Questions on proving and manipulating inequalities.

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-2
votes
0answers
19 views

Show that this function is a distance function

I'm not quite familiar with valuation. Any help is highly appreciated. Thank you
0
votes
2answers
20 views

proof by induction summation inequality

show by induction that: $$\sum_{i=1}^n i^2 = O(n^3)$$ what I have so far: $$\sum_{i=1}^n i^2 <= n^3$$ base case: for n=1 $$\sum_{i=1}^1 i^2 <= 1^3$$ ...
3
votes
3answers
83 views

Slick proof of exponential inequality

Today I saw that using taylor series, one can show that $e^x+e^{-x}\leq 2e^{x^2/2}$. Is there a slick proof using some sort of Jensen-type inequality or integral bound?
0
votes
3answers
66 views

Cannot follow proof that $n! \leq en(n/e)^n$

prove that $n! \leq en(n/e)^n$ skip proof for base (n=1)... Assume it holds for $n-1$, verify for $n$. We have $n! = n* (n-1)! \leq n * e(n-1)(\frac{n-1}{e})^{n-1} $ by inductive assumption. we ...
-1
votes
1answer
13 views

question on proving inequalities

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
0
votes
3answers
27 views

How to study positivity of $x\sqrt{4-x^2}-4\arcsin({\frac x2})$

I have to study where the function is positive/negative. What's the method to solve the inequality $x\sqrt{4-x^2}-4\arcsin({\frac x2})>0$ ?
0
votes
2answers
26 views

Olympiad minimum question, minimal value

If the numbers $A, B, C$ are such that the expression $\sqrt{A-B} + \sqrt{(B+3)^2} + C^2 - 4C + 4$ is as small as possible, then $A+B+C$ is? I thought start with, $A > B > C$ without loss of ...
2
votes
2answers
56 views

Prove, inequality ,positive numbers

$$\frac{a}{e+a+b}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{c+d+e}+\frac{e}{d+e+a}<2$$ Prove that for positive numbers $a,b,c,d,e$ there is such inequality
-1
votes
3answers
22 views

How can we make this expression small? [on hold]

How can we make the following expression small: $$(bx-ay)^2+(cx-az)^2+(cy-bz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2$$, where $a,b,c,x,y,z$ are nonnegative reals? Note: I'm not looking for an exact answer, ...
1
vote
0answers
11 views

Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
0
votes
1answer
19 views

Proving that a system of equalities and inequalities is inconsistent (Vol. 4)

I am studying sign pattern matrices and I have to prove this statement. Prove that the system $$\begin{cases} a,b,d,e,f,g,h,i>0 \\ -a+e-i>0 \\ -ae+ai+bd-ei+fh>0 \\ aei−hfa-bdi−gbf>0 \\ ...
1
vote
1answer
55 views

Find the value of $ ( ab + bc + ca )^2 $

If $a,b,c$ are real numbers which satisfy $a^2+b^2+ab = 9$ $b^2+c^2+bc = 16$ $c^2+a^2+ca = 25$ find the value of $ ( ab + bc + ca )^2 $
2
votes
1answer
27 views

Counterexample Poincaré Inequality for $H_0^1$ in 2D

Is there any counterexample to the Poincaré inequality $$\int_\Omega|f|^2dx\leq C(\Omega)\int_\Omega|\nabla f|^2dx $$ for $f\in H_0^1(\Omega)$, $C(\Omega)>0$ and $\Omega\subset\mathbb{R}^2$? I ...
1
vote
2answers
54 views

If $G(x)=P[X\geq x]$ then $X\geq c$ is equivalent to $G(X)\leq G(c)$ $P$-almost surely

Suppose $[\Omega,\mathcal{F},P]$ denotes a probability triplet and $X:\Omega\to\mathbb{R}$ is a real-valued random variable. Define $$ G(x)=P[X\geq x]. $$ My current reading material claims ...
1
vote
5answers
76 views

If $a^2+b^2+c^2=1$ then prove the following.

If $a^2+b^2+c^2=1$, prove that $\frac{-1}{2}\le\ ab+bc+ca\le 1$. I was able to prove that $ ab+bc+ca\le 1$. But I am unable to gain an equation to prove that $ \frac{-1}{2}\le\ ab+bc+ca$ . Thanks in ...
-4
votes
2answers
33 views

Prove and disprove the following inequality.

