Let $n$ be a positive integer greater than $2$. Let $X(1),X(2),\cdots,X(n)$ and $ Y(1),Y(2),\cdots,Y(n)$ be two strictly increasing sequences of n real numbers each, listed in order of increasing size. This question asked whether there always existed a polynomial $P(X)$ such that
$Y(m)=P(X(m))~~~~~~~~~~~~~~~~ \forall m=1,2,\cdots,n$
$P(X)$ is strictly increasing throughout the closed interval of real numbers whose end-points are $X(1)$ and $X(n)$.
There were numerous responses to this question which gave an affirmative answer and discussed a number of related results in this area.
I have been trying to prove whether the following additional constraints can also be imposed on the polynomial $P(X)$. Let $T$ be the triangle which is the convex hull of the points $(X(1),Y(1))$, $(X(1),Y(n))$ and $(X(n),Y(n))$. If all of the points $(X(m),Y(m))$ are in the interior of $T$ for each positive integer $m$ which is greater than $1$ and less than $n$, can the point $(X,P(X))$ always be in the interior of $T$ for each real number $X$ in the open interval whose end-points are $X(1)$ and $X(n)$?
My second question is: If the points $(X(m),Y(m))~~\forall m=1,2,\cdots,n$ are all points of $T$, can the graph of $P(X)$-in the closed interval of real numbers whose end-points are $X(1)$ and $X(n)$ always be a subset of $T$? I have been unable to find proofs or counter-examples to answer either of these questions.
The triangle $T$ becomes of interest when dealing with data of the following sort. The $X$ values represent fractions or percentages of a population, each member of which has some yearly income. If the total yearly income of all the members of the population is $Q$, then the $Y$ value corresponding to each $X$ value represents that fraction of $Q$ which is the total yearly income of the poorest $X$-percent of the population.