Questions on proving, manipulating and applying inequalities.

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4
votes
0answers
20 views

Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
0
votes
1answer
15 views

L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of ...
0
votes
0answers
25 views

Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
0
votes
0answers
22 views

In order to show or refute: Given a real function $f$ and $a, b \in R$ then $a\leq b \Rightarrow f(a)\leq f(b)$, what should I regard?

Is it enough to show that $f$ is increasing or decreasing in any interval $I$ that contains both numbers $a$ and $b$?
1
vote
1answer
106 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
0
votes
0answers
8 views

To clear for variable 'a' in a sum of dependent products

I can't seem to find a way to clear this equation for variable $a$: $E[k] = \displaystyle\sum_{k=1}^nk\frac{a}{n+a-k}\displaystyle\prod_{i=0}^{k-1}1-\frac{a}{n+a-i}$ Do you think it's possible? Any ...
0
votes
2answers
40 views

Inequality involving exponential function (base $2$ and $3$) [on hold]

Show that the following inequality holds for every real number x: $$3^x+0.5>2^x$$
1
vote
1answer
23 views

L2 Matrix Norm Upper Bound in terms of Bounds of its Column

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1} $ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a ...
-3
votes
2answers
81 views

Investigating Nicolas' criterion for the Riemann Hypothesis. [on hold]

Throughout this note, $N_k$ denotes the $k$-th primorial number (the product of the first $k$ primes), $\varphi(n)$ the Euler totient function, and $\gamma$ is the Euler-Mascheroni constant. By the ...
1
vote
1answer
23 views

Proof for $x\le -1 \implies x^3-x\le 0$?

Here is my proof: Let $x\in \mathbb{R}$, assume $x\le -1$ Then $x^2\ge 1$ Then $x^3\le -1$ Since $x\le -1$ $x^3\le x$ Then $x^3-x\le 0$ Therefore $x\le -1 \implies x^3-x\le 0$ Therefore ...
6
votes
1answer
63 views

Basic question $|x^2| < 9$

I have a rather basic question. Let's assume that $|x^2| < 9$, where $x\in \mathbb{R}$. Then everyone knows that $x \in$ (-3,3). However, I have trouble arriving at the answer based on basic ...
2
votes
3answers
48 views

$a+\frac{1}{a}\ge 2$ for $a\in\mathbb{R}_{+}$

This inequality is more than obvious: $$a+\frac{1}{a}\ge 2 $$ But my question is: is this only a special case of some "bigger" lemma (like e.g. $\frac{a+b}{2}\ge\sqrt{ab}$ is a special ase of the ...
2
votes
2answers
65 views

$|\sin(\sin( \cdots \sin(x)\cdots))|$ ($N$ times) is always $\leq|\sin( \cdots \sin(1)\cdots)|$ ($N-1$ times)

Is this inequality always true? $$ \bigl\lvert\,\underbrace{\sin(\sin(\cdots \sin}_{N\text{ times}}(x)\cdots))\bigr\rvert\le\bigl\lvert\,\underbrace{\sin(\sin( \cdots \sin}_{N-1\text{ ...
0
votes
3answers
45 views

Prove that $\frac{a^2}{b}+\frac{c^2}{d}\ge\frac{(a+c)^2}{b+d}$

I'm looking for hints, not for a complete solution: prove that for $a,b,c,d\in\mathbb{R}_{+}$ the following inequality holds: $\frac{a^2}{b}+\frac{c^2}{d}\ge\frac{(a+c)^2}{b+d}$
0
votes
0answers
26 views

Which of the following sets is compact, bounded, closed or open and why? [on hold]

Which of the following sets is compact, bounded, closed or open and why? $M1= [-1,42]$ $M2= (-1,42]$ $M3= (-1,42)$ $M4= (-\infty, +\infty)$ $M5= \{z \in \mathbb C: 0 < \operatorname{Re} z + ...
1
vote
0answers
32 views

Prove that: $(x_1+…+x_k)^2\leq 2(x_1^2+…+x_k^2)$. [duplicate]

Prove that: $$(x_1+...+x_k)^2\leq 2(x_1^2+...+x_k^2)$$ for all $x_1$, ... $x_k\in\mathbb R$. Is it also true that $$\left|x_1+...+x_k\right|^2\leq 2(|x_1|^2+...+|x_k|^2)$$ for all $x_1$, ... ...
2
votes
3answers
84 views

Does $\sqrt{a+b} \le \sqrt a + \sqrt b$ hold for all positive real numbers a and b?

