New Inequalities for the p-Angular Distance in Normed Spaces with Applications - Dragomir, S. S. (Ukrains’kyi Matematychnyi Zhurnal, 2015)
For nonzero vectors x and y in the normed linear space (X, ‖ ⋅ ‖), we can define the p-angular distance by α_p[x,y]:=x^p-1x-y^p-1y. We show (among other results) that, for p ≥ 2, arraylα_p[x,y]py-x01(1-t)x+ty^p-1dt3.36em py-x[x^p-1+y^p-1 2+x+y2^p-1]3.36em py-xx^p-1+y^p-1 2py-x[ {x, y}]^p-1,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.