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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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2answers
53 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 ...
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0answers
45 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=...
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2answers
46 views

Inequality in double integral [on hold]

Given The square $D=\{(x,y)\mid0\le x\le1 ; 0\le y\le 1\}$, prove that $$\iint_D \left(x^3+y^3\right)^{1/3} \ dA \le \iint_D \left(x^2+y^2\right)^{1/2} \ dA.$$
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0answers
21 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
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3answers
38 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
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3answers
63 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+...
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1answer
40 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
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2answers
45 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-...
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1answer
47 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 ...
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6answers
55 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 ...
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1answer
34 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 ...
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1answer
21 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 ...
2
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2answers
48 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
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0answers
40 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, ...
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0answers
21 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
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1answer
29 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
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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 ...
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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. $...
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1answer
37 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 ...
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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
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1answer
34 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? $...
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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
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1answer
27 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)| \...
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0answers
22 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 ...
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1answer
25 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
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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-...
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0answers
20 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 $\mathbb{R^2}$. ...
1
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1answer
132 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
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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 $\...
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0answers
25 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))^{\...
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1answer
27 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}...
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0answers
47 views

New inequality $(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\sqrt{\frac{\prod_{cyc}(49x-7y+z)}{129(x+y+z)}}\leq \frac{344}{8}$ [on hold]

I'm interested by this problem : Let $x,y,z>0$ then we have : $$(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\sqrt{\frac{\prod_{cyc}(49x-7y+z)}{129(x+y+z)}}\leq \frac{344}{8}$$ I think this problem ...
0
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0answers
69 views

Prove $2(x+2y+z)(x+z)\geqq\frac{(y+z)(3x+2y-z)(x^2-y^2+2yz-2z^2)}{(x+y-z)(x-z)}$ for $x,\,y,\,z\geqq 0$ [on hold]

For $x,\,y,\,z\geqq 0$ and $\left ( x+ y- z \right )\left ( x- z \right )\neq 0$, prove $$2\left ( x+ 2\,y+ z \right )\left ( x+ z \right )\geqq \frac{\left ( y+ z \right )\left ( 3\,x+ 2\,y- z \right ...
3
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2answers
99 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
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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
55 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
57 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://...
7
votes
7answers
108 views

How to solve $\sqrt{x+2}\geq x$?

How do you solve the inequality $$\sqrt{x+2}\geq{x}?$$ Now since ${x+2}$ is under the radical sign, it must be greater than or equal to ${0}$ to be defined. So, ${x+2}\geq{0}$ Thus ${x}\geq{-2}$ ...
2
votes
1answer
56 views

a < b if and only if a++ ≤ b.

I have to prove that a < b if and only if a++ ≤ b. I am using the book analysis 1 by Terence Tao, which unfortunately has no section for solutions of exercises. both a and b are natural numbers, ...
2
votes
3answers
106 views

How to prove $\left|\sqrt{2} - \frac{m}{n}\right| > \frac{1}{3n^2}$ inductively? [duplicate]

I saw a problem online (Orig) as follows. I'm curious if there's a straightfoward way to prove it using induction. It's easy to prove that (Orig) holds when $n=1$ or $m=1$ , which seems like a good ...
0
votes
0answers
51 views

Help with inequality with one unknown

Please could you help how to solve the inequality $(\sqrt{x-9})(2^{x-8}+3^{x-9}-9)\geq 0$
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votes
0answers
57 views

Inequality for $a+b+c=3$ $3\geq (ab)^{bc}+(bc)^{ca}+(ca)^{ab}$

It's a new problem that I find interesting : Let $a,b,c>0$ such that $a+b+c=3$ then we have : $$3\geq (ab)^{bc}+(bc)^{ca}+(ca)^{ab}$$ My try : If $ab\leq 1$ , $bc\leq 1$ , $ca\leq 1$ the ...
0
votes
2answers
21 views

Interrelated constraints via linear combinations

Given $x$ and $y$ are real variable such that: $\left| x \right|\le \alpha ,\left| y \right|\le \alpha ,$ where $\alpha$ is a positive constant. I want to determine bounds of $u,v$ where $u,v$ are ...
1
vote
1answer
44 views

Determine a Constant

Let $\lambda \in C$ and that $\lambda \in B(\mu,r)$, a closed ball centered at $\mu \in C$, with radius $r < \mu$. I am trying to determine the value of a constant $\tau$ to guarantee that: $$ |1-\...
0
votes
2answers
55 views

Inequality conjecture for ordered numbers

Consider the following inequality. Let $x_1>x_2>...>x_n>0$ be some positive numbers. Then $\sum_{i=1}^n x_i+\sum_{i=1}^n\sum_{j\neq i} \frac{x_ix_j}{x_i+x_j}\leq M\left [\sum_{i=1}^n i\...
0
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2answers
51 views

Is it correct that if (1) then (2)?

Is it correct that if $$3 < a < b$$ then $$a^b > b^a?$$
0
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1answer
57 views

Inequality $\frac{abc}{\sqrt{3}\sqrt{a^2+b^2+c^2}}\Big(\sum_{cyc}\frac{1}{7a^2+b^2}\Big)\leq \frac{1}{a+b+c}\Big(\sum_{cyc}\frac{a^3}{7a^2+b^2}\Big)$

Let $a,b,c>0$ then we have : $$\frac{abc}{\sqrt{3}\sqrt{a^2+b^2+c^2}}\Big(\sum_{cyc}\frac{1}{7a^2+b^2}\Big)\leq \frac{1}{a+b+c}\Big(\sum_{cyc}\frac{a^3}{7a^2+b^2}\Big)$$ My try : Here ...
-1
votes
1answer
44 views

To check that if this inequality is an equivalence relation on $\mathbb{Z}$ [closed]

I proved this inequality ; Which is a relation on $\mathbb{Z}$ s.t a and b belongs to $\mathbb{Z}$ $$a^2 - b^2 \le 7$$ is reflexive , I'm stuck at the symmetry of this relation, can anyone help? ...
0
votes
0answers
14 views

How to check satisfiability of a large number of “lorenzian” quadratic inequalities

Given a list of $m$ vectors $x^i=(x^i_t,\textbf{x}^i)\in\mathbb{R}^{n+1}$, $i\in \mathbb{Z}_m$ and two disjoint sets of vector pairs $A,B\subset \mathbb{Z}_m\times\mathbb{Z}_m$ as well as a set $C\...