I thought of this a while ago, but can't make up a proof or a counterexample. Does anyone know more about this? $$\sqrt{a+b} \le \sqrt a + \sqrt b , \forall a,b \in \mathbb R_+$$ Moreover, what ...
4
votes
2answers
48 views

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$? Because of the nature of the square root function, its derivative monotonically decreases. so ...
9
votes
0answers
56 views

A curious triangle inequality

Let $ABC$ be a triangle. Pick a point $P$ inside the triangle. How would you show that \begin{equation} |PA|+|PB|+|PC|+\min\{|PA|,|PB|,|PC|\}\leq |AB|+|BC|+|CA|. \end{equation}
0
votes
2answers
37 views

What is my mistake

Spot my mistake: $$\frac{\left(\text{P}_1+\text{P}_2+\dots+\text{P}_n\right)-\left(\text{Z}_1+\text{Z}_2+\dots+\text{Z}_n\right)}{n-m}\le-\ln(50)$$ ...
0
votes
1answer
40 views

$1+xy+yz+xz-x-y-z>0$ where $x,y,z \in (0,1)$

$f(x,y,z)=1+xy+yz+xz-x-y-z$, where $x,y,z \in (0,1)$. Show that: $f(x,y,z)>0$. $\begin{equation} \begin{cases} \frac{\partial f}{\partial x}=y+z-1=0 \\ \frac{\partial f}{\partial y}=x+z-1=0 ...
6
votes
1answer
110 views

Is there a series to show $22\pi^4>2143\,$?

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{(2k+2)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, when ...
1
vote
1answer
15 views

Trouble with an inequality between magnitudes of complex numbers

We are supposed to show that $$|ab^* + a^*b| \leq 2|ab|$$ where a and ba re complex numbers and a* and b* are their respective conjugates (so $a = x_1+iy_1$, $a^* = x_1-iy_1$, $b = x_2+iy_2$, $b^* = ...
2
votes
0answers
24 views

Choosing three integers to satisfy an equation under a specific condition

Find three integers $(a,b,c)$ such that: $x*a + y*b + z*c = a + b$ only when $x = 1, y = 1, z = 0$ where $x, y$ and $z$ can be chosen as any non-negative integers. For example, choosing $a = 1$; $b = ...
2
votes
1answer
27 views

Proof of Cauchy-Schwarz Inequality 1

In my lecture notes I've written the proof of Cauchy-Schwarz inequality as: Let t $\in$ R and $\langle x+ty, x+ty\rangle \geq 0$, then $\langle x+ty, x+ty\rangle $ = $\langle x, x+ty \rangle + ...
0
votes
3answers
43 views

How do you prove $\frac{u}{v} < \frac{z}{w} \implies \frac{u+z}{v+w} < \frac{z}{w}$

The bounds for the variables are $\forall u,v,w,z \in \mathbb{R}^+$ What I've got so far: $\frac{u}{v} < \frac{z}{w}$ $\frac{u}{v+w} < \frac{z}{w}$ I'm not sure where to go from here...
2
votes
4answers
82 views

Prove that $a+\frac{1}{b}>2$ or $b+\frac{1}{a}>2$ for two strict positive numbers

Another Olympiad Problem, let $x$ and $a$ and $b$ be strictly real positive numbers. Prove that $x$+$\frac{1}{x}$$>$$2$ (proven) Than conclude that $a$+$\frac{1}{b}$$>$$2$ or ...
1
vote
1answer
56 views

prove the inequality $0< \frac{1}{m}+\frac{1}{n}+\frac{1}{p}< \frac{47}{60}$

I have an Olympiad Problem, let $m$, $n$ and $p$ denote three natural numbers where: $$m>n>p>2$$ prove that : $$0< \frac{1}{m}+\frac{1}{n}+\frac{1}{p}< \frac{47}{60}$$ I've been ...
1
vote
0answers
15 views

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. [duplicate]

Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$. I'm having real trouble proving this inequality. I'd greatly appreciate any help.
0
votes
0answers
17 views

Solving $f(x) \leq 10 f(kx) + 10kg(x)$ for $f, g$ nonnegative on $(0, 1]$

Suppose we are given two nonnegative functions $f$ and $g$ on $(0,1]$ that satisfy $f(x) \leq x^{-1/2}$ and $$f(x) \leq 10 f(kx) + 10kg(x)$$ for all $k$ sufficiently large. Is it possible to reduce ...
1
vote
1answer
44 views

inequality with a positive matrix

Let $$ A=\left[ \begin{array}{cc} a & b\\ \overline{b} & c\\ \end{array} \right]$$ be a positive semi-definite positive of $M_2(\mathbb{C})$. How prove the inequality $ac \geq ...
-1
votes
1answer
58 views

Do there exist $a,b,c,d,e,f$ such that $ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1$ and…

Do there exist $a,b,c,d,e,f$ satisfying: \begin{cases} ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1\\ a+b+c+d+e+f \le 1\\ a+d+f \le 0\\ b+e+f \le 0\\ f\le 0 \end{cases}? ...
0
votes
2answers
35 views

