Questions on proving, manipulating and applying inequalities.

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0
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0answers
20 views

Rank of product of two rectangular matrices

Given $A_{m \times n}$ matrix with rank $m$, and $B_{n \times p}$ matrix with rank $p$, where $n > p \geq m$. We know that $$ \operatorname{rank}(AB) \leq ...
3
votes
2answers
130 views

How to apply the Gronwall lemma

Consider $x'=f(x)$ such that $(x_1,x_2)\mapsto(-x_1+2x_2,-2x_1-x_2)$. Show that for two solutions $x(t)$ and $y(t)$ of the above differential equation, we have: $$\lVert x(t)-y(t)\rVert \leq ...
1
vote
0answers
33 views

“Transference” argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
3
votes
2answers
72 views

Proof of multivariate regression plane maximizes correlation in normals

I am doing a homework sheet as practice for an upcoming course in multivariate statistics and been stuck on the following problem: Let ...
-1
votes
1answer
50 views

The choice of scalar factors in the proof of the Schwarz inequality

In this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for $r$ and ...
1
vote
1answer
27 views

$f(x) = x^{p}(1-x)^{q}$ for all $x\in \left[0,1\right]\;,$ Where $p,q\in \mathbb{Z^{+}}$, Then Max. of $f(x)$ at $x=$

The function $f(x) = x^{p}(1-x)^{q}$ for all $x\in \left[0,1\right]\;,$ Where $p,q$ are positive integers, has maximum value for $x=$ $\bf{Using\; Derivative}$ Let $$f(x) = ...
1
vote
4answers
66 views

Prove that $1^2 + 2^2 + … + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + … + n^2$

I'm having trouble on starting this induction problem. The question simply reads : prove the following using induction: $$1^{2} + 2^{2} + ...... + (n-1)^{2} < \frac{n^3}{3} < 1^{2} + 2^{2} + ...
2
votes
2answers
53 views

If $a,b,c,d,e,f$ are non negative real numbers such that $a+b+c+d+e+f=1$, then find maximum value of $ab+bc+cd+de+ef$

$(a+b+c+d+e+f)^2=$ sum of square of each number (X)+ $2($ sum of product of two numbers (Y) $)$ $ab+bc+cd+de+ef \le Y$ since all are positive. Therefore $1\ge X+(ab+bc+cd+de+ef)$ Edit: From AM GM ...
3
votes
1answer
76 views

Difficulty to prove this inequality in Binomial Coefficient.

This inequality is found in a book titled as Randomized Algorithms, by Rajeev Motwani and Prabhakar Raghavan, in Chapter 3, during explaining Occupancy Problems, to see the book click here PP. 43-44 ...
4
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1answer
64 views

Find the maximum and minimum of $\sum \limits_{i=1}^n x_i ^3$

Let $x_1,x_2, \dots ,x_n$ be a sequence of integers such that $i) -1\le x_i\le 2$ for $i=1,2,\dots,n$ $ii)x_1+x_2+\dots+x_n=19$ $iii){x_1}^2+{x_2}^2+\dots +{x_n}^2=99$ Determine the minimum and ...
30
votes
1answer
691 views
+50

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
3
votes
0answers
36 views

Matrix product bound

Consider the following inequality \begin{align*} AB^{-1}A^\top \preceq cI \end{align*} where $A\in\mathbb{R}^{n\times m}$, $B\in\mathbb{R}^{m\times m}$, $c\in\mathbb{R}$ (given), and $I$ is the ...
0
votes
2answers
51 views

generalized Cauchy-Schwarz inequality

How to prove $A'B(B'B)^{-1}B'A \leq A'A$, where $A$,$B$ are $n\times k$ matrices and $B'B$ is assumed to be positive definite? I don't see why it is a Cauchy-Schwarz inequality.
1
vote
3answers
36 views

Prove that from the equalities, $\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$ follows $x^yy^x=y^zz^y=z^xx^z$.

Problem : Prove that from the equalities, $$\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$$ follows $$x^yy^x=y^zz^y=z^xx^z$$. My approach : $$\frac{x(y+z-x)}{\log ...
0
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0answers
22 views

Upper bound for incomlete Gamma function

It is well-known, that for real arguments $a \geq 0$ and $x \geq 0$ the upper incomplete Gamma function $$\Gamma(a,x) = \int_x^\infty e^{-t} t^{a-1} \, \mathrm{d} t$$ behaves for sufficiently large ...
0
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0answers
25 views

Is there any smart way to check triangle inequality for a matrix?

