# Tagged Questions

For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.

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### $\mathbf{E}\left[\frac{(U_1+c)^2}{\max((U_1+c)^2, U_2^2)} \right] \ge \mathbf{E}\left[\frac{U_2^2}{\max((U_1+c)^2, U_2^2)} \right]$

We consider two i.i.d. random variables $U_1$ and $U_2$ such that $\mathbf{E}[U_1] = \mathbf{E}[U_2] = 0$ and $\textrm{Var}[U_1] = \textrm{Var}[U_2] < \infty$. Prove that for any $c > 0$ the ...
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### Proving a Definite Integral Inequality without Geometrical Intuition

I solved an integral inequality problem using geometrical methods. However, I just cannot satisfy with them and want a without-geometrical-intuition proof, and I couldn't find one. Proof the ...
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### Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
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### Divergence of an integral

I am sure this is a familiar example to many, as it comes up as an example of a function which belongs to $L^p$ for $p=1$ but not $L^p$ for $p < 1$, but I am having a hard time seeing why it is "...
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### Integral inequality for a Cauchy exponential series product

My goal is to get an inequality $\forall t>0$ for the following integral $$\int_0^t \left(\sum_{n=1}^\infty \exp(-n^2 t_0)\right)^2\,\mathrm{d}t_0 \le f(t).$$ The goal is to at least lose the ...
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### How does this inequality imply this one?

I am having a little trouble understanding this part of a proof. There is an integral $\text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx$ Now, $\text{J}_0 = \frac{\pi ^3}{24}$ The part of ...
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Suppose the random variables $X_i$ are independent and satisfy $E[X_i] = 0$. Then the following inequality holds: E\left[\left(\sum \limits_{i = 1}^n X_i\right)^4\right] = \sum \limits_{i = 1}^n E[... 1answer 475 views ### continuity of norms with respect to p Let f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu). Then w(p)=||f||_p is continuous function of p for any p\in [1,\infty). How to prove this? I have obtained the proof that \... 1answer 32 views ### How to Proceed in Solving this Equation Let f: [0,\infty)\to \mathbb{R} a non-decreasing function. Then show this inequality holds for all x,y,z such that 0\le x<y<z. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-... 1answer 2k views ### A function of two cumulative probability distributions with same first 2 moments Let \Phi_1 and \Phi_2 be cumulative probability distribution functions with domain [L, \infty), L\geq 0, both distributions having the same expectation \mu and the same second moment (hence ... 1answer 47 views ### Reference for theorem? Inequality of integrals of increasing function over two distributions I have a monotone increasing function H(x) and two distributions with CDFs F_1 and F_2, where F_1(x) \leq F_2(x) everywhere. The domain is [0,\infty). This seems like it must be true: \...
A question says: Prove Theorem 1.7 (Uniqueness). Hint: suppose that $x$ and $x^*$ are distinct solutions to the same IVP (from the same initial point). Consider the function \$\nu(t)=||x(t)-x^*(t)||...