Repeated integrals of increasing functions

Abstract

Motivated by a problem on comonotone approximation of $C^n$ functions by entire functions, for increasing functions $f\colon[0,1]\to[0,1]$, we characterize the possible values of $(a,b,c)$, where $a=I(f)(1)$, $b=I^2(f)(1)$, $c=I^3(f)(1)$ ($I$ is the integral operator $I(f)(x)=\int_0^xf(t)\,dt$), as those which satisfy the conditions $0\leq a\leq 1$, $a^2/2\leq b\leq a/2$, $2b^2\leq 3ac$, $a^2 + 4b^2 + 6c\leq 6ac +2ab+2b$, and $0\leq c\leq a/6$. Our main theorem states that if $a,b,c$ are real numbers for which the inequalities are strict, then there is a function $f$ satisfying $a=I(f)(1)$, $b=I^2(f)(1)$, $c=I^3(f)(1)$ which is $C^\infty$ with $f(0)=0$, $f(1)=1$, $Df(x)>0$ for $0<x<1$, and whose derivatives $D^jf(0)$ and $D^jf(1)$, $j\geq 1$, are arbitrary as long as they are consistent with the increasing nature of $f$. The construction of $f$ proceeds by starting with a continuous parametrization $s\mapsto \rho_s\in C^\infty([0,1])$ defined on an open subset of $\mathbb{R}^4$, and composing with successive continuous transversals through the open set to fix the values of $I^j(\rho_s)(1)$ for $j=0,1,2,3$. Addressing the aforementioned problem on comonotone approximation, we examine the set $V_n\subseteq\mathbb{R}^{2(n+1)}$ of possible values $D^jf(0)$, $D^jf(1)$, $j=0,\dots,n$, of the derivatives of a $C^n$ function at the endpoints when $D^nf$ is increasing but not constant. We make a conjecture about the nature of this set and prove our conjecture for $n\leq 3$ as a consequence of the theorem mentioned above.