Questions on proving, manipulating and applying inequalities.

learn more… | top users | synonyms (1)

1
vote
0answers
27 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 $x_1$, ... ...
0
votes
3answers
86 views

Norm of vectors inequality [duplicate]

I tried proving this with triangular inequality but i was not right can any one help me with this
4
votes
2answers
45 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 ...
1
vote
2answers
393 views

Upper bound on the entropy of a sum two random variables

Let $X$ be a random variable such that $|X| \leq A$ almost surely, for some $A > 0$. Let $Z$ be independent of $X$ such that $Z \sim {\cal N}(0, N)$. My question is: How large can the entropy ...
2
votes
3answers
81 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 ...
8
votes
0answers
39 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
33 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 ...
3
votes
2answers
563 views

Show that if $A\subseteq B$, then inf $B\leq$ inf $A\leq$ sup $A \leq$ sup $B$

Is my logic correct or am I missing something? Show that if $A\subseteq B$, then inf $B\leq$ inf $A\leq$ sup $A \leq$ sup $B$ Case 1. If $A \subset B$ then there exists $b_1,b_2\in B$ such that ...
30
votes
1answer
664 views
+500

Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$

Let $a$, $b$ and $c$ be non-negative numbers. Prove that: $$a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}.$$ I have a proof, but my proof is very ugly: it's ...
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 + ...
2
votes
2answers
84 views

What is the intuition behind the Cauchy-Schwarz inequality in the real numbers?

The Cauchy-Schwarz inequality states that $$\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$$ The proof, with the discriminant argument, is ...
5
votes
0answers
49 views

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

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{2^6(k+1)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, ...
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
19 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 = ...
0
votes
3answers
39 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
78 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
52 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 ...
4
votes
1answer
196 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
5
votes
0answers
151 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 ...
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
1answer
46 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
0answers
15 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
41 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 ...
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.
2
votes
0answers
26 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 $A,B$ $n\times n$ ...
-5
votes
2answers
211 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
1answer
108 views

Is this a valid proof for $\left | a+b \right | \leq \left |a \right| + \left | b \right | $?

$\left | a+b \right | \leq \left |a \right| + \left | b \right | \Rightarrow$ $\sqrt{{(a+b)}^2} \leq \sqrt{{a}^2} + \sqrt{{b}^2}$ ${(\sqrt{{(a+b)}^2})}^2 \leq ({\sqrt{{a}^2} + \sqrt{{b}^2}})^2 ...
1
vote
4answers
92 views

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$ Proof: $$\begin{align} \left|(|x|-|y|)\right| &\leq|x-y| \\ {\left|\sqrt{x^2}-\sqrt{y^2}\right|}&\leq \sqrt{(x-y)^2} ...
0
votes
2answers
33 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, ...
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 ...
-2
votes
2answers
87 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 ...
1
vote
1answer
48 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 ...
3
votes
1answer
39 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 ...
0
votes
1answer
23 views
0
votes
2answers
1k views

Combining inequalities into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
11
votes
5answers
284 views

This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$

let $0<a\le b\le c\le d$, and such $abcd=1$,show that $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$ it seems harder than This inequality $a+b^2+c^3\ge ...
2
votes
5answers
122 views

How do I prove $\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$

How do I prove the following inequality $$\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$$ without the help of induction? Thanks for any help!!
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 ...
2
votes
5answers
396 views

Proving $\pi^3 \gt 31$

$$\large \pi^3 \gt 31$$ Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand. It is extremely easy with the use of any calculator, so I was wondering ...
2
votes
3answers
65 views

Proving the inequality $ \frac {x+y}{x^2+y^2}\leq \frac 12 \left(\frac {1}{x}+\frac{1}{y}\right)$

Let x and y be definitely two positive numbers : Prove that : $$ \left( \frac {x+y}{x^2+y^2}\right) \leq \frac 12 \left(\frac {1}{x}+\frac{1}{y}\right)$$ I answered this one by squaring the two ...
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.
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 ...
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) ...
3
votes
2answers
58 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 ...