# Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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### Logarithm equation, maybe inequality related? [closed]

Solve on $\mathbb{R}$ the equation $\log_3(4 + 3x^4) + \log_5(1 + \sqrt{x^2}) + \log_2(1+x^2) = \log_3 4$.
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### Usefull inequality for p-laplacian study of form $|x-y|^{p-1}y + |x-y|^p \leq C |x|^{p-2}x(x-y)$

I've been looking for an inequality of the kind $|x-y|^{p-1}y + |x-y|^p \leq C |x|^{p-2}x(x-y) \$, for all $x, y\in \mathbb{R}$, $p>2$ and $C>0$ a constant. or results to help me demonstrate ...
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### Is it true that for $x>0$, one has $x\le\frac{1}{\log x}$?

I know that for $x>0$, one can quite easily prove that $$\frac{1}{x}\le\frac{1}{\log{x}},$$ which follows from the trivial identity that $x\le e^x\iff \log x\le x\iff\frac{1}{\log x}\ge\frac{1}{x}$,...
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### Solving simultaneous linear inequalities over the integers

Find the number of integral points $(x,y)$ such that $x_1<x<x_2$ and $y_1<ay+bx<y_2$. I encountered this pattern while studying linear congruence of type $a x + b y \equiv c \pmod m$.
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### Higher order Jensen-like expansion upper bound

If $Z$ is a random variable with fine moment generating function, what is a good way to upper bound $$|\log \mathbb{E}e^Z- \mathbb{E}Z- \frac{1}{2}\mathbb{E}Z^2|$$ This looks like a third offer Taylor ...
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### Inequality I'm sure to be true

While solving a bigger problem, I've reduced it to an inequality $$\left(1+2^{b^\frac{1}{b-1}-1}\right)^b < 1+2^{b^\frac{b}{b-1}-1}$$for $b>2$, which looks plausible when looking at the plots. I'...
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### Prove that $\liminf_{n\to\infty}a_{n}\leq\liminf_{n\to\infty}\frac{s_{n}}{n+1}\leq\limsup_{n\to\infty}\frac{s_{n}}{n+1}\leq\limsup_{n\to\infty}a_{n}$ [duplicate]

Suppose that $(a_{n})_{n=0}^{\infty}$ is a bounded sequence. Let $s_{n} = a_{0} + a_{1} + \ldots + a_{n}$. Show that \begin{align*} \liminf_{n\to\infty}a_{n}\leq\liminf_{n\to\infty}\frac{s_{n}}{n+1}\...
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### Cant obtain the following inequality

Given the following two inequalities, how do I obtain the third inequality? Sorry I could not get the images being displayed. Also $$f(O)\geq d$$ I have been sitting the whole day trying to solve it ...
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### solve an exponential inequality $2a^x-a^{x+1} \le b$

Given two scalars $0<a<1$ and $0<b<1$, how to solve the unknown $x$ in the following equation: $2a^x-a^{x+1} \le b$
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I have been working on a lemma (page 419 Understanding Machine Learning) and I understand every component of the proof but I do not understand how "the proof follows". Essentially I need ...
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### For $a,b,c>0$ prove that $abc(a+b+c) \le a^3 b + b^3 c + c^3 a$

if $a,b,c > 0$ then prove that $abc(a+b+c) \le a^3 b + b^3 c + c^3 a$ My attempt The hint given for this question was to use the Cauchy-schwarz inequality. But if you look at the expression given, ...
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### Why solving the inequality $\text{arcsec}(x)>\frac{\pi}{4}$ is giving only half of the answer

$$\text{arcsec}(x)>\frac{\pi}{4}$$ Taking $\sec$ on both sides yields $$\sec(\text{arcsec}(x))>\sec(\frac{\pi}{4})$$ $$x>2^{\frac{1}{2}}$$ But from this desmos graph one can easily see that ...
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### How to prove that for every integer $k \geq 2$ we have $k^{1/k} \leq e^{1/e}$ without using the first derivative test?

I just stumbled across this cool property, by doing some calculus I could prove it, the function $f(x) = x^{1/x}$ has a local maximum at $x = e$ and the derivative changes sign at that point, but I ...
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