Analytic formula for Huber-like function $f(x) := \sup_{|y| \le r}xy - \frac{1}{p}|y|^p$, where $p \ge 1$ and $r>0$ are fixedreal

Question

Let $x \in \mathbb R$, $r>0$, and $p \ge 1$.

Question. What is the analytic formula for $f_p(x;r) := \sup_{|y| \le r}xy - \frac{1}{p}|y|^p$ ?

N.B. I'm particularly interested in the case $p \in [1,2)$.

Examples

The case $p = 2$ is well-known $$ f_2(x;r) = \sup_{|y| \le r}xy - \frac{1}{2}y^2 = \begin{cases} x^2/2,&\mbox{ if }|x| \le r,\\ r |x| - r^2/2,&\mbox{ else.} \end{cases} $$

The case $p = 1$ writes $$ \begin{split} f_1(x;r) &= \sup_{|y| \le r}xy - |t| = \sup_{|y| \le r}xy-\sup_{|z| \le 1}xz = \inf_{|z| \le 1}\sup_{|y| \le r}y(x-z)\\ & = r\inf_{|z| \le 1}|x-z| = r(|x|-1)_+. \end{split} $$