On the Generalized Gap Identity for the Bergström Inequality

In this paper, we revisit the generalized gap identity for the Bergström inequality established by T. O. Pop in 2009. Our first contribution is methodological: we present a self-contained proof using explicit summation notation to clarify all algebraic details. Furthermore, we demonstrate the versatility of this identity by applying it to both classical and multivariable cyclic inequalities, explicitly quantifying their respective gap terms. Finally, we reformulate the core identity to incorporate auxiliary variables.