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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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1answer
17 views

prove by induction that 2^n/n! < 4/n

How do I do this? I know how to do the base case but I can't figure out how to do the next steps.
0
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1answer
25 views

Finding the Minimum Value of Sum of Positive Real Numbers

I'm not sure how to solve the following problem: $$d_1^2 + \ldots + d_n^2 = \sigma^2$$ $$d_i \geq 0$$ Find the minimum possible value of $$d_1 + \ldots + d_n$$ I have a hunch that its when every ...
0
votes
0answers
15 views

A question regarding inequality between weighted averages

Suppose you have three probability density functions $p_{1}(x)$, $p_{2}(x)$, and $p_{3}(x)$. Suppose further that the expectation values of $p_{1}(x)$ and $p_{2}(x)$ are unequal $\int_{0}^{1}p_{1}(x) ...
1
vote
1answer
18 views

How to use trace operator for inequalities dealing with an hermitian matrix and its inverse?

When I read a paper, I met these implications involving inequalities : $$R-a^Ha\ge0 \ \implies \ I-R^{-1/2}aa^HR^{-1/2}\ge0 \ \implies \ 1-a^HR^{-1}a\ge0$$ $R$ is an invertible Hermitian matrix ...
-1
votes
1answer
24 views

How to prove this inequality? It is true or false? [on hold]

I'm trying to prove this statement: Let $ \alpha, t >0 $, then there exists a positive constant $ c $ such that $$ \left( 1+\frac{t}{2}\right)^{-\alpha} \leq c \left( 1+t\right)^{-\alpha}. $$ Is ...
0
votes
1answer
15 views

Showing existance of tournament without transitive subtournament

Show that there exists a tournament on $n$ vertices that does not contain a transitive tournament on $2\log_2n+2$ vertices. My attempt: The number of tournaments of $n$ vertices is $2^{\binom{n}{2}}$...
1
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1answer
18 views

Proving $|E| \leq \frac{1}{2}(\sqrt{t-1}\;n^{3/2} + n$) in a graph with no copies of $K_{2,t}$

Show that if an $n$-vertex graph $G=(V,E)$ has no copy of $K_{2,t}$ then: $$|E| \leq \frac{1}{2}(\sqrt{t-1}\;n^{3/2} + n)$$ I know how to prove it for $n=2: \;$ Dentoe $|E|=m,$ $d(v)$ the degree of $...
1
vote
0answers
53 views

Proving $a^{ab}+b^{ab}\leq a^{a^2}+b^{b^2}$ and an identity .

I was working on this Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$ when I have discovered the following identity : $$\Bigg|\Big(\frac{1}{2}\Big)^{\frac{x}{8}}\pm\Big(\frac{1}{4}\Big)^{\frac{x}{8}}...
0
votes
2answers
36 views

Prove the inequality $x^2+4y^2<1$ when $x-y=x^3+y^3$

I have to prove the following inequality: $$ x^2+4y^2<1$$ with this constrain $$x-y=x^3+y^3$$ and where $x$ and $y$ are positive real numbers. From the constrain we have also $0<x<1$. I ...
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0answers
14 views

$\|f\|_{L^{\frac{10}{3}}(\mathbb T^3)} \leq C \|f\|_{H^1(\mathbb T^3)}^{3/5} \|f\|_{L^2(\mathbb T^3)}^{1-(3/5)}$? [on hold]

Let $\mathbb T^3$ be 3-dimensional torus. Can we expect that $\|f\|_{L^{\frac{10}{3}}(\mathbb T^3)} \leq C \|f\|_{H^1(\mathbb T^3)}^{3/5} \|f\|_{L^2(\mathbb T^3)}^{1-(3/5)}$?
0
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2answers
25 views

The number of integers 'n' which satisfy the inequality $(n^2 - 2)(n^2 - 20) < 0$ is? [on hold]

Tried hit and trail method . And not got the answer . Don't know anything about going forward onto the question .
0
votes
1answer
38 views

For which $a \in \mathbb{R}$ is $x^2 + 4|x-a| - a^2 \geq 0$ true for all $x \in \mathbb{R}$? [on hold]

As the title says, I have a problem answering the question: for which $a \in \mathbb{R}$ is $$ x^2 + 4|x-a| - a^2 \geq 0 $$ true for all $x \in \mathbb{R}$?
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votes
1answer
39 views

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$ $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to?

