Questions on proving and manipulating inequalities.

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0
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83 views

Which is true $A$ is subset of $B$ or $B$ is subset of $A$.

Consider the sets dened by the real solutions of the inequalities $$A=\{(x,y):x^2+y^4\le 1\}$$ and $$B=\{(x,y):x^4+y^6\le 1\}$$Then which is true $A$ is subset of $B$ or $B$ is subset of $A$. ...
2
votes
2answers
81 views

$ E\left( \left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \right) \le \left( \frac{1}{n}\sum_{j=1}^n E(|X_j|^p)^{1/p} \right)^p$

The following is problem 14 of section 3.2 from Chung's "A Course in Probability Theory". If $p>1$, we have $$\left| \frac{1}{n}\sum_{j=1}^n X_j \right|^{p} \le \frac{1}{n}\sum_{j=1}^n |X_j|^p$$ ...
2
votes
1answer
128 views

How to prove this inequality $xy\sin^2C+yz\sin^2A+zx\sin^2B\le\dfrac{1}{4}$

Let $x,y,z$ is real numbers,and such that $x+y+z=1$,and in $\Delta ABC$,prove that $$xy\sin^2C+yz\sin^2A+zx\sin^2B\le\dfrac{1}{4}$$ I think this inequality maybe use $x^2+y^2+z^2\ge ...
7
votes
2answers
104 views

How to show this inequality?

Show that $$-2 \le \cos \theta(\sin \theta+\sqrt{\sin^2 \theta +3})\le2$$ Trial: I know that $-\dfrac 1 2 \le \cos \theta\cdot\sin \theta \le \dfrac 1 2$ and $\sqrt 3\le\sqrt{\sin^2 \theta ...
0
votes
2answers
47 views

Prove that nearly all positive integers are equal to $a + b + c$ where $a | b$ and $b | c$, $a \lt b \lt c$

If a positive integer $n$ is equal to $a + b + c$ where $a | b$, $b | c$ and $a \lt b \lt c$, let it be called "faithful". Prove that nearly all numbers are faithful and list the non-faithful ...
8
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1answer
146 views

How prove this $ab+bc+cd\le\dfrac{5}{4}$

let $a,b,c,d\in \Bbb R$ and $a,b,c,d>-1,a+b+c+d=0$ prove that $$ab+bc+cd\le\dfrac{5}{4}$$ I have this solution if $b\le c$, then $$ab+bc+cd=a(b-c)-c^2\le ...
1
vote
2answers
1k views

Solving the domain and range of a region satisfying two inequalities?

The question I was provided was: "Find the domain and range of the region satisfied by the following inequalities: i) $y \ge (x-1)^2$ ii)$y \le2x+1$ Any help would be greatly appreciated. Would you ...
6
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1answer
128 views

How to prove this Stirling related equation

Here is what I need to prove, but have no idea were to start. I know there is some connection with the Stirling theorem. $$ \sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right )^{d} $$ for ...
3
votes
1answer
118 views

Find max and min of $IJ + FE + GH$

Let $D \in \triangle ABC$. Passing through D, contruct$\, FE \parallel AB, IJ \parallel AC, GH \parallel BC$. Find max and min of IJ + FE + GH Can this problem be solved by AM-GM ? I tried $IJ + ...
4
votes
1answer
87 views

Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function

In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions $\vartheta$ and $\psi$. I have a question about Theorem 2.27 on page 22. My ...
3
votes
1answer
97 views

looking for reference for integral inequality

Math people: I would like a reference for the following fact (?), which I proved myself (I am 99% sure the proof is valid) but which has probably been done before. My proof was a little messy. If ...
6
votes
5answers
104 views

Prove that $3^n>n^4$ if $n\geq8$

Proving that $3^n>n^4$ if $n\geq8$ I tried mathematical induction start from $n=8$ as the base case, but I'm stuck when I have to use the fact that the statement is true for $n=k$ to prove ...
1
vote
1answer
45 views

A simple question about a bounded function

Let $f$ be a function defined on $[0,\infty)$. If $|f(x)| \leq M$ for all $ x \in [0, \infty)$, then can I say $$ \exists C,R >0 : |f(x)| \leq \frac{C}{(1+x)^2}\;\;(x \geq R) $$ is equivalent to ...
6
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2answers
343 views

Show that $\frac {a_1^2}{a_2}+\frac {a_2^2}{a_3}+…+\frac {a_n^2}{a_1}\geq a_1+a_2+…+a_n$ using AM-GM.

