Inequalities for absolute value operators related to skew p-angular distance

In this paper we show some operator inequalities for absolute value operators. More precisely, we show that if $A,B\in B(H)$ such that $\lvert A\rvert$ and $\lvert B\rvert$ are invertible, and $p, r, s\in\mathbb{R}$ where $r, s 1$ with $\frac{1}{r}+\frac{1}{s}=1$, then \begin{equation*} \left\lvert A\lvert B\rvert^{p-1}-B\lvert A\rvert^{p-1}\right\rvert^{2}\leq r\lvert A\rvert^{p-1}\lvert A-B\rvert^{2}\lvert A\rvert^{p-1}+s\lvert B\rvert^{p-1}\left\lvert\lvert A\rvert-\lvert A\rvert^{p}\lvert B\rvert^{1-p}\right\rvert^{2}\lvert B\rvert^{p-1}. \end{equation*} In addition, we obtain some equivalent equality conditions, when the case of equality holds.