# reference request – Elementary functions (growing faster than exponential ones) with elementary Legendre–Fenchel transforms

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 $$c$$?