Questions on proving and manipulating inequalities.

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How prove this inequatity $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge 4+(x-y)^2$

let $x,y,z>0$,and such $$4\le x+y+z\le 5$$ show that $$\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 4+(x-y)^2$$ It seem $\dfrac{x^2}{y}+\dfrac{y^2}{z}+\dfrac{z^2}{x}\ge 4+(x-y)^2$ maybe is ...
8
votes
0answers
123 views

Verifying the inequality $\sum_{i=1}^n \frac{\cos y_i}{\sin x_i}≤ \sum_{i=1}^n\cot x_i $

Let $\sum_{i=1}^n x_i=\sum_{i=1}^ny_i= \pi$ , where $n>1$ and $x_i >0 , y_i>0 , \forall i=1,2,..,n$. How can we prove that $$\sum_{i=1}^n \frac{\cos y_i}{\sin x_i}≤ \sum_{i=1}^n\cot x_i $$ ? ...
8
votes
0answers
184 views

How prove $ \sum (a+b+c+d)\sum\frac{ab+ac+ad+bc+bd+cd}{a+b+c+d}\sum\frac{abc+abd+bcd+cda}{ab+ac+ad+bc+bd+cd}\leq \sum a\sum b\sum c\sum d$

let $a_{i}>0,b_{i}>0,c_{i}>0,d_{i}>0,i=1,2,\cdots,n $ show that ...
7
votes
0answers
159 views

Proving a geometric inequality without Lagrange multipliers

Let $e=(1,1,...,1)$ be the $n$-dimensional vector consisting only of ones. Let $r=\sqrt{\dfrac{n}{n-1}}$ and $\alpha \in (0,1)$ fixed. Given a vector $x=(x_1,x_2,...,x_n) \in \mathbb R^n$ such that ...
7
votes
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133 views

Question about an upper bound

Let each of the positive integers $a_1 ,\dots, a_n$ be less than $m$ such that the least common multiple of any two of the positive integers $a_1 ,\dots, a_n$ is greater than the integer $m$. Then ...
6
votes
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150 views

Solution set of inequalities in $\mathbb{R}^6$

Let $\theta\in (0,1)$ fixed. We define $A_1$ be the set of all $(a_1,a_2,a_3,a_4,a_5,a_6)\in (0,1)^6$ such that the following conditions hold: \begin{equation} (1) \quad a_2\le a_1, a_4\le a_3, a_6 ...
6
votes
0answers
126 views

How prove this inequality $(\sum a_{1}^{1.5})^2\ge \sum a_{1}\sum a_{1}a_{2}$

Now my question let $a_{1},a_{2},\cdots,a_{n}$ are positive numbers,and $a_{n+i}=a_{i},i=1,2,\cdots$,show that $$(\sum a_{1}^{1.5})^2\ge \sum a_{1}\sum a_{1}a_{2}$$ my teacher (tian275461) have ...
6
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387 views

How prove this inequality $a+b+c=3$

Let $a,b,c$ be nonnegative real numbers, no two of which are zero such that $a+b+c=3.$ Prove that $$ \dfrac{a}{5b+c^3}+\dfrac{b}{5c+a^3}+\dfrac{c}{5a+b^3} \geq \dfrac{1}{2}$$ I think this inequality ...
6
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0answers
178 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
6
votes
0answers
272 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
6
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154 views

Multivariate polynomial with all coefficients positive

Let $n\geq 3$ be an integer. Consider the following polynomials : $$ f(x_1,x_2, \ldots ,x_n)=\bigg(\frac{1}{n}\sum_{k=1}^n x_k^n\bigg)^{2n-2}- \bigg(\prod_{k=1}^n \frac{x_k^{2n-2}+\big(\prod_{j\neq ...
6
votes
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703 views

Farkas Lemma proof

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
6
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231 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
5
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85 views

Very hard inequality: $\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le k_p \sqrt{a+b+c}.$

Given $p>0$. Find the smallest real number $k_p$ such that the following inequality holds for any non-negative reals $a,b,c$: $$\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le ...
5
votes
0answers
104 views

How prove this inequality $\frac{1}{n!}\sum\limits_{k=0}^{\infty}\frac{k^n}{k!}\ge e(C\ln{n})^{-n}$

Show that: $$\dfrac{e^n}{(\ln{n})^n}\ge \dfrac{1}{n!}\sum_{k=0}^{\infty}\dfrac{k^n}{k!}\ge e(C\ln{n})^{-n},\ n\ge 2$$ where $C>e$ is constant. My try: ...
5
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0answers
185 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
5
votes
0answers
90 views

Binomial asymptotic.

