Questions on proving, manipulating and applying inequalities.

learn more… | top users | synonyms (1)

0
votes
1answer
22 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? $$\left\lvert\underbrace{\sin(\sin( \cdots \sin}_{N\text{ times}}(x)\cdots))\right\rvert\le\left\lvert\underbrace{\sin(\sin( \cdots \sin}_{N-1\text{ ...
0
votes
3answers
38 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
23 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
82 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
47 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 ...
8
votes
0answers
48 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
36 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
84 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
22 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
26 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
41 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
81 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
54 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
16 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
49 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
34 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, ...
2
votes
0answers
28 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
29 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
160 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
52 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 ...
1
vote
2answers
70 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
70 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 $$ f'(x) \ge g'(x)$$ ...
0
votes
1answer
34 views

Name of the inequality $|x|+|y| \geq |x+y|$?

What is the name of the inequality $|x|+|y| \geq |x+y|$? I remember seeing this inequality and thinking it was the triangle inequality, but that only holds if $x,y,z$ are the side lengths of a ...
5
votes
7answers
203 views

Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis

I studying in Real Analysis 2, but I have no idea how to solve this problem. My guess is to use Mean Value Theorem or a similar theorem? Could any one help me? Thanks.
1
vote
1answer
64 views

Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far: induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates ...
4
votes
0answers
78 views

Prove that $a+b^2+c^3+d^4 \ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$ [duplicate]

If $0 < a \le b \le c \le d$ and $abcd = 1$ prove that $$a+b^2+c^3+d^4 \ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$$ I first thought of multiplying both sides with ...
2
votes
0answers
33 views

Prove that $(n-1)!S_m\geq (n-m)!m!P_m.$

If $a_1, a_2,\cdots a_n$ be all positive rationals such that $S_n=a_1^m+a_2^m+\cdots +a_n^m$, $P_m=\sum a_1a_2\cdots a_m$ (the sum of products m taken m at a time). Prove that $$(n-1)!S_m\geq ...
-5
votes
2answers
218 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
2
votes
6answers
91 views

Prove $\frac{1}{3}(x+y+z)^2 \geq xy + yz + xz.$

Prove that for nonnegative $x,y,z$ that $\frac{1}{3}(x+y+z)^2 \geq xy + yz + xz.$ I saw this result in a problem but didn't know how to prove it. I tried expanding and collecting to get the ...
0
votes
1answer
31 views

Estimate $|f(x)| \le \frac C{|x|^3}$

Let $$f(x) = \frac{\sin x}x+\frac{\sin(x-1)}{2(x-1)}+\frac{\sin(x+1)}{2(x+1)}.$$ Find the common denominator and use common trigonometric identities to establish that $$|f(x)| \le \frac ...
0
votes
0answers
11 views

Norm and Inner Product Inequality in Hilbert spaces

Let $H$ be a Hilbert space, and suppose that $C \subset H$ is closed, convex and nonempty. Then, for $y_{j}=P_{C}(x_{j})$, $j=1,2$ where $P_{C}$ is the metric projection onto $C$ and $x_{1},x_{2} \in ...
1
vote
0answers
27 views

Criteria for inequality

I am working with an inequality and I need to prove something of the shape $$c\cdot a+d\cdot b \leq a\cdot b$$ The numbers $a$ and $b$ have a specific form, but for the $c$ and $d$ I only know that ...
3
votes
0answers
19 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq ...
0
votes
0answers
9 views

Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...