On the $L_{\mathrm{YJ}}(\xi, \eta, X)$ constant for the Bana\'s-Fr\k{a}czek space

TL;DR

This paper calculates the exact value of a specific constant, $L_{YJ}(\xi, \eta, X)$, for Banaś-Frączek spaces, providing precise formulas through detailed computation.

Contribution

It provides the exact value of the $L_{YJ}(\xi, \eta, X)$ constant for Banaś-Frączek spaces, a result not previously established.

Findings

Exact formula for $L_{YJ}(\xi, \eta, R_\lambda^2)$ derived.

Demonstrates detailed computation techniques for Banach space constants.

Clarifies the dependence of the constant on parameters $\xi$, $\eta$, and $\lambda$.

Abstract

In this paper, for any $\lambda \geq 1, R_\lambda^2$ is the Bana\'s-Fr\k{a}czek space. The exact value of $L_{\mathrm{YJ}}(\xi, \eta, X)$ for this space will be calculated. Specifically, $L_{\mathrm{YJ}}\left(\xi, \eta, R_\lambda^2\right)=1+\frac{2 \xi \eta}{\xi^2+\eta^2}\left(1-\frac{1}{\lambda^2}\right)$ is the result thereafter through meticilous computation.