The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval.
Let $F\colon[0,1]\to[0,1]$ be a nondecreasing function. For $t\in[0,1]$, let $$G(t):=\int_t^1 F(x)\,dx.\tag{1}$$ Take any $a,b,c$ such that $0\le a<b<c<1$. Find the best upper and lower bounds on $G(b)$ in terms of $a,b,c,G(a),G(c)$.
I think the answer to this question is not altogether trivial, and it might be useful in some research in probability. Indeed, changing, if necessary, values of the function $F$ on an at most countable set and extending the resulting function to $\mathbb R$, without loss of generality we may assume that $F$ is the cumulative distribution function (cdf) of a random variable (r.v.) $X$ with values in $[0,1]$, and then for all $t\in[0,1]$ $$G(t)=\int_t^1 F(x)\,dx=\int_t^1 P(X\le x)\,dx=\int_t^1E1(X\le x)\,dx=E\int_t^1 1(X\le x)\,dx=E\int_0^1 dx\,1(x\ge t)1(X\le x)=E\int_0^1 dx\,1(x\ge t\vee X)=E(1-t\vee X)=1-E(t\vee X),$$ where $u\vee v:=\max(u,v)$. So, the question can be restated as the one about the best upper and lower bounds on $H(b)$ in terms of $a,b,c,H(a),H(c)$, where $H(t):=E(t\vee X)$.
Therefore, an answer will be given below.