  Based on a suitable improvement of a triangle inequality, we derive new
mutual bounds for $p$-angular distance $\alpha_p[x,y]=\big\Vert \Vert
x\Vert^{p-1}x- \Vert y\Vert^{p-1}y\big\Vert$, in a normed linear space $X$. We
show that our estimates are more accurate than the previously known upper
bounds established by Dragomir, Hile and Maligranda. Next, we give several
characterizations of inner product spaces with regard to the $p$-angular
distance. In particular, we prove that if $|p|\geq |q|$, $p\neq q$, then $X$ is
an inner product space if and only if for every $x,y\in X\setminus \{0\}$,
$${\alpha_p[x,y]}\geq \frac{{\|x\|^{p}+\|y\|^{p} }}{\|x\|^{q}+\|y\|^{q}
}\alpha_q[x,y].$$
