Assume to have a smooth (or at least $C^2$) and convex real function $f$ such that:
$f(x)>0$ for $x\in (a,b)$
$f'(x)>0$ for $x\in (a,b)$
$f''(x)>0$ for $x\in (a,b)$
Now, fix an interval $[x_1,x_2] \subset(a,b)$, namely $a<x_1<x_2<b$. We are given a set of 6 numbers: $(f_1,f'_1,f''_1,f_2,f'_2,f''_2)$, representing the values of $f$ and its first and second derivatives at $x_1$ and $x_2$.
Question: Under what conditions can we conclude that the 6 numbers $(f_1,f'_1,f''_1,f_2,f'_2,f''_2)$ are inconsistent with the three properties of $f$ listed above?
Partial answer: We aim to determine the inequalities the six numbers must satisfy. If any of the 6 numbers $(f_1,f'_1,f''_1,f_2,f'_2,f''_2)$ is negative, then it's game over: we need to require that they are all positive.We can refine this broad requirement by asking $0<f_1<f_2$ because of the monotonicity of $f$. Similarly, we must also have $0<f'_1<f'_2$. However, this is not enough: a stronger requirement would be$$0<f'_1< \frac{f_2-f_1}{x_2-x_1} <f'_2$$meaning that the average slope over $(x_1,x_2)$ lies between the slopes at the endpoints of the interval.Is$$0<f_1<f_2 \quad \& \quad 0<f'_1<f'_2 \quad \& \quad 0<f'_1< \frac{f_2-f_1}{x_2-x_1} <f'_2\quad \& \quad f''_{1,2}>0$$the most restrictive set of requirements for the 6 numbers to be compatible with the 3 properties of $f$? If not, how to proceed? If yes, how can we prove it?