Bounds for the p-Angular Distance and Characterizations of Inner Product Spaces

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