Prove: $ 0 \le a \lt b$ implies $ 0 \le a^2 \lt b^2 $ and $0 \le \sqrt{a^3} \lt \sqrt{b^3}$. Now show that the statement is false if the hypothesis $a \ge 0$ or $a \lt 0$ is removed. EDIT: Someone ...
3
votes
0answers
47 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ In Doob's weak $L^1$ ...
0
votes
1answer
15 views

Showing that $\Re z \le |\Re z| \le |z|$ and $\Im z \le |\Im z| \le |z|$

What I'm wanting to show is that $$\Re (z) \le |\Re (z)| \le |z|$$ and also $$\Im(z)\le |\Im(z)| \le |z|$$ So what I've done so far is to consider $$z=x+iy$$ Using the above $z$ I also said that $$\Re ...
-1
votes
1answer
16 views

Prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|))

For a,c, and x in the reals, prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|)). I'm trying to practice these kinds of questions, and any help or suggestions are greatly ...
-1
votes
2answers
25 views

Prove that if $y>1$, then $\forall M\in\mathbb{R}$, there exists an $N$ in the natural numbers s.t. $n\geq N$ implies $y^n>M$. [on hold]

For $y\in\mathbb{R}$, prove that if $y>1$, then $\forall M\in\mathbb{R}$, $\exists N\in\mathbb{N}$ such that $$ n≥N \implies y^n>M. $$ I'm not used to proving these kinds of questions so any ...
0
votes
0answers
32 views

Prove a statement involving Big-O notation [on hold]

Let $z \in \mathbb{C}$. We adopt the convention that $\sqrt{z} = \sqrt{r} e^{\frac{i \theta}{2}}$ if $z = re^{i \theta}$. I want to show that $$ \sqrt{z^2 +4t} = z + \frac{2t}{z} + O (|z|^{-2} ), ...
-2
votes
0answers
14 views

How would I get Maple to display all integer solutions to this system of inequalities? [on hold]

I need to find all the integer solutions satisfying: $$20+x\geq0;\space2x+5y\geq;\space-x-2y\geq0.$$ I'm not sure which Maple functions would work and whatnot. A guy can only google this stuff for so ...
0
votes
2answers
26 views

Spivak's Calculus, chapter 1 problem 19 (inequalities)

I'm having trouble with problem 1-19 in Spivak's Calculus. I have to prove that if $|x-x_0| < \frac{\epsilon}{2} $ and $ |y-y_0| < \frac{\epsilon}{2} $ then $ |(x-y)-(x_0-y_0)| < \epsilon $. ...
0
votes
2answers
28 views

Show using inequality of means that $a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$

Show using inequality of means that for $a>0$ and $n\in\mathbb{N}$: $$a\cdot n \cdot \frac{1}{n} \le a^2n^2+\frac{1}{n^2}$$ I'm sure it's not that complicated, but I'm probably missing ...
0
votes
1answer
50 views

Suppose $f(x,y)=c(x^{m}+y^{n})-dx^{a}y^{b}+sin(x+y^{2})$, show that f is greater than

Suppose $f(x,y)=c(x^{m}+y^{n})-dx^{a}y^{b}+\sin(x+y^{2})$, where m,n are positive even integers, a,b are positive integers, c,d are positive real numbers and $\frac{a}{m}+\frac{b}{n}<1$. Show that ...
2
votes
1answer
49 views

How to show without calculator that $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor$

By wolfram alpha, I get $\left\lfloor\, \log_{10}{999^{999}}\right\rfloor =\left\lfloor\, \log_{10}{999^{999}}+\log_{10}2\right\rfloor=2996$. How to prove that $\left\lfloor\, ...
0
votes
1answer
40 views

Which one is greater?

For any $x\in\mathbb{R}^+$, let $x\diamond 1=x$ and $x\diamond (n+1) = x^{x\diamond n}$ for $n\in\mathbb{N}$. For example, $2\diamond 3 = 2^{2^2}=16$. If $t$ be an unique positive real number such ...
-1
votes
0answers
21 views

Can a strict inequality be derived from a weaker one?

Suppose P and Q are two statements, with P being the stronger one. Let us denote the set of statements derived from P and Q be A and B respectively. Then can the strongest statement belonging to A be ...
0
votes
3answers
73 views

Proof of an inequality by induction

Let $n \in \mathbb N^+$. Show that if $x_1, x_2, ... , x_n$ are $n$ real numbers such that $-1 \le x_i \le 0$ for each $1 \le i \le n$, then $$(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + ...
2
votes
2answers
55 views

Show $\lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + \lambda_na_n\rvert < 1$ when $\lvert a_i\rvert < 1$ and $\lambda_i\geq 0$

If $\lvert a_i\rvert < 1$, $\lambda_i\geq 0$ for $i = 1,\ldots,n$ and $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$, show that $$ \lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + ...
0
votes
1answer
42 views

Which is bigger, the number of neurons in the brain or the all the stars in the observable universe?

In other words, is 100 billion larger than $10^{22}$ or ....? Are there also other interesting comparisons of systems with large number of members? i.e. the sand on the beach, the atoms in the air, ...
3
votes
1answer
40 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
0
votes
1answer
16 views

Prove that $E(t)$ satisfies the following differential inequality.

It is given that $u_t=2u_{xx}-3u ,\hspace{0.3cm} u_x(0,t)=0=u_x(1,t)$ Use the Young's inequality to show that the energy $$E[u,u_x](t):=\frac{1}{2}\int_0^1(|u|^2+|u_x|^2)dx$$ satisfies the ...
-1
votes
0answers
17 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
1
vote
0answers
26 views

Is there a name for the inequality $\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$?