Here is the description of the problem: We have a matrix with: all (i,i) cells are 0; Some cells are filled with certain number while others are left blank. Now, we want to fill the blanks with ...
10
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0answers
136 views
+100

cyclic three variable inequality

Let $a,b,c$ be nonnegative real numbers and $a+b+c=3$. Prove the inequality $$ \sqrt{24a^2b+25}+\sqrt{24b^2c+25}+\sqrt{24c^2a+25}\le 21 $$ I have tried to find the solution using classical ...
0
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1answer
31 views

What exactly does this inequality do?

I this paper which is titled "KSVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation", in the section about "kmeans algorithm for vector quantization", there is the ...
0
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0answers
31 views

What is (if there is) the generic term for equalities and inequalities

I'm writing a text about a particular linear programming (LP)I optimization problem, that is described using a mixture of inequalities (, ...
0
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1answer
13 views

An inequality for power of positive functions

Let $f,g,h$ be positive real vlaued functions on a finite set $\mathbb{X}$. Let $p >1$. I am wondering whether the following should be true? $$\sum_{x\in ...
6
votes
4answers
101 views

Showing that $\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x$

Show that $$\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x,\forall x\in \left ( 0;\frac\pi2 \right )$$ Firstly, I had use the differentiation of $f(x)=\left (\frac{\sin x}{x} \right )^3- ...
1
vote
2answers
55 views

Can you verify this inequality $\binom {m^2} {m-1} \geq m^{m-1} \geq 2^{n/2}/n$

$N \geq \binom {m^2} {m-1} \geq m^{m-1} \geq 2^{n/2}/n$, given $n = 2 m\log m$. Can you prove it? Where N is the number of subfunction. This question is part of proof on finding lower bound on the ...
3
votes
2answers
273 views

Inequality, Vojtěch Jarník Competition 2006

This is the problem from Vojtěch Jarník Competition 2006. Given real numbers $0=x_1,x_2<\dots<x_{2n}<x_{2n+1}=1$ such that $x_{i+1}-x_{i}\leq h$ for $1\leq i \leq 2n$, show that ...
2
votes
5answers
158 views

Proving that $\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}$ [closed]

Proving that $$\sin x > \frac{(\pi^{2}-x^{2})x}{\pi^{2}+x^{2}}, \qquad\forall x>\pi$$
0
votes
1answer
29 views

Why is the integral of a square always larger than the square of an integral?

I learned in physics that $\langle x^2 \rangle - \langle x \rangle ^2 = \sigma_x^2 \ge 0$ and thus $\langle x^2 \rangle \ge \langle x \rangle ^2$. In the case of continuous distribution, it becomes ...
0
votes
2answers
37 views

Is it true that $|f(x)|\leq |f^2(x)|$?

Is the following true for all $x\in\mathbb{R}$ and for all real functions f? $$\left| f(x)\right| \leq \left| f^2(x)\right|$$ Also, is it true that $|f(x)|\leq |f^3(x)|$?
5
votes
2answers
205 views

If $f(k)=\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{\infty}\dfrac{1}{t^2}$ then $f(k+1)>f(k)$

Let $$f(k)=\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{\infty}\dfrac{1}{t^2}.$$ Prove $$f(k+1)>f(k).$$ My idea: ...
3
votes
3answers
81 views

Proving an inequality between $\frac 1{n+1}$ and $\frac 1n$ and a definite integral

For all natural numbers $n$, prove that $$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$$ I have tried working with $\frac 1{t+1} < \frac 1t < \frac 1{t-1}$ but this doesn't ...
-3
votes
0answers
39 views

$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$ [closed]

For all natural numbers $n$, prove that $$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$$ (Do not use induction.) Please help me on the first step. :)
5
votes
2answers
413 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
4
votes
0answers
58 views

Prove $\cos(\sin x)>\sin(\cos x)$ [duplicate]

Prove that $\cos( \sin x)>\sin(\cos x), \forall x\in\mathbb{R}$. I have thought that we should consider their difference and show it is positive for all x, so: Let $$A=\cos\sin x-\sin\cos ...
4
votes
2answers
124 views

A singular Gronwall inequality

Let $f : [0,T] \to R^+$ be a continuous function such that $f(0)=0 $ and : $$ f(t)\le C\int_0^t s^{-1}f(s) ds,\; \forall t\in [0,T] $$ for some constant $C>0.$ Is it true that $f(t)=0,\; \forall ...
22
votes
13answers
763 views

Which is greater, $98^{99} $ or $ 99^{98}$? [duplicate]

Which is greater, $98^{99} $ or $ 99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an ...
0
votes
2answers
41 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
3
votes
4answers
84 views