If $\sin(x)+\sin(y)\ge \cos(\alpha) \times \cos(x)$, $\forall x\in \mathbb R$, then $\sin(y)+\cos(\alpha)$ is equal to ? My thinking:- I have break the left hand side on $sinC + sinD$ and right hand ...
0
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0answers
24 views

How to get inequality ‎$‎0‎\leq ‎\gamma‎_g + ‎log ‎g(1)\leq ‎\frac{g^‎\prime_{-}(1) + g^‎\prime_{+}(1) ‎}{2g(1)}‎$‎?

‎Let ‎‎$‎g:‎\mathbb{R^+}‎‎‎\rightarrow‎‎\mathbb{R^+}‎$ ‎be a‎ ‎real‎ ‎function ‎such ‎that ‎‎$\log g(x)$ ‎is ‎concave and ‎$‎\displaystyle{\lim_{x\to\infty}}‎\frac{g(x+w)}{g(x)} = 1‎$ ‎for each ‎$‎w&...
0
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1answer
25 views

A complex integral with boundary on the norm

I was asked this question. I couldn't even manage to find a starting point on the question. Any help is welcomed. P.S: I know it is from a book but I do not know from which one. If you have any idea ...
0
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2answers
72 views

maximum value of $\sum (a-b)^2$

If $a^2+b^2+c^2=5$ and $a,b,c \in \mathbb{R},$ find the maximum value of $(a-b)^2+(b-c)^2+(c-a)^2$. My Try: $(a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2)-2(ab+bc+ac)$ $$=10-2(ab+bc+ac)$$ Now this ...
0
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1answer
19 views

Algebraic manipulation and inequality

Given real-valued terms $a,b,c \in {R}$ with the following conditions on them: $ 0 \leq a \leq 1 $, $|b|<1$, and $|c|< 1$. And given the terms $ X= \frac{1}{1-( (1-a)b + ac )}$ and $ Y =\frac{1}...
1
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1answer
30 views

$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v \in \{-1,1\}^n }{sup} |\underset{i,j}{\sum} a_{ij}u_iv_j|$?

let $(a)_{ij}$ be a $M\times N$ Matrix with real entries ,is that possible to prove that: for any $x \in [-1,1]^n, y \in [-1,1]^m$ we have: $$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v ...
-1
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0answers
43 views

If (x − 1)(x − 3)(x + 3)(x + 5) < 0, the range of its roots is included by [on hold]

It's an sat 2 question. I don't know where to start with. As the question talks about inequality how can it have a roots and it mention about the range. im my opinion the range of the roots always ...
2
votes
1answer
80 views

Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$

I have to prove the inequality $$ \frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}} $$ when $x^2+y^2=1$, using Cauchy-Schwarz Inequality. The RHS is equal to $\frac{12}...
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votes
1answer
16 views

How do i Solve this Linear Efficiency Problem in math?

I dont understand how to do these, anyway someone could explain to me and show me how to do it? Thankyou The environmental and health department of a country runs two refuse centers. Center 1 costs ...
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0answers
90 views

Questions concerning economics [on hold]

Windies Bats produces two models of cricket bats the Layle and the Gara. They are produced on two separate assembly lines. Producing a Layle requires 2 hours on Line I and 1 hour on Line II, while ...
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1answer
17 views

Space explosion [on hold]

A shell of mass M is at rest at a point in space. It bursts into two fragments with a combined energy of E (ignore radiation losses). 1. Show that the relative speed of the two fragments after ...
0
votes
3answers
28 views

Infern sign of quadratic equation from coefficients

Consider the quadratic function $f(x)= ax^2 + bx + c$ where $x \in \{1,2,\ldots\}$. Presume that $b^2 - 4ac < 0$ holds and $a \neq 0$. We know that there does not exist a $x \in \mathbb R$ such ...
0
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0answers
13 views

Data processing inequality - KL divergence before and after a stochastic matrix

I'm studying the Blahut Arimoto algorithm using these notes and towards the end of section 6, an interesting quantity arises. The author does not talk about how to compute it so I was hoping I could ...
0
votes
1answer
26 views

Bound $|x^TAy|$ in terms of $\|A\|$ and $|x^Ty|$

Under what conditions on a square matrix $A$ of size $n$ do we have $|x^TAy| \le |x^Ty|$ for all $x,y \in \mathbb R^n$ ? Notes The above inequalities hold for $A \in \{0, I\}$, and so by simple ...
1
vote
2answers
67 views

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Let $a,b,c$ be non-negative real numbers such that $c \geq 1$ and that $a+b+c=2$. Find the minimum and maximum values of $$P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$$ To find the minimum of $P$ ...
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votes
1answer
47 views

