Questions tagged [mean-value-theorem]

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Mean values of $\zeta(s)$ for $\Re(s)=1/2$ vs $\Re(s)\ne 1/2$

Say I have a good estimate for the $L^2$ mean of the Riemann zeta function $\zeta(s)$ for $\Re s = 1/2$, $|t|\leq T$: $$\int_0^T |\zeta(1/2+i t)|^2 = T \log T - T (1 + \log 2 \pi - 2\gamma) + O(T^\...
1
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0answers
88 views

Mean Value Inequality with Linear Term

I am having trouble proving this modified mean-value inequality. Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$ Prove that there exists constants $r_0,C>0$ depending only on $c$...
2
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0answers
186 views

The collection of mean value abscissas in the Mean value theorem

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 ...
1
vote
1answer
143 views

Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it. Problem definition: Let $f(\xi) \in \...
4
votes
1answer
1k views

How to understand the integral?

In order to understander the nonlinear elliptic equation with natural boundary condition, $$\sigma_2(D^2u)=0 \text{ in } \Omega$$ I wish to understand the following integral, $$E(u,\Sigma)=\int_\Sigma ...
1
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0answers
135 views

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 ...
2
votes
1answer
203 views

Mean value theorem in terms of Wirtinger calculus?

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 \...
15
votes
1answer
449 views

Bull's-eye Riemann sum

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 ...