Is there a name for the inequality $$\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$$? And does anyone have any nice examples or applications, especially with an economic flavor? The transposed multivariate ...
3
votes
1answer
46 views

Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$

For $x, y \ge 0$ prove that: $$x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$$ What I think would apply is the AM-GM Inequality, so first, $$(x^2 + y^2 + 1)^2 \le (x^3 + y + 1)(y^3 + x + ...
2
votes
1answer
27 views

Symmetric and homogeneous three variable inequality with radicals.

While trying to solve a problem, I got the following inequality which appears correct, but I cannot prove. For positive $x, y, z$, $$\sum_{cyc} \frac{x}{y^2+z^2} \ge \sum_{cyc} ...
3
votes
1answer
37 views

Finding the lowest upper bound of product of two number using Young's inequality

Young's inequality for product can be stated as follows: $ab \leq \frac{1}{p}a^p + \frac{1}{q}b^q$ where a and b are nonnegative real numbers and p and q are positive real numbers such that 1/p + 1/q ...
1
vote
2answers
38 views

Prove the inequality $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ by using derivative

The problem: show that $x \le x+(1-x) \sin^2(x) \le 1$ for $x \in (0,1)$ I tried to solve it with the derivative and the inequality $\sin(x) \le x$ for $x>0$ thanks for helpers
2
votes
2answers
221 views

Is this logically valid?

$$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1} > ln(n)$$ and so, necessarily, $$1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n-1}+\frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}.....+\frac{1}{n} > ln(n)$$ ...
3
votes
0answers
98 views

Solve an inequality using Cauchy-Schwarz Inequality

Le $a,b,c,d \in \mathbb{R^{+}}$. Using Cauchy-Schwarz Inequality prove that the following inequality holds: $$\frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} \ge \frac{1}{\frac 1a + \frac 1b} + ...
-1
votes
1answer
24 views

Ordering of real numbers compatible with n-th powers/reciprocal powers (induction)

I have to use induction to prove that $$0 \leq a < b \implies 0 \leq a^n < b^n$$ for all natural n. Also (perhaps very similarly) that $$0 \leq a < b \implies 0 \leq a^{1/n} < b^{1/n}.$$ ...
-1
votes
1answer
71 views

Maximize $x + y + z$

(PUMaC 2006 Algebra #10) If $x, y, z$ are real numbers and \begin{alignat*}{9}2x+\ &&y+\ &&z\leq&66\\ x+\ &&2y+\ &&z\leq&60\\ x+\ &&y+\ ...
-5
votes
0answers
41 views

Find integers which satisfy the following inequality [on hold]

Let $a_1$, $a_2$, ..., $a_n$ be $n$ real numbers whose squares sum to $1$. Prove that for any integer $k \geq 2$, there exist $n$ integers $x_1$, $x_2$, ..., $x_n$, each with absolute value $\leq k-1$ ...
1
vote
1answer
26 views

Proving that a system of equalities and inequalities is inconsistent (Vol. 3)

I am studying sign pattern matrices and this is (hopefully!) the last of the systems that I have to prove inconsistent. Prove that the system $$\begin{cases} a,b,d,e,f,g,h,i>0 \\ -a+e-i=0 \\ ...
1
vote
1answer
29 views

Proving that a system of equalities and inequalities is inconsistent [on hold]

Prove that the system $a,b,d,e,f,g,h,i>0$ $ae+ai−bd+ei−fh=0$ $aei−hfa-bdi−gbf=0$ is inconsistent. I tried using some standard techniques such as factoring, or multiplying an equality and ...
0
votes
1answer
16 views

Inequalities and arithmetic operations

I'm reading a paper with some math involved, and on a demonstration the author makes this assumptions: $a/b < e/f$ and $c/d < e/f$ And after that and without stating anything else it ...
1
vote
1answer
20 views

How can I prove this inequality using Cauchy's inequality?

Cauchy's inequality is given by: for real numbers, $a_1,...,a_n$, $b_1,...,b_n$, $(a_1^2,...,a_n^2)^{1/2}(b_1^2,...,b_n^2)^{1/2} \geq |a_1b_1+a_2b_2+...+a_nb_n|$. Assuming this, prove that ...
3
votes
4answers
70 views

Find all $x$ for which $x+3^x<4$

Find all $x$ for which $x+3^x<4$ I'm stuck at this one...how does one solve for $x$? I've tried: $x+3^x<4$ $3^x<4-x$ $x<\log_3({4-x})$ But I don't know where to go from there. If I ...
3
votes
2answers
29 views

An upper bound for an average exponentiated weighted sum of a vector from $\{-1,1\}^n$

Suppose $\mathcal{S}=\{\mathbf{x}:\mathbf{x}\in\{-1,1\}^n\}$, that is, $\mathcal{S}$ contains all $2^n$ vectors of length $n$ containing -1 and 1. I am interested in the following average: ...