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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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 $\Re s = 1/2$, $|t|\leq T$:
$$\int_0^T |\zeta(1/2+i t)|^2 = T \log T - T (1 + \log 2 \pi - 2\gamma) + O(T^\...
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Mean Value Inequality with Linear Term
I am having trouble proving this modified mean-value inequality.
Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$
Prove that there exists constants $r_0,C>0$ depending only on $c$...
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The collection of mean value abscissas in the Mean value theorem
The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have
$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$
for at least one $c\in [a,b]$. We ...
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Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?
For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it.
Problem definition: Let $f(\xi) \in \...
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How to understand the integral?
In order to understander the nonlinear elliptic equation with natural boundary condition,
$$\sigma_2(D^2u)=0 \text{ in } \Omega$$
I wish to understand the following integral,
$$E(u,\Sigma)=\int_\Sigma ...
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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 ...
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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 \...
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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 ...
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