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for any ,  ∈ . This improves a result of Maligranda from [“Simple norm inequalities,” ., , 256–260 (2006)] who proved the inequality between the first and last terms in the estimation presented above. The applications to functions  defined by power series in estimating a more general “distance” ‖(‖‖) − (‖‖)‖ for some ,  ∈  are also presented.

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