How to prove if $5/2 < x < (5/4)(1+\sqrt2)$, then $25/(x(2x-5)\ge 8$

if $\frac52 < x < \frac54(1+\sqrt2)$, then $\frac{25}{x(2x-5)} \ge 8$ First I unpacked the conclusion to: $$ 16w^2-40w-25 \le 0 $$ I attempted to solve by manipulating the interval (squaring, ...
3
votes
0answers
35 views

Proving Holder's inequality for Schatten norms

Sticking to the finite dimensional case, Holder's inequality for Schatten norms is given by $$\left\|AB\right\|_{S^1}\leq\left\|A\right\|_{S^p}\left\|B\right\|_{S^q}$$ for $A,B$ $n\times n$ ...
0
votes
1answer
32 views

Optimizing the area of a rectangle with one side against a wall using the am-gm inequality

Given 300 meters of fence, how can I find the dimensions of a rectangle that is built against a wall the encloses the maximum area. I found this question in a calculus book and saw a simple solution ...
0
votes
1answer
23 views

Is the following inequality true? $\sup\limits_{2T\leq t\leq 4T}f(t)\leq \sup\limits_{2T\leq t\leq 3T}f(t).\sup\limits_{3T\leq t\leq 4T}f(t)$

$\sup\limits_{2T\leq t\leq 4T}f(t)\leq \sup\limits_{2T\leq t\leq 3T}f(t).\sup\limits_{3T\leq t\leq 4T}f(t)$
3
votes
1answer
44 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
-2
votes
2answers
89 views

Prove or disprove that $(a_1+a_2+\ldots+a_n)\leq n\sqrt{a_1^2+\ldots+a_n^2}$, by showing that $RHS-LHS\geq 0$ if possible. [on hold]

Prove or disprove that $$\left|a_1\right|+\left|a_2\right|+\ldots+\left|a_n\right|\leq n\sqrt{a_1^2+\ldots+a_n^2}$$ Where $a_1,\ldots,a_n\in\mathbb{R}$ and $n\in\mathbb{N}$. EDIT: I was hoping there ...
5
votes
0answers
174 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$. I. $\pi$ and $\pi^2$ There are series with all terms positive for ...
2
votes
1answer
32 views

Finding the maximum value of a divergent series [on hold]

I came across this divergent sum- $$\sum_{n=1}^\infty\frac{1}{n+1}$$ Now,a divergent sum does not a limit.So is it possible to get a maximum value for the sum or more specifically prove that ...
2
votes
0answers
54 views

I am trying to show an inequality involving the product of three inner product terms

Define the inner product $\langle\cdot,\cdot\rangle$ for continuous functions defined on $[0,1]$ as: $$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$ where $\rho$ is a real number. I ...
1
vote
1answer
49 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
0
votes
4answers
61 views

Proof that $|x|+|y|\leq\sqrt{2(x^2+y^2)}$

How do I prove that for $x,y\in\mathbb{R}$ we have $|x|+|y|\leq\sqrt{2(x^2+y^2)}$? I thought that $(|x|+|y|)^2=x^2+y^2+2|x||y|\leq2(x^2+y^2)$, but I'm not sure why that holds.
0
votes
2answers
34 views

If $\left| x \right| \ge \left| y \right|$ then Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?

Let $x,y\in \mathbb{R}$ and $\left| x \right| \ge \left| y \right|$. Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?
1
vote
3answers
48 views

Cauchy like inequality $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$

Problem: Prove that for real $x, y, \alpha, \beta$, $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$. I am looking for an elegant (non-bashy) ...
4
votes
2answers
61 views

Finding the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$

If $x,y \in \mathbb {R}$, find the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$ and $xy>0$. This problem was inspired by a problem which asked if $x,y \in \mathbb {R}$ and $xy \neq ...
2
votes
3answers
77 views

Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.

I am having trouble with my homework problem, it says: Suppose that $n$ is a positive integer and that $x > 0$. Show that $$\left(1+\frac{x}{n}\right)^n < e^x.$$ I have proved the base ...
1
vote
7answers
72 views

Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction?

I have the inequality $\frac{n}{n+1} < \frac{n+1}{n+2}$ I'm not sure how to go about proving it. I've started by testing with n = 1, which results in $\frac{1}{2} < \frac{2}{3}$ which is true ...
1
vote
3answers
56 views

proving an inequality related to $AM\ge GM$

$$a^2+ab+b^2\ge 3(a+b-1)$$ $a,b$ are real numbers using $AM\ge GM$ I proved that $$a^2+b^2+ab\ge 3ab$$ $$(a^2+b^2+ab)/3\ge 3ab$$ how do I prove that $$3ab\ge 3(a+b-1)$$ if I'm able to prove the ...
1
vote
3answers
63 views

Prove $\ln x \ge \frac{x-1}{x}$

Prove that for every $x>0$: $$\ln x \ge \frac{x-1}{x}$$ What I did: $$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$ $$f(1) = g(1) = 0 $$ So it's enough to prove that $$ f'(x) \ge g'(x)$$ ...