Conjugate Convex Functions without Infinity

Let $B_{r}(E)$ be the closed ball of radius $r$ around the origin in a real Banach space $E$ and $\mathcal{F}_{r}(E)$ be the set of all $r$-Lipschitz continuous convex functions defined on $B_{r}(E)$. Suppose $f$ is a real-valued and bounded below function on $B_{r}(E)$. We define the $I$-conjugate function $f^{I}$ of $f$ to improve the Fenchel inequality and investigate the properties of $f^{I}$. In particular, $(f^{I})^{I}$ coincides with $f$ on $B_{r}(E)$ if and only if $f$ is in $\mathcal{F}_{r}(E)$. Excluding the value $+\infty$, the transformation from $f$ to $f^{I}$ enlarges the potentiality of the contribution to numerical computation for convex analysis.