# Questions tagged [mean-value-theorem]

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### Mean value theorem for Dirichlet series - optimize?

Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality $$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq \sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$ ...
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### A discrete-to-continuous approach to the Dirichlet principle?

Dirichlet principle: Let $\Omega \subset R^n$ be a compact set with $C^1$ boundary. Then, there exists a unique solution $f$ satisfying $\Delta f = 0$ in $\Omega$ and $f=g$ on $\partial \Omega$. We ...
The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as \begin{equation} f(x)-f(y)=\int_{0}^{1}\nabla f(x(\tau))d \tau \...
Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane. Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$ In other ...