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Results tagged with inequalities
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user 169474
for questions involving inequalities, upper and lower bounds.
2
votes
Accepted
Positive, monotone decreasing function, with derivative limit in 0 equal to ∞ submultiplicat...
Define $f$ on $]0,e^{-2}]$ by
$$f(x) = \frac{1}{-\sqrt{x}\ln(x)}.$$
Then $f$ is positive, decreasing since
$$\frac{\mathrm{d}}{\mathrm{d}x}\big(-\sqrt{x}\ln(x)\big) = \frac{-\ln(x)-2}{2\sqrt{x}} \ge 0 …
- 3,808
12
votes
Trigonometric inequality
I give a way that may work for odd numbers. This is too long for a comment.
First, the quantity
$$\cos\Big(\frac{2m\pi}{p}\Big)+\cos\Big(\frac{2n\pi}{q}\Big)
= 2\cos\Big(\pi\Big(\frac{m}{p}+\frac{n}{ …
- 3,808
1
vote
Accepted
Equality cases in a certain case of Jensen's inequality
Since the distribution of the r.v. $X$ is zero-mean, $X$ integrable (hence also the copy $Y$).
The equality is attained iff the distribution of $X$ is carried by at most two points.
Indeed, by indepen …
- 3,808