On geometrical characteristics and inequalities of new bicomplex Lebesgue Spaces with hyperbolic-valued norm

Abstract

In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic ( 𝔻 {\mathbb{D}} -valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -convexity, 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -strict convexity, and 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -uniform convexity. Moreover, the basic inequalities such as 𝔻 {\mathbb{D}} -Hölder’s inequality and 𝔻 {\mathbb{D}} -Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties.