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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

2
votes
1answer
36 views

Proving an inequality given conditions.

Let real numbers $x_1, x_2, x_3, x_4, x_5, x_6$ satisfy $x_1+x_2+x_3+x_4+x_5+x_6=0, $ and $x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=6.$ Prove $x_1x_2x_3x_4x_5x_6\leq\frac{1}{2}.$ I am trying to figure out ...
1
vote
0answers
40 views

How does Feller get this inequality?

I am currently loosing my mind trying to understand the proof Feller gives in his Book "An Introduction to Probability Theory and it's Applications" Vol2 about the general Berry-Esseen Theorem. What I ...
2
votes
3answers
25 views

logarithmical inequality with equality case proof

If 10$\leq$a$_1$$\leq$a$_2$$\leq$...$\leq$a$_{2020}$$\leq$100 with a$_i$ are real numbers, i=$\overline{1,2020}$ then prove that $$\sum_{i=1}^{2019} log_{a_i}a_{i+1}\leq2020 $$ and the equality case ...
2
votes
2answers
51 views

proof that if $a,b \ge 2$ then $ab \ge a+b$

how is my proof that if $a,b \ge 2$ then $a+b \le ab$ so if $a \ge 2$ and $b \ge 2$ then $a-1 \ge 1$ and $b-1 \ge 1$ $(a-1)(b-1) \ge b-1$ $(a-1)(b-1) \ge 1$ $(a-1)(b-1) - 1 \ge 0$ $ab -a -b\ge 0$ ...
1
vote
1answer
39 views

lower bounding logarithm of sums

Is it true that $$ \log\left(\sum_{i=1}^{n} \alpha_i\right) = \log\left(n \frac{1}{n}\sum_{i=1}^{n}\alpha_i\right) = \log(n) + \log\left(\frac{1}{n}\sum_{i=1}^{n} \alpha_i\right) \\ \geq \log(n) + \...
4
votes
0answers
35 views

Proving these inequalities for a symmetric random walk

I would like to prove the following inequalities. Here $S_n = \sum_{i=1}^n X_i$ where $x_i$ are i.i.d and symmetric. $$P(|S_n|>x) \geq \frac{1}{2}P(\max_{k\leq n}|X_k|>x)\geq \frac{1}{2}(1-e^{...
1
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0answers
24 views

Matrix inequality related to Minkovski space.

$M=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{pmatrix}$ $L\in M_4(\mathbb{R})$ s.t. $L^tML=M$ ($L^t$ is the transposition of $L$) Please ...
0
votes
0answers
24 views

Proving equivalence with Farkas lemma

The problem I am trying to solve: trying to show that the two systems are equivalent. $$\exists x : Ax=a$$ $$Bx \leq b $$ second system: $$\nexists y,z: y^TA+z^TB=0$$ $$y^Ta+z^Tb<0$$ $$z\geq 0$$...
3
votes
3answers
71 views

Help understanding Polya's proof of the Arithmetic-Geometric Mean Inequality

Background Steele, in The Cauchy-Schwarz Master Class, states the general Arithmetic-Geometric Mean Inequality as follows (pg 23): $a_{1}^{p_{1}}a_{2}^{p_{2}} \ldots a_{n}^{p_{n}} \leq p_{1}a_{1} ...
1
vote
1answer
34 views

Inequality of double sum with max operator and squared

I am trying to understand a proof, however one inequality remains unclear: $\mathbb{E} \big( \frac{1}{k_1 k_0} \sum_{i=1}^{k_1} \sum_{j=1}^{k_0} a_{j,i} -b_{j,i} \big)^2 \leq \max_i \frac{1}{k_m^2}\...
2
votes
6answers
80 views

Prove that $5x^2− 2xy− 8x+2y^2− 2y+ 5\geqq 0$ for all $x, y\in \mathbb{R}$. When does the equality occur?

I tried grouping the $x$'s and the $y$'s but that didn't get me anywhere. I know that $5x^2, 2y^2$, and $5$ are always positive. I am not sure what to try next.
2
votes
2answers
76 views

How many tuples of {$a, b, c, …$} satisfy $abc… \leq n$?

Let $n$ and $k$ be positive integers. Let $a, b, c, ...$ be $k$ positive integers such that $abc... \leq n$. How many tuples of {$a, b, c, ...$} satisfy the inequality? Note that the tuples {$a=1, ...
-1
votes
1answer
33 views

integral inequality: midpoint of interval vs. expectation over interval

I have been thinking about an inequality that should be self-evident, but which I have difficulties proving formally. It looks like the following: Take a function $f(x)$ with $f'(x)>0$ and scalars ...
1
vote
2answers
47 views

Proving $\sqrt{a^2+c^2}+\sqrt{b^2+d^2}\ge \:\sqrt{\left(a+b\right)^2+\left(c+d\right)^2}$ [duplicate]

The inequality: $$\sqrt{a^2+c^2}+\sqrt{b^2+d^2}\ge \:\sqrt{\left(a+b\right)^2+\left(c+d\right)^2}$$ But can someone help me with a nice elegant solution. This is an olympiad question I was trying to ...
1
vote
2answers
76 views

How do I prove the inequality $(\sum a^3)^2 \leq (\sum a^2)^3$?

Let $a_1, \dots, a_n \in \mathbb{R}.$ I wish to show that $(\sum_{i=1}^n a_i^3)^2 \leq (\sum_{i=1}^n a_i^2)^3$ in order to prove another statement. But I cannot see how to prove this, if at all the ...
0
votes
1answer
61 views

How can I prove this general case of the inequality?

I think I need some help with this problem. According to p.544 of Feller's An Introduction To Probability Theory. Vol II if $|a_k|\leq c_k$, $|b_k|\leq c_k$ then we have $|\prod_{i=1}^na_i -\prod_{i=...
0
votes
2answers
51 views

Inequality in double integral [on hold]

Given The square $D=\{(x,y)\mid0\le x\le1 ; 0\le y\le 1\}$, prove that $$\iint_D \left(x^3+y^3\right)^{1/3} \ dA \le \iint_D \left(x^2+y^2\right)^{1/2} \ dA.$$
1
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0answers
29 views

$A$ is continuous in $0_X$ => $\exists c \in \mathbb{R}$, $ c \geq 0$ : $||Ax||_Y \leq c ||x||_X $

the map $A : X \rightarrow Y $ is linear, where $(X,||.||)$ and $(Y,||.||)$ are normalized vector spaces. I already have a solution, which is correct but a friend of mine showed me his solution and ...
-1
votes
3answers
43 views

Prove that $n(n+2)$ lies between $n^2$ and $(n+1)^2$, given n is a positive integer

I'm currently trying to prove the inequality $$n^2<n(n+2)<(n+1)^2$$ Is it possible to solve this without induction?
2
votes
3answers
74 views

Find minimum and maximum of $P=a+b+c$

Let $a,b,c\ge 0$ such that $a^2+b^2+c^2+abc=4$. Find minimum and maximum of $$P=a+b+c$$ +)Maximum: Let $x=\frac{2\sqrt{ab}}{\sqrt{\left(c+a\right)\left(c+b\right)}};y=\frac{2\sqrt{bc}}{\sqrt{\left(a+...
-2
votes
1answer
44 views

Prove $\sum\limits_{cyc}\frac{ab}{b^{\,2}+ c^{\,2}}\geqq \frac{3}{2}$

For $a\geqq b\geqq c> 0$. Prove $$\frac{ab}{b^{\,2}+ c^{\,2}}+ \frac{bc}{c^{\,2}+ a^{\,2}}+ \frac{ca}{a^{\,2}+ b^{\,2}}\geqq \frac{3}{2}$$ I used discrim to find and I want to see a solution ...
1
vote
2answers
48 views

Prove $2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$ [on hold]

For $a,\,b,\,c\geqq 0$ and $b\equiv {\rm mid}\,\{\,a,\,b,\,c\,\}$. Prove $$2\sum\limits_{cyc}\,a^{\,3}+ 3\,abc\geqq 3\sum\limits_{cyc}\,a^{\,2}b$$ Inspried from $\lceil$ Prove $k=0$ is the only non-...
-2
votes
1answer
58 views

Prove $\sum\limits_{cyc}\,\frac{a^{\,2}+ k\,b^{\,2}}{k\,a^{\,2}+ b^{\,2}}\geqq 3$

For $a\geqq b\geqq c> 0$. Prove $$\sum\limits_{cyc}\,\frac{a^{\,2}+ k\,b^{\,2}}{k\,a^{\,2}+ b^{\,2}}\geqq 3 \tag{SHED}$$ with $k= \frac{b}{c}\geqq 1$. I used SHEDtechniQ to find and I want to ...
1
vote
6answers
56 views

Prove $5^n + 3^n - 2^{2n+1} > 0$ by induction

I am not sure how to deal with the $-2^{2n+1}$ term. I did the basis proof for n=1 I am stuck at this step: $$ 5^{k+1}+3^{k+1}-2^{2(k+1)+1} = 5\cdot 5^k + 3 \cdot 3^k -2^3 \cdot 2^{2k} $$ Any ...
0
votes
1answer
37 views

Satisfying an inequality involving complex numbers

here's a question I haven't been able to solve. Let $z_{n}=(-\frac{3}{2}+\frac{\sqrt{3}}{2}i)^{n}$. Find the least positive integer $n$ such that $|z_{n+1}-z_{n}|^{2}>7000$. Ok so far I've done ...
0
votes
1answer
24 views

Norm 2 against norm inf

We know from basic linear algebra that $\forall x \neq 0, \frac{||x||_2}{||x||_{\infty}} \leq \sqrt{n}$ (where $n$ is the dimension).We also know that the equality occurs if and only if all ...
3
votes
2answers
52 views

Given $3$ positive reals $a$, $b$ and $c$ such that $a+b+c = 1$, show that $a^a b^b c^c + a^b b^c c^a + a^c b^a c^b \le1$.

Good Day! How are you doing? I was learning about the awesome A.M - G.M. inequality from the Brilliant Wiki. There was a question in the exercises: Given $3$ positive reals $a$, $b$ and $c$ such ...
1
vote
0answers
42 views

$(x+y)^r \le x^r+y^r$ when $r \in (0,1)$ and $x,y$ are real positive numbers. [duplicate]

I'm sorry for the silly question, but I have a doubt. Given two positive real numbers $x,y$ and taking $0<r<1$, is it true that $$ (x+y)^r \le x^r+y^r? $$ In all the examples I considered, ...
1
vote
0answers
24 views

Minimize $E(m)=\sum_{i = 0}^{n-1} {m \choose i} \cdot [c \cdot (\frac{N}{m})^2]^i \cdot[1-c \cdot (\frac{N}{m})^2]^{m-i}$

Minimize over $m$ the expression: $E(m) = \sum_{i = 0}^{n - 1} {m \choose i} \cdot \left[ c \cdot \left(\frac{N}{m} \right)^2 \right]^i \cdot \left[1 - c \cdot \left(\frac{N}{m} \right)^2 \right]^{m ...
2
votes
1answer
32 views

Gaussians and Young's inequality for convolutions

Consider a simplified version of Young's inequality: $$ ||f\ast g||_p\leq ||f||_1||g||_p, \quad 1< p\leq\infty $$ $$ f\ast g\equiv \int_{\mathbb R}dy f(y)g(x-y). $$ What strategy one should follow ...
0
votes
1answer
45 views