Show $\frac{\sin(x)}{x}>\cos(x)$ for $0<x<\pi$ using the Mean Value Theorem

I'm trying to show the inequality $$\frac{\sin(x)}{x}>\cos(x)$$ by for $0<x<\pi$ using the Mean Value Theorem, but I don't know how to start. I can show that $\sin(x)<x$, but I can't see ...
0
votes
0answers
21 views

helping inequality for cyclic three variable inequality

Let $a\ge b \ge c\ge 0$ be reals and $a+b+c=3$ .Then prove $$c(24a^2b+25)(b^2+ac)+50b(a^2+c^2)+5bc^2\le 200+3b^2c^4$$ this one has a proof replacing $b=3-a-c$ and then using calculus but uggly ...
4
votes
2answers
49 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
0
votes
2answers
29 views

Proving inequality between $p$-norm and $q$-norm

How does one show that for $1\leq p<q<\infty$, and $x_i\geq 0$, $(\sum_{i=1}^n x_i^p)^{1/p}\leq n^{1/p-1/q}(\sum_{i=1}^n x_i^q)^{1/q}$?
22
votes
4answers
4k views

Purely “algebraic” proof of Young's Inequality

Young's inequality states that if $a, b \geq 0$, $p, q > 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$). Back when I ...
6
votes
1answer
1k views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and ...
2
votes
1answer
20 views

Log-determinant ordering for sum of positive definite symmetric matrices

If, for real positive definite symmetric $A, B, C$, $$\log\det (A+B) \geq \log\det(A+C)$$ then can it be said that $$\log\det(B) \geq \log\det(C)?$$ NOTE: A crude form of the reverse is certainly ...
0
votes
1answer
41 views

An inequality $a_1\leq a_2\leq a_4 , a_1\leq a_3\leq a_4$

$a_1 , a_2, a_3 , a_4 , b_1 , b_2 , b_3 , b_4\in\Bbb R , p\in(0, 1)$. $a_1\leq a_2\leq a_4 , a_1\leq a_3\leq a_4 , b_1\leq b_2\leq b_4 , b_1\leq b_3\leq b_4 $. Show that $$ ...
1
vote
1answer
31 views

Inequality for the gradient of a power of absolute value

Let $U \subset \mathbb{R}^2$ be open, and let $f : U \to \mathbb{C}$ be a smooth complex-valued function which does not vanish anywhere on $U$. Let $r > 0$ be a real constant. Does the ...
1
vote
1answer
13 views

Are binomial coefficients with fixed “denominator” log-concave?

I'm working on a problem and began suspecting that the following inequality holds. Let $k\in\mathbb{N}$ be fixed, and define $f(n)={n\choose k}$. Then $f(n)$ is log-concave in $n$, in particular if ...
2
votes
1answer
104 views

Inequality deduced from martinagles

Let $c$ be a positive real constant, and let $x_i,i \in \{1,2,...,n\}$ be real numbers such that $$ |x_i|\le c,\forall i \in \{1,2,...,n\}.$$ Let $p_i,i \in \{1,2,...,n\}$ be positive reals ...
1
vote
2answers
26 views

Create some new numbers using $n$ arbitrary positive real numbers

Known facts: Let $a_i$, $b_i$, $i=1, \ldots, n$ be positive real numbers such that $a_1+ \cdots + a_n = a_1b_1 + \cdots +a_nb_n = 1$. Then $$b_1^{a_1}b_2^{a_2} \cdots b_n^{a_n} \leq 1.$$ ...
-1
votes
2answers
49 views

Rank of the product of two full rank matrices

I have searched for the above topic and found some results, but the answer I am looking for is not found anywhere. Here is my question: Given $A_{m \times n}$ matrix with rank $m$, and $B_{n ...
2
votes
7answers
106 views

How is $\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \leq \frac{\sqrt{|xy|}}{\sqrt{2}}$

$$\left|\frac{xy}{\sqrt{x^2+y^2}}\right| \leq \frac{\sqrt{|xy|}}{\sqrt{2}}$$ Does this apply in general, because here in this example i have $x \to 0, y \to 0$. Some inequality is used I believe to ...
2
votes
2answers
36 views

Integral values satisfying a inequality

Consider the following inequality : $$\frac{x^2+a^2}{a(4+x)} \ge 1$$ I am trying to find the range of integral values of $a$ for which this inequality holds for all $x$ belongs to $(-1,1)$ I ...
2
votes
1answer
59 views

Chernoff-like bound for small intervals in tail distribution

I am searching for a Chernoff-like bound that controls the probability of small intervals in the tail distribution. More specifically, let $X_1, \ldots, X_n$ be independent random variables with ...