New Inequalities for the $p$-Angular Distance in Normed Spaces with Applications

Abstract

For nonzero vectors $x$ and $y$ in the normed linear space $(X, ‖ ⋅ ‖)$, we can define the $p$-angular distance by $${\alpha}_p\left[x,y\right]:=\left\Vert {\left\Vert x\right\Vert}^{p-1}x-{\left\Vert y\right\Vert}^{p-1}y\right\Vert .$$ We show (among other results) that, for $p ≥ 2$, $$\begin{array}{l}{\alpha}_p\left[x,y\right]\le p\left\Vert y-x\right\Vert {\displaystyle \underset{0}{\overset{1}{\int }}{\left\Vert \left(1-t\right)x+ty\right\Vert}^{p-1}dt}\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \left[\frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}+{\left\Vert \frac{x+y}{2}\right\Vert}^{p-1}\right]\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}\le p\left\Vert y-x\right\Vert {\left[ \max \left\{\left\Vert x\right\Vert, \left\Vert y\right\Vert \right\}\right]}^{p-1},\hfill \end{array}$$, for any $x, y ∈ X$. This improves a result of Maligranda from [“Simple norm inequalities,” Amer. Math. Month., 113, 256–260 (2006)] who proved the inequality between the first and last terms in the estimation presented above. The applications to functions f defined by power series in estimating a more general “distance” $‖f(‖x‖)x − f(‖y‖)y‖$ for some $x, y ∈ X$ are also presented.