Questions on proving and manipulating inequalities.

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2
votes
1answer
44 views

What is the most elementary proof of these inequalities?

Let $p$ be a non-zero integer, and let $x_1$, $\ldots$, $x_n$ be $n$ positive real numbers. Then we define the $p$-th power mean $M_p$ of these numbers as $$ M_p \colon= (\frac{x_1^p + \ldots + ...
0
votes
1answer
36 views

How to get n from n-1

I know this question is probably trivial, but I'm having great difficulty with it for some reason. So, I want to solve for $p$: $n-1 \geq 2(n-p)$ I know that the answer is $n \leq 2p -1 ...
1
vote
2answers
69 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
6
votes
2answers
186 views

How prove this Stronger AM-GM inequality $\frac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le A_{n}-G_{n}$

let $a_{i}>0,i=1,2,\cdots,n,n\ge 3$,show that $$\dfrac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}-\sqrt[n]{a_{1}a_{2}\cdots ...
6
votes
4answers
204 views

Inequality for harmonic means

Prove that for real numbers $a_1 ,a_2 ,...,a_n >0$ the following inequality holds $$\frac{1}{a_1 } +\frac{2}{a_1 +a_2 } +...+\frac{n}{a_1 +a_2 +...+a_n }\leq 4\cdot \left(\frac{1}{a_1} ...
0
votes
2answers
49 views

What does | mean in this exercise? And how do I solve it?

I was doing a practice exam for my SATs and I stumbled across this problem in the inequality section of the Algebra part. And I don't know what that symbol means and how to solve the problem with that ...
2
votes
1answer
39 views

How to prove this integral inequality?

Here is a problem: Let $B_r=\{ (x_1,x_2,\cdots,x_n)\in \mathbb{R}^n: x_1^2+x_2^2+\cdots+x_n^2<r^2\}.$ Let $f$ be a $C^1$ real function on $B_2$. Prove that $$\inf_{a\in R}\int_{B_2} ...
0
votes
3answers
52 views

How to derive this inequality?

How to derive the following inequality for all positive integers $n \geq 2$? $$ \frac{n!}{n^n} \leq \left(\frac{1}{2}\right)^k,$$ where $k$ denotes the greatest integer less than or equal to $\dfrac ...
0
votes
2answers
40 views

How to derive these inequalities?

I can derive the inequalities $$ n^p < \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^p $$ for any positive integers $p$ and $n$. These actually follow from the identity $$b^p - a^p = (b-a)(b^{p-1} + ...
0
votes
0answers
25 views

How about integral version of Holder's inequality?

In light of the fact that Minkowski's inequality have integral version, I thought there might be one for Holder's as well. I cannot find any through searching (there is an infinite product version in ...
2
votes
2answers
19 views

How to obtain this upper bound on the summation from this inequality?

I can show that $$ \frac{1}{\sqrt{n}} < 2 (\sqrt{n} - \sqrt{n-1} ) $$ for $n \geq 1$. Now from this how to derive the following inequality? $$ \sum_{n=1}^m \frac{1}{\sqrt{n}} < 2\sqrt{m} - 1 $$ ...
-2
votes
1answer
49 views

basic inequality proof needed [on hold]

I would be really thankful if you could come up with some solution to this inequality.
1
vote
0answers
29 views

Can we prove this inequality in another way?

As explained here, I've managed to prove the following inequality: $\sigma(n)\geq\sqrt n(d(n)-2)+n+1$. This can be proved easily in two cases (one for $n$ being a perfect square and one for otherwise) ...
4
votes
3answers
123 views

How prove this $x+\sin{x}-2\ln{(1+x)}\ge 0$

Question: let $x>-1$, show that $$x+\sin{x}-2\ln{(1+x)}\ge 0$$ this is true,because : http://www.wolframalpha.com/input/?i=x%2Bsinx-2ln%281%2Bx%29 My try: since ...
2
votes
3answers
74 views

Proving $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$

Prove $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$. I tried to do this in two ways, I'm not sure about CMVT and I have a problem with the other way. Using Cauchy's MVT: RHS: ...
12
votes
2answers
153 views

$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
3
votes
2answers
51 views

Trigonometric Triangle Equality

$A, B, C$ are the angles of a triangle then $tan^2(A/2)+tan^2(B/2)+tan^2(C/2)$ is always greater than what integral value.
6
votes
2answers
84 views

Advice for self-studying Inequalities and Calculus

I'm interested in self-studying the following books over the next year or so: Spivak's Calculus (I'm already in Ch. 5 and it is very slow going) The Cauchy-Schwarz Master Class by J. Michael Steele ...
4
votes
3answers
86 views

Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$

Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0<a\le a_n\le b <\infty.$ Prove: $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 ...
1
vote
1answer
21 views

