The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have $$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$ for at least one $c\in [a,b]$. We want to study the properties of such $r\in C_{f,a,b}:=\{c: c \text{ that satisfy (*)} \}=f^{-1}\big(\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)\big)$.
In our particular problem, we want to compare to it a different $c'$ from $$\frac{1}{\mu[0,a]}\int_{0}^{a}f(x)d\mu(x)=f(c')$$, by estimating their difference from below: $$c-c'>\delta.$$ (Our f is positive continuous but nondifferentiable and non-monotone and so we cannot simply invert).
Q1: How to express the condition $c-c'>\delta$ in terms of properties of f and $\mu$? What are some distinct requirements on $f,\mu$ for this lower bound to be true? Conversely, suppose that the MVT set is contained within a small neighbourhood $C_{f,a,b}\subset [x_{0}-\epsilon,x_{0}+\epsilon]$. What does that imply for the behaviour of f (at least when $\mu=$Lebesgue)?
Q2: But I am curious if there is a framework for studying the properties of $C_{f,a,b}$? Is there any literature out there on putting bounds on the inf and sup of this set for various types of functions?
Q3: A geometric interpretation for such c is as the values that give matchings area in (*). I am curious if there have been any more theoretical results using that idea. Maybe studying some minimization problem: $$arginf\{|\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)-f(r)|^{2}\}$$ using some optimization algorithm.
(from MVT's wiki)
- If $|c-c'|\leq \delta$ for all $c\in C_{f,a,b},c'\in C_{f,0,a}$ then by the geometric interpretation and the Second-MVT, we intuitively see that the function must be like a trapezoid reaching a plateau inside the $\delta-$neighbourhood of $x=a$. Inside that plateau f must have a few high peaks that have enough area underneath them to give a mean value equality. This is assuming we work with $\mu$ Lebesgue; for more general measures, we can get different shapes.
- Some work on exploring this same reverse question: given c what can we say about f: "On Functions Whose Mean Value Abscissas Are Midpoints, with Connections to Harmonic Functions" and "When Are There Continuous Choices for the Mean Value Abscissa?".
- Partial Converse to the mean value theorem