On inequality of exponentially activated, unit transform of vector

Let $x$ be a vector and $A$ a matrix. Let $$ y := \dfrac{Ax}{\lVert A \rVert} $$ bet a unit transform of vector $x$, and $$ \widehat{y}:= y/\lVert x\rVert $$ the same transformation with ...
-1
votes
2answers
57 views

Where is $x=\frac{a+b+c+d}{4}$ in $0<a<b<c<d<e$? [on hold]

Where is the exact position of $x=\frac{a+b+c+d}{4}$ in $0<a<b<c<d<e$? The real problem I have a box-and-whisker plot. It shows $L$ : the min value. $Q_1$ : the lower quartile. $...
0
votes
1answer
41 views

Telling the greatest angle of the triangle when slopes are given

When coordinates or position vectors (2D or 3D) of vertices A,B,C of a triangle ABC or its side vectors AB, BC, CA are given we can find the largest angle of $\Delta$ ABC by finding the angle ...
0
votes
0answers
19 views

Is it possible to use Log Sum Exp inequality to the below integral?

$ \int_0^{\infty} \sum_{j=1}^N \frac{1}{\sigma_j^2} \hspace{0.06cm} exp(\frac{-x}{\sigma_j^2}) \hspace{0.06cm} log_2{\big\{\sum_{k=1}^N \frac{1}{\pi \sigma_k^2} \hspace{0.06cm} exp(\frac{-x}{\sigma_k^...
1
vote
1answer
36 views

Does the following result hold? [duplicate]

Suppose that $A$ and $B$ are symmetric and non-negative matrices. Let $\rho(A)$ denote the spectral radius of $A$ and $\rho(B)$ denote the spectral radius of $B$. Does the following result hold? $...
-3
votes
2answers
42 views

Trigonometry and inequalities [on hold]

Hi can somebody please guide me through as to how did we arrive at the step after "So" Regards
0
votes
1answer
28 views

Find the maximum of $f(x, y) = \sum_{k = 1}^{n} x_k y_k$ subject to $\sum_{k=1}^{n} x_k^2 = 1$ and $\sum_{k=1}^{n} y_k^2 = 4$

Find the maximum of $f(x, y) = \sum_{k = 1}^{n} x_k y_k$ subject to $\sum_{k=1}^{n} x_k^2 = 1$ and $\sum_{k=1}^{n} y_k^2 = 4$ I just applied Cauchy Schwarz inequality to find $$-4 \leq |f(x, y)| \...
0
votes
0answers
23 views

Inequality involving periodic functions and Sobolev space.

Set $n\in\mathbb{N}$. For $\psi=u+iv$ in $H^1_T(\mathbb{R}^n)=\lbrace \psi\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)$, I wonder if $$ \int_Q |\nabla u|^2\geq C\int_Q (u-1)^3 ...
0
votes
1answer
28 views

Number of integral values of x satisfying the inequality

What is the number of integral values of $x$ satisfying the inequality: $$\frac{(e^x-1)(\sin(x)-2)(x^2-5x+4)}{x^2(-x^2+x-2)(2x+3)}\le 0$$ I was able to find three solutions: $0$, $1$ and $4$. Is ...
1
vote
4answers
71 views

Prove $\frac{b-a}{6}≤\sqrt{1+b} - \sqrt{1+a}≤\frac{b-a}{4}$ if $3\leq a<b\leq 8$ [on hold]

I don't really know if I should use brute force or some kind of theorem, it comes on a calculus past exam and it says: suppose: $3≤a<b≤8$ prove that $$\frac{b-a}{6}≤\sqrt{1+b} - \sqrt{1+a}≤\frac{b-...
1
vote
0answers
21 views

prove $ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$

$$ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$$ for all $u \in H^1(\Omega)$ for the open unit circle $\Omega$ in $\mathbb{R^2}$. ...
2
votes
1answer
143 views