Probably simple application of Holder or Minkowski for integrals

This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose $U$ and $V$ are integrable over measure space ...
3
votes
2answers
80 views

$\displaystyle \frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$

Let $a,b,c$ be positive reals . Prove that $\displaystyle \frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$ I found this one in a book, no hints mentioned, but marked as very ...
0
votes
3answers
91 views

How Prove $4x^3+8y^3+15xy^2-27x-54y+54\ge 0$

let $x,y\ge 0$,show that $$4x^3+8y^3+15xy^2-27x-54y+54\ge 0$$ when $x=y=1$ is equality. see: http://www.wolframalpha.com/input/?i=4x%5E3%2B8y%5E3%2B15xy%5E2-27x-54y%2B54 this inequality is ...
3
votes
0answers
42 views

Inequality (Minkowski)

Given $T:\mathbb{R}^{2}\to \mathbb{R}$ s.t. \begin{eqnarray*} |T(x)|\leq \sum_{k=1}^{2}|x_{k}|\,and\,|T(x)-T(y)|\leq \sum_{k=1}^{2}|x_{k}-y_{k}| \end{eqnarray*} for all ...
1
vote
0answers
26 views

Radical Inequality related to three variables

Prove the following. $$\sum_\text{cyclic}\sqrt[4]{\dfrac{(a^{2}+b^{2})(a^{2}-ab+b^{2})}{2}}\leq\dfrac{2}{3}\left(\sum_\text{cyclic}\dfrac{1}{a+b}\right)\left(\sum_\text{cyclic}a^{2}\right)$$
1
vote
1answer
29 views

Proof or disproof $Y = 2\sum_{i=1}^{n} a^{2}_{i} b^{2}_{i} + \sum_{i \not=j}^{n} a^{2}_{i}b^{2}_{j} - 2\sum_{i=1}^{n} a_{i} b_{i} \ge 0$

In my attempt to answer this question I came cross this question : if $$ \sum_{i=1}^{n}a^2_{i}\le 1,\sum_{i=1}^{n}b^2_{i}\le 1 $$ do we have ? $$Y = 2\sum_{i=1}^{n} a^{2}_{i} b^{2}_{i} + \sum_{i ...
3
votes
2answers
86 views

QM-AM-GM-HM proof help

Out of interest, I am trying to proof QM-AM-GM-HM inequality. If you don't know it, it's something like this... Let there be $n$ numbers $x_1, x_2, x_3...x_n$, where $x_1, x_2, ...,x_n>0$. Proof ...
6
votes
1answer
74 views

A formula for $\pi$ and an inequality

For any $n\in \mathbb{N}$ prove the identity : $$\pi =\sum_{k=1}^{n}\frac{2^{k+1}}{k\dbinom{2k}{k}}+\frac{4^{n+1}}{\dbinom{2n}{n}}\int_{1}^{\infty}\frac{\mathrm{d}x}{(1+x^2)^{n+1}}\tag{1}$$ and thus ...
1
vote
1answer
24 views

Proof of the Bergström inequality using Cauchy

$\sum_{i=1}^n \frac{a_i^2}{b_i} \geq \frac{(a_1+...+a_n)^2}{b_1+...+b_n}$ for real $a_i, b_i > 0$. I'm told it follows from Cauchy-Schwarz, but I just don't see it.
0
votes
1answer
36 views

Approximation in Normal distribution random variable

Let ${X_n : n \geq 1}$ be independent $\mathcal{N}(0,1)$ random variables. How do we get the following approximation?
1
vote
2answers
95 views

How to prove this special inequality?

Let $x,y,z>0$ s.t. $$\underset{cyc}{\sum} xy=1$$ Prove that $$\underset{cyc}{\sum}\frac{1}{xy+z}>3$$ I see that when $x=1,y\rightarrow1,z\rightarrow0$ $$\underset{cyc}{\sum}\frac{1}{xy+z} ...
2
votes
1answer
55 views

Inequality $(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$

For positive real numbers $a,b,c$ prove that $$(a+b+c)(ab+bc+ca)(a^3+b^3+c^3)\le (a^2+b^2+c^2)^3$$ My try : Rewrite this as $$\frac{a^2+b^2+c^2}{ab+bc+ca}-1\ge ...
2
votes
0answers
36 views

Find Max : $P=\frac{1}{x^2+1}+\frac{4}{y^2+4}+\frac{3z}{9+z^2}$

Let $x,y,z>0$ and satisfying $3xy+yz+2zx=6$ Find Maximum of this expression: $P=\frac{1}{x^2+1}+\frac{4}{y^2+4}+\frac{3z}{9+z^2}$
1
vote
1answer
42 views

