# 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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### Inequality $\{a,b,c,d\}\subset[0,1]\Rightarrow\sum\limits_{cyc}(a^4+a^2b^2)+8\prod\limits_{cyc}(1-a)\geq1$

Let $\{a,b,c,d\}\subset[0,1]$. Prove that: $$a^4+b^4+c^4+d^4+a^2b^2+b^2c^2+c^2d^2+d^2a^2+8(1-a)(1-b)(1-c)(1-d)\geq1$$ I tried convexity, the substitution $a=\frac{x}{x+1}...$ and more, but without ...
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### Prove $\sqrt{\frac{a+3}{a+3b}}+\sqrt{\frac{b+3}{b+3c}}+\sqrt{\frac{c+3}{c+3a}} \ge 3$

Let $a,b,c$ are positives such that $ab+bc+ca=3$. Prove that: $$\sqrt{\frac{a+3}{a+3b}}+\sqrt{\frac{b+3}{b+3c}}+\sqrt{\frac{c+3}{c+3a}} \ge 3$$ Once, I see this problem in AoPS:https://...
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### $\frac{1}{2a^2+3}+\frac{1}{2b^2+3}+\frac{1}{2c^2+3}\geq\frac{3}{5}.$

Let $a, b, c \in\mathbb{R}, a, b,c\geq \frac{1}{4}$ and $a+b+c=3$. Show that $\frac{1}{2a^2+3}+\frac{1}{2b^2+3}+\frac{1}{2c^2+3}\geq\frac{3}{5}.$ My idea: $(4a-1)(a-1)^2\geq 0$. Then I tried to divide ...
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### Prove that $f(u+v)\le f(u) +f(v)$ being $f(u)= Au\cdot u\in\mathbb R$

Let $A\in\mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Consider the function $$f: u\in\mathbb R^n\mapsto f(u)= (Au\cdot u)^{1/2}\in\mathbb R.$$ Let $\|\cdot\|_1$ denote a norm ...
1 vote
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### Factorial equation and no solution existing :A conjecture

Conjecture : Let $x<0,n>2$ ,$n$ an integer such that $|x|<(n!+1)(n^2+1)$ then it seems that : $$1/(x/(n!+1))!-1/(x/(n^2+1))!=0$$ Have no integer solution in $x$ where : $$x!=\Gamma(x+1)$$ Or :...
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### Proof of Doob's Maximal Inequality for Positive Supermartingales

I am working through a course on Stochastic Processes and am looking for a proof verification for an alternative to the one that I have been presented. My proof, as noted below, currently contains a ...
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### shorter proof of generalized mediant inequality?

Show $\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\frac{a_{i}}{b_{i}}$, where $b_{i}>0$. Attempt Assume the largest is $\frac{a_{n}}{b_{n}}\Rightarrow$ ...
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### The sum of five natural numbers, two of which are greater than 40, is 127. Accordingly, what is the largest of these numbers?

What I think the question wanted to convey is that we should find the largest possible value in $a,b,c,d,e$ so that their sum is $127$, but so that two of them are greater than $40$. The given (...
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### On a step on a complex inequality with summation.

To make it short I have a doubt in the last step of a complex number inequality, the problem is the next one Let complex numbers $z_1,z_2,z_3,...,z_n$ all modulus $1$ and $z_1+z_2+z_3+...+z_{2012}=0$ ...
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### inequality about sequences satisfied$S_n\le \lambda n$

A nonnegative sequence $\{a_n\}$ satisfies that at least $n$ of the first $n^2$ terms of the sequence $\{a_n\}$ are greater than $n$, and the sum of the first $n$ terms $S_n\le \lambda n$ holds for ...
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### Prove that $\mathbb{E}\exp{\lambda\xi} \le \exp\left(\lambda^2 \Vert\xi\Vert_{\psi_2}^2\right)$

Problem: Let $\xi$ be a real random variable. We say that $\xi$ is $\psi_2$ when $\exists \lambda >0$ such that $\mathbb{E}\exp(\xi^2/\lambda^2) \le e$. We denote by $\Vert \xi \Vert_{\psi_2}$ the ...
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### Prove that $\mathbb{P}\left(\Vert A \Vert \ge cK(\sqrt{N} + \sqrt{K} + t)\right) \le 2\exp(-t^2)$

Problem: Let $A = (a_{ij}): l_2^k \to l_2^N$ be a random matrix whose coefficients $a_{ij}$ are independent, centered, $\psi_2$ (the definition is in this post) and satisfy. \begin{align*} \...
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1 vote
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### Does the solution exist for the linear matrix inequality $A^TP+PA-Q<0$, where $P=P^T>0$, $Q=Q^T>0$? [closed]

I am studying the consensus problem of a linear multi-agent system, and a control gain needs to be designed to reach the consensus property. But the design procedure depends on the solution of the ...
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### Hardy's Inequality: Problems $3.14$ and $3.15$ in Rudin's RCA

In Problem $3.14$, we prove (a) Hardy's inequality, (b) the condition for equality, and I shall talk about (c), (d) below. Problem $3.15$ is the discrete case of Hardy's inequality. I have asked three ...
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I'm currently reading through Roman Vershynin's High Dimensional Probability and working through one of the exercises (7.6.1). Consider a set $T \subseteq \mathbf{R}^n$ and define its Gaussian width $... 2 votes 2 answers 48 views ### Prove that$a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$the question Let$a,b,c$be positive numbers such that$a+b+c=1$. Prove that$a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$. the idea After i put some values to$a$,$b$, ... 1 vote 2 answers 99 views ### How to solve the following mixed exponential inequality:$3^x + x < 4\$? (from Spivak's Calculus)
I have encountered this inequality in Spivak's Calculus (first chapter exercises), which I'm not sure how to solve. $$3^x + x < 4$$ I might be wrong but my gut feeling says the inequality holds ...