Let $F$ be the set of all convex functions $fcolon(0,infty)to(0,infty)$ with $f(0)=0=f’_+(0)$ and $f_+(infty-)=infty$, where $f’_+$ is the right derivative of $f$. For any function $fin F$, its Legendre–Fenchel transform $g_fcolon(0,infty)to(0,infty)$ (also known as the convex conjugate of $f$) is defined by the formula

$$g_f(y):=sup_{xge0}(xy-f(x))$$

for real $yge0$.

E.g., if $f(x)=x^p/p$ for a real $p>1$ and all real $xge0$, then $fin F$ and $g_f(y)=y^q/q$ for $q:=1/(1-1/p)$ and all real $yge0$.

A couple of other pairs $(f,g_f)$ of “explicit” functions with $fin F$ can be obtained from this table, including the one with $f(x)=e^x-1-x$ for all real $xge0$ and $g_f(y)=(1+y)ln(1+y)-y$ for all real $yge0$.

Are there any pairs $(f,g_f)$ of “explicit” (say elementary, in some sense) functions with $fin F$ such that $f$ increases faster than any exponential function: $f(x)/e^{cx}underset{xtoinfty}longrightarrowinfty$ for any real $c$?