arXiv subjects

Kallal Pal

Publications and source records attributed to Kallal Pal.

2 recordsLinked to original sources

arXiv2025

Some inequalities related to Heinz mean constant with Birkhoff orthogonality

Motivated by the work of Baronti et al. [J. Math. Anal. Appl. 252(2000) 124-146], where they defined the supremum of an arithmetic mean of the side lengths of a triangle, summing antipodal points on the unit sphere, we introduce a new geometric constant for Banach spaces, utilizing the Heinz means that interpolate between the geometric and arithmetic means associated with Birkhoff orthogonality. We discuss the bounds in Banach spaces and find the values of constant in Hilbert spaces. We obtain the characterization of uniformly non-square spaces. We investigate the correlation between our notion of the Heinz mean constant and other well-known terms, viz., the modulus of convexity, modulus of smoothness, and rectangular constant. Furthermore, we also give a characterization of the Radon plane with an affine regular hexagonal unit sphere.

math.FA↗

arXiv2023

Geometric properties of a novel type of orthogonality via norm derivatives

In this article, we generalize the notion of orthogonality as a linear combination of norm derivatives in order to give a novel concept that we refer to as $ρ_{α,β}$-orthogonality. Also, we discuss some of its geometric properties in a real normed linear space and present some sufficient criteria for the smoothness of a normed space by using $ρ_{α,β}$-orthogonality. We provide a few examples to show that the $ρ_{α,β}$- orthogonality cannot be compared to other well-known orthogonalities in any way. In addition to this, we offer a characterization of inner product spaces by making use of the functional notation $ρ_{α,β}$. In addition, we show that any $ρ_{α,β}$-orthogonality that preserves linear mapping between two normed linear spaces must necessarily be a scalar multiple of an isometry. Also, using the $ρ_{α,β}$-functional, we define the idea of an angle between two vectors and talk about their characteristics in normed spaces.

math.FA↗