How is $\log(x) \leq x-1$ for all $x>0$?

sorry if this is a very obvious question, but I'm writing a proof for the Kulback-Leibler inequality and the first step is to state $\log(x) \leq x-1$ for all $x>0$. I get it for $x>1$, but in ...
0
votes
1answer
68 views

Underapproximating the exponential function from below [duplicate]

I think that for positive natural numbers t and n we have $$ \left(1+\frac{n}{t}\right)^t\ \le\ e^n\,. $$ Is this true? I have constructed a proof of it (which would probably take some lengthy ...
0
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0answers
10 views

Inequality bounding $\|Qp\|_2^2$ where $Q$ is a Gaussian matrix and $p$ an independent Gaussian vector

I got the following expression, representing the r.h.s. of a tail bound on $Pr[\|Q^Tp\|_2^2 > tk]$, where the $k\times r$ matrix $Q$ has i.i.d. standard Gaussian entries, the vector $p$ is drawn ...
1
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4answers
73 views

Proving $\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca}$ for positive $a$, $b$, $c$

I'm at the end of an inequality proof that started out complex and I was able to simplify it to: $$\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca} \quad\text{where}\quad a, b, c ...
1
vote
1answer
44 views

If $x_1\geq x_2\geq…\geq x_n\geq0,$ $n\geq2$, $m\geq k\geq1$ then $x_1^mx_2^k+x_2^mx_3^k+…+x_n^mx_1^k\geq x_1^kx_2^m+x_2^kx_3^m+…+x_n^kx_1^m.$

Is the following inequality is true?? If $x_1\geq x_2\geq...\geq x_n\geq0,$ $n\geq2$, $m\geq k\geq1$ then $$x_1^mx_2^k+x_2^mx_3^k+...+x_n^mx_1^k\geq x_1^kx_2^m+x_2^kx_3^m+...+x_n^kx_1^m.$$ My ...
1
vote
1answer
33 views

Inequality in Sobolev Space ($L^p$ norm)

I want to find a constant $C$ that depends on the parameters $a$ and $p$ that satisfies the inequality $$\|f\|_p \leq a\|f'\|_1+C\|f\|_1$$ for all $f \in W^{1,1}(0,1)$. This is for arbitrary $p \in [...
0
votes
1answer
33 views

Proving $\sup_{\Bbb{R}^{n} \times (0,\infty)} |u_{x_{i}}| \leq \sup_{\Bbb{R}^n} |g_{x_{i}}|$ in case of solution of PDE $u(x,t)$?

$g \in C^{1}(\Bbb{R}^{n})$, $g$ and $Dg$ are bounded in $\Bbb{R}^{n}$. $$u(x,t) = \frac{1}{(4\pi t)^{\frac{n}{2}}} \int_{\Bbb{R}^n} e^{-\frac{|x-y|^2}{4t}} g(y) dy$$ $(x,t) \in \Bbb{R}^n \times (0,\...
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votes
0answers
31 views

Function continuous and twice differentiable [duplicate]

f is continously twice differentiable function on real line $(f(x))²\le1$ ,$x$ real $(f'(x))²+(f’’(x))²\le1 $for all real $x$ Prove :$-(f(x))²+(f'(x))²\le1$ for all real $x$
1
vote
1answer
29 views

Bound on tail probability with knowledge of a bound on expected value

I'm trying to solve the following problem: If the random variables $X$, $Y$ satisfy: $$E[(X-a)_+]\leq E[(Y-a)_+],\qquad \forall a\in R$$ and $\forall t>0$ we have: $$P[Y\geq t]\leq ke^{-bt},\...
2
votes
2answers
45 views

$\sqrt{x^2+xy+y^2} + \sqrt{u^2+uv+v^2} \geq \sqrt{(x+u)^2+(x+u)(y+v)+(y+v)^2}$

Let $x,y,u,v \in \mathbb{R}.$ Prove that $$\sqrt{x^2+xy+y^2} + \sqrt{u^2+uv+v^2} \geq \sqrt{(x+u)^2+(x+u)(y+v)+(y+v)^2}$$ Proof 1: $$\sqrt{x^2+xy+y^2} + \sqrt{u^2+uv+v^2} \geq \sqrt{(x+u)^2+(x+u)(y+v)...
0
votes
2answers
31 views