Is there any "direct" proof of the following asymptotic inequality: let $\alpha\le 1$ and consider $$Q_n(x)=\sum_{k=1}^n\frac{\alpha(\alpha+1)(\alpha+2)\cdots(\alpha+k-1)}{k!}x^k$$ Then, $$\int_0^1 ...
5
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0answers
148 views

Evaluate integral $ I_s(x) \leq \frac{C}{(\pi(1-2 \alpha s))^{d/2}}\exp\left(\frac{\alpha}{1-2 \alpha s }|x|^2\right) $

For all $ x \in \mathbb{R}^n ,\hspace{5mm} 0 \leq s<t ,\hspace{5mm} t \in \mathbb{R}^+$ $$ I_s(x)=\int_{\mathbb{R}^n}\left|v\left(y \sqrt{2s}+x\right)\right|\exp(-|y|^2) \, \mathrm dy. $$ How we ...
5
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126 views

Primes of the form $\dfrac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
5
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140 views

Bounding function involving Beta functions

Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + ...
4
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0answers
41 views

Soft Question: Inequalities like this

I am studying signed and complex measure and at a point in a proof the following lemma is being used: Lemma. If $z_1,...,z_n$ are complex numbers, then there exists a subset $S\subset\{1,2,...,n\}$ ...
4
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0answers
71 views

How prove this inequality $\sin^2{\frac{A}{2}}+\sin^3{\frac{B}{2}}+\sin^4{\frac{C}{2}}\ge\frac{7}{16}$

in $\Delta ABC$,such $$5\cos{A}+6\cos{B}+7\cos{C}=9$$ show that $$\sin^2{\dfrac{A}{2}}+\sin^3{\dfrac{B}{2}}+\sin^4{\dfrac{C}{2}}\ge\dfrac{7}{16}$$ By the way: This inequality is my favourite ...
4
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89 views

Is $(tr(A))^n\geq n^n \det(A)$ for a symmetric positive definite matrix $A\in M_{n\times n} (\mathbb{R})$

If $A\in M_{n\times n} (\mathbb{R})$ a positive definite symmetric matrix, Question is to check if : $$(tr(A))^n\geq n^n \det(A)$$ What i have tried is : As $A\in M_{n\times n} (\mathbb{R})$ a ...
4
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0answers
206 views

How prove this inequality $a+b+c+d=4$

let $a,b,c,d$ be postive numbers,and such $a+b+c+d=4$, show that ...
4
votes
0answers
92 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...
4
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0answers
68 views

Conditions to satisfy trigonometric inequality

I'm looking for sufficient (and necessary would be good too) conditions on $a,b,c$ such that \begin{align} a\cos\phi + b \cos 3\phi + c \cos 5\phi \geq -1 \hspace{20pt} (\forall \phi) \end{align} ...
4
votes
0answers
134 views

Prove that:$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geqslant a^n+b^n+c^n$

Find $n\in\mathbb{N}^+$ For all Positive real numbers $a,b,c$ sastifying $a+b+c=3$ $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\geqslant a^n+b^n+c^n$
4
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294 views

Proving the Power Mean Inequality using Chebyshev's sum inequality

Question: Can Chebyshev's sum inequality be used to prove the generalized mean inequality, or at least a portion of it? If we let $$P(r) = \sqrt[r]{\frac{x_1^r + x_2^r + \cdots +x_n^r}{n}}$$, we have ...
4
votes
0answers
86 views

how prove $\phi(n)\ge \frac{n}{6\log \log (n)} $ $\forall n\ge5 $

How to prove$\forall n\ge5 $ $$\phi(n)\ge \frac{n}{6\log \log (n)} $$ $\phi$ is Euler function Thanks in advance
4
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0answers
109 views

Inequality problem with factorials

I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you: Let $a,b,c$ be nonnegative integers. Prove that $$ ...
4
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0answers
69 views

Find Lipshitz constant

Let $x, y$ be two vectors in $R^n$. Let $\pi$ be permutation on $\{1, \ldots, n\}$ with uniform distribution. Find Lipshitz constaants of the following functions: $$ f(\pi)=\sum_{i=1}^nx_{\pi(i)}y_i ...
4
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0answers
284 views

prove that this inequality

Let $a_{0},a_{1},\cdots,a_{n}\ge 0,n\ge 1,$ and let $S_{k}=\displaystyle\sum_{i=0}^{k}\binom{k}{i}a_{i}$ with $k=0,1,2,\cdots,n$. We Assume that $\binom{0}{0}=1,\binom{n}{k}=\dfrac{n!}{k!(n-k)!}$. ...
4
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0answers
156 views

Solving a system of linear inequalities

I have a personal problem I want to solve. I have a system of linear inequalities with 97 unknowns and 150000 ineqaulities, I think formal notation of the problem should be something like this ...
4
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0answers
77 views

Polynomial bound

Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that $$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$ Suppose that $P(x)> 0$ for all ...
4
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0answers
72 views

Dominated convergence on $e^{-n^2 t} t^{s/2-1}$

I am trying to apply the Dominated Convergence Theorem to show that $$\sum_{n\ge 1} \int_0^1 e^{-n^2 t} t^{s/2-1}dt= \int_0^1 \sum_{n\ge 1}e^{-n^2 t} t^{s/2-1}dt$$ as soon as $s>1$. I've ...
4
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0answers
196 views

Proof of an Inequality

Given a sequence $(b_n)_{n=1}^p$ of positive numbers such that $b_1>b_2>\cdots>b_p>0$, define $$\beta=\bigg(p!\frac{p}{p-1}\bigg)^{\frac{1}{\min\limits_{1\leq k\leq ...
4
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0answers
239 views

How can I prove this inequality?

