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Results tagged with inequalities
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user 104330
for questions involving inequalities, upper and lower bounds.
16
votes
Accepted
Bounding a Fourier coefficient of a non-negative periodic function in terms of its $L^2$-norm
Your conjecture is indeed correct. The proof is based on the following simple yet powerful trick I learned many years ago: $|z| = \sup_{|v| = 1} \Re (zv)$. Therefore it is enough to only bound from ab …
- 6,476
13
votes
Lower bound $\int_0^1 \frac{|f'(x)|^2}{f} \,\mathrm{d} x$ by $\int_0^1 |f-1|^2\, \mathrm{d} x$
First of all such a Csiszar–Kullback-Pinsker inequality or whatever cannot possibly be true since $x^2$ explodes faster than $x\log x$ so you can make a local adjustment so that the right-hand side is …
- 6,476
12
votes
Is there a universal bound for this ratio of expectations?
Yes, there is a bound for this ratio -- it is always between $\frac{1}{2}$ and $1$. Upper bound is obvious since $|X+Y| \le |X| + |Y|$ so let us prove the lower bound.
Let $X$ be positive with probab …
- 6,476
7
votes
Accepted
Comparing two limsup's
No, this equality does not hold in general. What is true is that rhs is at most twice as large as lhs and this is sharp.
For the sake of brevity I will only show an example where lhs is $1$ and rhs …
- 6,476
3
votes
Accepted
Is this probability inequality true?
This is false:
Let the probability distribution be $P(X = 0, Y = 0) = P(X = 1, Y = 1) = \frac{1}{2}$ and let $A = \{ 0\}$, $B = \{ 0, 1\}$. Then the left-hand side of your inequality is $1$ while th …
- 6,476
3
votes
Accepted
About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$
I will show that $a \ge \sqrt{\pi}$ which I assume is what you wanted to ask (and it complements counterexample of Iosif Pinelis which appeared while I was writing this answer, showing that this is sh …
- 6,476
0
votes
Estimating the integral $\int_{\epsilon}^1 \Bigl\lvert \int_0^x \frac{f(y)}{\lvert x-y\rvert...
Isn't it just a convolution operator with $h(x) = \chi_{[0, 1]}(x)\frac{1}{\sqrt{x}}$? So, by the Young's convolution inequality, $f*h$ is in $L^p([0, 1])$ for all $p < \infty$ (in particular for $p = …
- 6,476
0
votes
Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'...
Moreover, it is known that Sobolev's and Morrey's inequalities are essentially sharp (in our case, we can not have a uniform estimate with a better modulus of continuity, say). … So informally we are asking if there're inequalities which work for Lipschitz functions and fail for $\frac{1}{2}$-Hölder, and for me it seems natural that it is a real possibility, since the Lipschitz …
- 6,476