UPPER AND LOWER BOUNDS FOR THE p–ANGULAR DISTANCE IN NORMED SPACES WITH APPLICATIONS
نویسنده
چکیده
For nonzero vectors x and y in the normed linear space (X ,‖·‖) we can define the p -angular distance by αp [x,y] := ∥∥‖x‖p−1 x−‖y‖p−1 y ∥∥ . In this paper we show among others that 1 2 ∣∣ ∣∣‖x‖p−1−‖y‖p−1 ∣∣‖x+ y‖− ( ‖x‖p−1 +‖y‖p−1 ) ‖x− y‖ ∣∣ αp [x,y] 1 2 ∣∣‖x‖p−1 −‖y‖p−1 ∣∣‖x+ y‖+ ( ‖x‖p−1 +‖y‖p−1 ) ‖x− y‖ ] for any p ∈ R and for any nonzero x,y ∈ X . Some reverses of the triangle and the continuity of the norm inequalities are given as well. Applications for functions f defined by power series in estimating the more general “distance” ‖ f (‖x‖)x− f (‖y‖)y‖ for certain x,y ∈ X are also provided. Mathematics subject classification (2010): Primary 46B20, 46B99; Secondary 51M16.
منابع مشابه
INEQUALITIES FOR THE p-ANGULAR DISTANCE IN NORMED LINEAR SPACES
New upper and lower bounds for the p-angular distance in normed linear spaces are given. Some of the obtained upper bounds are better than the corresponding results due to L. Maligranda recently established in the paper [Simple norm inequalities, Amer. Math. Monthly, 113(2006), 256-260].
متن کاملEstimating Upper and Lower Bounds For Industry Efficiency With Unknown Technology
With a brief review of the studies on the industry in Data Envelopment Analysis (DEA) framework, the present paper proposes inner and outer technologies when only some basic information is available about the technology. Furthermore, applying Linear Programming techniques, it also determines lower and upper bounds for directional distance function (DDF) measure, overall and allocative efficienc...
متن کاملTwo Equivalent Presentations for the Norm of Weighted Spaces of Holomorphic Functions on the Upper Half-plane
Introduction In this paper, we intend to show that without any certain growth condition on the weight function, we always able to present a weighted sup-norm on the upper half plane in terms of weighted sup-norm on the unit disc and supremum of holomorphic functions on the certain lines in the upper half plane. Material and methods We use a certain transform between the unit dick and the uppe...
متن کاملOn Some Inequalities in Normed Algebras
Some inequalities in normed algebras that provides lower and upper bounds for the norm of ∑n j=1 ajxj are obtained. Applications for estimating the quantities ∥∥∥∥x−1∥∥x± ∥∥y−1∥∥ y∥∥ and ∥∥∥∥y−1∥∥x± ∥∥x−1∥∥ y∥∥ for invertible elements x, y in unital normed algebras are also given.
متن کامل