How can we have $k$ strong enough such that $\sqrt{x^{\,2}+ 3\,x+ 1}+ x= k\in \mathbb{Q},\,x\in \mathbb{Q}\,?$ [on hold]

Prove $$\sqrt[4\,]{x^{\,4}+ 1}= \sqrt{x^{\,2}+ 3\,x+ 1}+ \sqrt{2\,x+ 10} \tag{1}$$ has no real root. By W$\mid$A $$\sqrt[4\,]{x^{\,4}+ 1}> \sqrt{x^{\,2}+ 3\,x+ 1}+ \sqrt{2\,x+ 10}\Leftrightarrow ...
1
vote
1answer
41 views

Where did this definition come from?

Let $C,D > 0$. We call a function $f : \Bbb R \to \Bbb R$ pretty if $f$ is a $\Bbb C^2$-class, $|x^3 f(x)| \leq C$ and $|xf''(x)| \leq D$. (i). Show that if $f$ is pretty, then, given $\...
0
votes
0answers
26 views

Inequality involving determinant and matrices?

Here is the statement : Let $A\in \mathcal{S}_n^{++}(\mathbb{R})$ and $B \in \mathcal{S}_n^{+}(\mathbb{R})$ then we have the following inequality : $(\det(A+B))^{\frac{1}{n}}\ge (\det(A))^{\...
1
vote
1answer
28 views

Is this generalization of Minkowski's inequality for sums right?

Could we write $$f^{-1}\left(\sum_{i=1}^nf(|a_i+b_i|)\right)\leq f^{-1}\left(\sum_{i=1}^nf(|a_i|)\right) +f^{-1}\left(\sum_{i=1}^nf(|b_i|)\right)$$ instead of Minkowski's inequality $$\left(\sum_{i=1}...
3
votes
2answers
100 views

Prove that the following limit exists $\lim_{n\to\infty}\left(\int\limits_{0}^{1}\vert f(x)\vert^{n}dx\right)^{\frac{1}{n}}$ [duplicate]

Let $f:[0,1]\rightarrow\mathbb{R}$ be continuous. Prove that the following limit exists $$\lim_{n\to\infty}\left(\int_{0}^{1}| f(x)|^{n}\,dx\right)^{\!\!1/n}$$ I tried like this: $$\left(\int_{0}^{1}...
1
vote
4answers
29 views

Least value of $\alpha \in \mathbb R$ ($\forall x >0)$ for which $4 \alpha x^2 + \frac{1}{x} \geq 1$

What is the least value of $\alpha \in \mathbb R$ ($\forall x >0)$ for which $$4 \alpha x^2 + \frac{1}{x} \geq 1?$$ I've tried applying the A.M $\geq$ G.M inequality- $$\dfrac{4\alpha x^2 +\...
0
votes
1answer
57 views

Using Young's inequality to prove $\frac{|x|^{a_1}|y|^{a_2}}{|x|^{b_1}+|y|^{b_2}} \geq 1$ where $\frac{a_1}{b_1}+\frac{a_2}{b_2}=1$

Use Young's Inequality prove that if $a_1$, $a_2$, $b_1$,$b_2$ are all positive and $\frac{a_1}{b_1} + \frac{a_2}{b_2} = 1$ then $$\frac{|x|^{a_1}|y|^{a_2}}{|x|^{b_1}+|y|^{b_2}} {\leq} 1$$ for ...
0
votes
0answers
37 views

Probability problem about a parking lot

We want to design a parking lot for a group of 200 apartments still under construction. It is known that for each department (from city statistics) the number of cars will be 0, 1 and 2 with ...
0
votes
0answers
58 views

Prove $\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$ with $a,\,b,\,c> 0$

Let $a,\,b,\,c$ be positive numbers. Prove that $$\sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,a+ b\,)}}\geqq \sum\limits_{cyc}\,\frac{a}{\sqrt{b(\,c+ a\,)}}$$ I tried Holder and $\lceil$ https://...