For nonzero vectors x and y in the normed linear space (X, ‖ ⋅ ‖), we can define the p-angular distance by
We show (among other results) that, for p ≥ 2,
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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 1, pp. 19–31, January, 2015.
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Dragomir, S.S. New Inequalities for the p-Angular Distance in Normed Spaces with Applications. Ukr Math J 67, 19–32 (2015). https://doi.org/10.1007/s11253-015-1062-8
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DOI: https://doi.org/10.1007/s11253-015-1062-8