Basic question about convexity

A convex function is defined as one that satisfies the following condition for $p_1 + p_2 = 1$. $$f(p_1x_1 + p_2x_2) \leq p_1f(x_1) + p_2f(x_2),$$ Does this imply that for all $\lambda \leq 1$ $$...
2
votes
1answer
62 views

Let $f:[0,a]\rightarrow[0,\infty)$ be a continuous function such that $f(t)\leq e^{\int_{0}^{t}f(s)ds}-1$ for all $t\in[0,a]$. Prove that $f\equiv0$

Let $f:[0,a]\rightarrow[0,\infty)$ be a continuous function such that $$f(t)\leq e^{\int_{0}^{t}f(s)ds}-1$$ for all $t\in[0,a]$. Prove that $f\equiv0$. I have thought like this: Assume $F(t)=\int_{0}...
1
vote
1answer
52 views

If $3a^2+2b^2=3a+2b$ then find the minimum value of $\sqrt{\frac{a}{b(3a+2)}}+\sqrt{\frac{b}{a(3b+2)}}$

Suppose $a,b$ are positive reals.If $3a^2+2b^2=3a+2b$ then find the minimum value of $\sqrt{\frac{a}{b(3a+2)}}+\sqrt{\frac{b}{a(3b+2)}}$ I feel this can be done by AM-GM.I failed to use that $3a^2+2b^...
0
votes
3answers
44 views

Exponential Equations and Inequalities and logarithms

So this is one kind of task: $2\times 8^{x}-7\times4^{x}+7\times2^{x}-2=0$ And I don't get any of these tasks, so I am asking for some kind of literature with introduction about this and good ...
0
votes
3answers
30 views

Inequality involving absolute values.

I want to ask is whether there is a method to solve following inequality more easily and compactly or it is the only method. $$|x-2|+|x-8|\le x-2$$ What I know is taking $x<2,8>x>2,x>...
1
vote
1answer
75 views

Finding a better bound in an inequality [on hold]

Consider points $(x,y)$ on the curve $\sqrt{x^2-3x}+\sqrt{y^2-3y}=1$. Prove that for all such pairs: $$x^2+y^2\lt2(x+y)+8.$$ NOTE.- This problem was proposed by two mathematicians, from Romania and ...
0
votes
1answer
33 views

Inequality involving log-sum-exp, variance, and mean

Fix $z_1,\ldots,z_n \in \mathbb R$. Let $\mu_n:= mean(z_1,\ldots,z_n):=\frac{1}{n}\sum_{i=1}^n z_i$, $lse_n(z_1,\ldots,z_n)=\log(\sum_{i=1}^n e^{z_i})$, and $\sigma^2_n := variance(z_1,\ldots,z_n):=\...
0
votes
3answers
22 views

Why does this procedure work?

We know that $$c\geq 0 \implies (|a|<c \iff -c<a<c).$$ May I know why does the following procedure $$|x|<2x \iff -2x<x<2x \iff (0<3x)~\wedge~(0<x) \iff x>0$$ work despite $...
2
votes
2answers
71 views

Prove inequality using M.V.T.

For $x>2$, prove that $(x-1)e^{\tfrac{2}{x}}-(x-2)e^{\tfrac{1}{x}}<e$ using the mean value theorem. I can't find the function to applying the mean value theorem. Give some advice or hint. ...
1
vote
1answer
39 views

Intuition for the “Wavy Curve” method for Rational Inequalities

I am studying inequalities and I saw this method by the name of Wavy Curve method. Since it was listed as an algorithm, I wanted to know its mathematical formulation. I could only see two part of this ...
1
vote
1answer
24 views

Lower bound given expectation and standard deviation.

A random variable X with integer values only has mean 3 and standard deviation 2. Under those assumptions, which is the best lower bound for $P[0\leq X \leq 6]?$. By my calculations, it is $\frac{5}{...
4
votes
2answers
75 views

Solving $ 2< x^2 -[x]<5$

How to solve inequalities in which we have quadratic terms and greatest integer function. $$ 2< x^2 -[x]<5$$ [.] is greatest integer function. Do we need to break into the cases as [0,1), [1,...
-1
votes
1answer
34 views

Find an integer $N$ such that $2^n > n^4$ whenever n is an integer greater than N.

Was just hoping to validate my proof: Let $n=16\Rightarrow 2^{16}>16^4\Rightarrow 65536>65536$. Prove $P(n)=2^n>n^4$ whenever $n>16$. Basis Step: $P(17)=2^{17}>17^4\Rightarrow 131072&...