Bessel's inequality for expected value

Let $X_1, X_2,\ldots$ be independent random variables with expected value $\mathbb{E}[X_i]=0$ and variance $V[X_i]=1$. Let $Y$ be another random variable, such that $\mathbb{E}[Y^2] < \infty$. I ...
1
vote
2answers
28 views

Inequality involving modulus

If $\vert x\vert\leqslant a$ and $\vert y\vert\leqslant b$ can we create some inequality that contains $\vert\vert x\vert-\vert y\vert\vert$?
0
votes
0answers
42 views

Help with an inequality of probability distribution functions

There are six random variables $X_{1}$, $X_{2}$, $X_{3}$, $Y_{1}$, $Y_{2}$, and $Y_{3}$ on $[0,c]$. Their cumulative distribution functions are $% F_{1}(t)$, $F_{2}(t)$, $F_{3}(t)$, $G_{1}(t)$, ...
3
votes
2answers
78 views

$2^x + 3^x - 4^x + 6^x - 9^x ≤ 1$ $\forall x \in R$

How can i prove that $2^x + 3^x - 4^x + 6^x - 9^x ≤ 1$ $\forall x \in R$. I tried $log(2^x + 3^x - 4^x + 6^x - 9^x) = log (1296^x) = x log(1296)$ i don't know if im correct here i stuck some help ...
2
votes
0answers
30 views

Inequality in matroid theory

Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct ...
2
votes
2answers
64 views

Prove that $(1+x)\ln(1+x) >\arctan(x)$

My solution is that slope of $\ln(1+x)$ is greater than $\arctan(x)/(1+x)$ & at $x=0$ both of these are equal and hence inequality proved. What i am looking for is the restrictions on $x$ in which ...
0
votes
1answer
13 views

Show $P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$

Show that, if $h:\mathbb R\to[0,b]$ and $0\le a< b$ then, $\displaystyle P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$ So $h$ is nonnegative and bounded. If $a=0$ then the inequality holds. because ...
2
votes
1answer
126 views

How to prove inequality $\frac{a}{a+bc}+\frac{b}{b+cd}+\frac{c}{c+da}+\frac{d}{d+ab}\ge 2$

Question: Let $$a,b,c,d>0,a+b+c+d=4$$ show that $$\dfrac{a}{a+bc}+\dfrac{b}{b+cd}+\dfrac{c}{c+da}+\dfrac{d}{d+ab}\ge 2$$ when I solved this problem, I have proved the following three ...
0
votes
2answers
30 views

Equations,inequalities and absolute values.

I really am confused when I am supposed to change the $<$ and $>$ symbol. For example this unsolved question in my reference book: solve $\displaystyle \frac{x+1}{x-1}>0$ One ...
1
vote
0answers
27 views

max value of difference of square roots of sums

Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the ...
2
votes
2answers
159 views

If $abc\neq 0$, then $ \frac{(a+b)^2}{c^2}+\frac{(a+c)^2}{b^2}+\frac{(b+c)^2}{a^2}\geq2 $

Let $a$, $b$ and $c$ be real numbers such that $abc\neq0$. Prove that: $$ \frac{(a+b)^2}{c^2}+\frac{(a+c)^2}{b^2}+\frac{(b+c)^2}{a^2}\geq2 $$ I checked for some values and it seems to be true. But no ...
0
votes
1answer
53 views

Coefficients in $\pm 1$

Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: $$\displaystyle ...
2
votes
2answers
30 views

Linear inequalities with one unknown

$$3x+\frac{2}{4} \leq x+\frac{7}{2}$$ What is the solution to the inequality? I started multiplying both sides by 4 which gave me $3x+2\leq14$. Then I subtracted two from both sides obtaining ...
6
votes
6answers
138 views

Show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$

This is a pretty straightforward question. I want to show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$. One way would be this. WLOG, suppose $x \leq y$. Then: $(1+\frac{x}{y})^a ...
1
vote
1answer
20 views

linear inequalities using LP solutions not from simplex

I am trying to solve a set of inequalities using linear programming (LP) with objective function set as a constant. Usually this set of inequalities would have many solutions all of them in the ...
0
votes
0answers
23 views

Miklos Schweitzer 2013, Strong lower bound on sumset $|A+qA|$,

Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: $$|A+qA|\ge (q+1)|A|-C_q.$$ This is a problem from ...
6
votes
4answers
152 views

How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand?

Is there any way we can prove this definite integral inequality by hand: $$ \int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4 $$ I don't where to start even, please help. That ...
0
votes
0answers
20 views

ML Inequality - don't understand this step?

I'm going through an example question but I don't understand one of the steps they've made. Let $C_R$ be the circle with radius R, centret at 0 and positively oriented. Show $$lim_{R \to ...