I have a pretty nasty looking function $$\sigma (t,y) = \sqrt{\frac{\sum_{i=1}^N \lambda_i \sigma_i \exp \left (-\frac{1}{2 t \sigma_i^2}\left(\ln{\frac{y}{S_0}} - \left(r - \frac{\sigma_i^2}{2} ...
4
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0answers
279 views

Inequalities involving the probability density function and variance

I am wondering whether anyone knows of any any inequalities involving the probability density function of an unknown distribution (as opposed to the cumulative distribution function) and its known ...
4
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0answers
552 views

Deriving Inequalities for Binomial Terms

Assume $\alpha + \beta = 1$, and one is trying to find a lower bound on the $k$th term in the binomial expansion of $(\alpha + \beta)^m$. The terms are of the form $\binom{m}{k} \alpha^k \beta ...
3
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0answers
61 views

Finding a uniform bound for this ugly expression

I am asked to find a positive bound $M$ independent of $t$ and $n$ for the following expression : ...
3
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0answers
30 views

How prove this inequality $\sum_{cyc}\frac{1-2\sin{\frac{C}{2}}}{\sin{\frac{B}{2}}}\ge 0$

in $\Delta ABC$,prove or disprove $$\dfrac{1-2\sin{\dfrac{C}{2}}}{\sin{\dfrac{B}{2}}}+\dfrac{1-2\sin{\dfrac{A}{2}}}{\sin{\dfrac{C}{2}}}+\dfrac{1-2\sin{\dfrac{A}{2}}}{\sin{\dfrac{A}{2}}}\ge 0$$ My ...
3
votes
0answers
122 views

Question concerning an inequality

I need the proof of the following inequality. $$‎\prod‎_{i=1}^{n} {(b_i -a_i)^{1/n}} +1‎\geq \prod‎_{i=1}^{n} {(b_i)^{1/n}} +\prod‎_{i=1}^{n} {(1-a_i)^{1/n}},$$ when $0<a_i<b_i<1$.
3
votes
0answers
99 views

Putnam Competition 2003 A2 - Question

I just wanted to have your opinions on my solution to this question. Any criticism would be welcome, especially with LaTeX formatting. I'm new to this website and still can't seem to get my LaTeX code ...
3
votes
0answers
39 views

How prove $\left(\sum_{i=0}^{n-1}a_{i}\cos{\frac{2k\pi i}{n}}\right)^2\le (n^2-1)\left(\sum_{i=0}^{n-1}a_{i}\sin{\frac{2k\pi i}{n}}\right)^2$

let $n$ is odd number, and $a_{i},(i=0,1,\cdots,n-1)$ is $\{0,1,2,\cdots,n-1\}$ arrangement,and $k=0,1,2,\cdots,n-1$ prove or disprove $$\left(\sum_{i=0}^{n-1}a_{i}\cos{\dfrac{2k\pi ...
3
votes
0answers
17 views

inequality involving lifts of a positive oriented homeomorphism of the circle

Let $\pi: \mathbb R \to S^1$ be the natural projection and let $f:S^1 \to S^1$ be a positive oriented homeomorphism. We say that $F: \mathbb R \to \mathbb R$ is a lift of $f$ is $\pi \circ F = f \circ ...
3
votes
0answers
41 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
3
votes
0answers
67 views

Prove this inequality for all integers $m>2$

Prove this inequality for all integers $m$: $$∑_{n=2}^{m} \frac{1-n^{2α-1}}{n^{\alpha}} > \frac{1-(m+1)^{2α-1}}{(m+1)^{\alpha}}$$ for all $0<α<1/2$ and $m>2$.
3
votes
0answers
147 views

How prove this sum equality?$\le\prod_{i=1}^{n}\left(\sum_{k=1}^{m}a_{k,i}\right)$

consider this matrix $$\begin{bmatrix} a_{1,1}&a_{1,2}&\cdots,&a_{1,n}\\ a_{21}&a_{2,2}&\cdots&a_{2,n}\\ \cdots&\cdots&\cdots&\cdots\\ ...
3
votes
0answers
63 views

Find the minimum value of: $P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$

Let $a,b,c\ge0$ such that: $(a+b)(b+c)(c+a)=1$. Find the minimum value of: $$P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$$. I've tried many things but all failed. Please ...
3
votes
0answers
52 views

Inequality $\sum_{k=1}^{n}\frac{k}{a_{1}+\cdots+a_{k}}\le\left(2-\frac{7\ln{2}}{8\ln{n}}\right)\sum_{k=1}^{n}\frac{1}{a_{k}}$

$n\geq2$ is a integer, $a_n\gt0, n=1,2,\dotsc$, then $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le\left(2-\dfrac{7\ln{2}}{8\ln{n}}\right)\sum_{k=1}^{n}\dfrac{1}{a_{k}}$$ I don't know how to ...