Vert T\Vert _A$$\end{document} denote the A-numerical radius and the A-operator seminorm of an operator T acting on the semi-Hilbert space (H,⟨·,·⟩A)\documentclass[12pt]{minimal}
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\begin{document}$$({\mathcal {H}},\langle \cdot ,\cdot \rangle _A)$$\end{document} respectively, where ⟨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}. It is well-known that 14‖T♯AT+TT♯A‖A≤ωA2T≤12‖T♯AT+TT♯A‖A,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{1}{4}\Vert T^{\sharp _A} T+TT^{\sharp _A}\Vert _A\le \omega _A^2\left( T\right) \le \frac{1}{2}\Vert T^{\sharp _A} T+TT^{\sharp _A}\Vert _A, \end{aligned}$$\end{document}where T♯A\documentclass[12pt]{minimal}
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\begin{document}$$T^{\sharp _A}$$\end{document} denotes a distinguished A-adjoint operator of T. In this paper, we aim to give some new refinements of the above inequalities. Furthermore, we establish an A\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {A}}$$\end{document}-seminorm inequality involving 2×2\documentclass[12pt]{minimal}
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\begin{document}$$2\times 2$$\end{document} operator matrices, where A=diag(A,A)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {A}}=\text {diag}(A,A)$$\end{document}. This generalizes a recent result of Bani-Domi and Kittaneh. As an application, a refinement of the triangle inequality related to ‖·‖A\documentclass[12pt]{minimal}
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\begin{document}$$\Vert \cdot \Vert _A$$\end{document} is given.