Given $a_1,a_2,...,a_n$ be positive reals. Show that $\displaystyle\frac {a_1^2}{a_2}+\frac {a_2^2}{a_3}+...+\frac {a_n^2}{a_1}\geq a_1+a_2+...+a_n$ using AM-GM. I know how to slve it using ...
4
votes
1answer
66 views

$P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$

If $E(X^2)=1$ and $E(|X|)\ge a >0$, then $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$. I can see from the well known inequality $E(|X|) \le E(|X|^2)^{1/2}$ that it must be the ...
1
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3answers
141 views

$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?

Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities: $$\|f*g\|_q\leq ...
5
votes
1answer
129 views

How prove this $\left(\sqrt{a^2+b^4}-a\right)\left(\sqrt{b^2+a^4}-b\right)\le a^2b^2$

let $a,b\in R$,and such that $$\left(\sqrt{a^2+b^4}-a\right)\left(\sqrt{b^2+a^4}-b\right)\le a^2b^2$$ prove that $$a+b\ge 0$$ I think this is very beatifull problem, have you nice methods? Thank ...
2
votes
3answers
288 views

Spivak problem on Schwarz inequality

I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality: $$ x_1y_1 + x_2y_2 \leq ...
14
votes
2answers
327 views

How to find $1+\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{n}}$

Let $a_{i}>0,i=1,2,\cdots,n$, If $$1+\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{n}}\ge ...
10
votes
1answer
220 views

Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that $$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$ Prove that $$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$ Where should ...
3
votes
1answer
115 views

Find max of $x^7+y^7+z^7$

Find max of $x^7+y^7+z^7$ where $x+y+z=0$ and $x^2+y^2+z^2=1$ I tried to use the inequality:$$\sqrt[8]{\frac {x^8+y^8+z^8} 3}\ge\sqrt[7]{\frac {x^7+y^7+z^7} 3}$$ but stuck
5
votes
1answer
153 views

How to find the minimum of $x+y^2+z^3$?

let $x,y,z>0$, and $x+3y+z=9$, find the minimum of $$x+y^2+z^3$$ I think this problem is very interesting. I have found this when ...
2
votes
1answer
172 views

Inequality for norms

Let g(x, y) be function on $X\times Y$. Show that for all $p\geq q$ $$ \|\,\|g\|_{L^q(Y)}\,\|_{L^p(X)}\leq \|\,\|g\|_{L^p(X)}\,\|_{L^q(Y)} $$ Thsnk you.
2
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1answer
60 views

Prove that $n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}$.

Let $c \not= 1$ be a real positive number, and let $n$ be a positive integer. Prove that $$n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}.$$ My initial thought was to try and induct on $n$, but the ...
2
votes
3answers
856 views

sin(x) inequality

This should be fairly straightforward but the proof seems to be alluding me. I want to show $x - \frac{x^3}{3!} < \sin(x) < x$ for all $x>0$. I recognize this shouldn't be too difficult ...
1
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1answer
40 views

How prove this equality $\sum\limits_{r=0}^{n_{k}}C_{n}^{r}M^{N-n_{k}+r}x^{n_{k}-r}\le(1+1)^N(M+x)^N$

prove that $$\sum_{r=0}^{n_{k}}C_{n}^{r}M^{N-n_{k}+r}x^{n_{k}-r}\le(1+1)^N(M+x)^N$$ and $x\in R,x>0, n_{k},N \in N^{+},N-n_{k}+r>0,n_{k}-r>0$ where $C_{n}^{m}=\dfrac{n!}{m!(n-m)!}$
3
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2answers
75 views

How to prove that there exists a $z_0 \in U_{1} [0]$ such $ \prod_{k=1}^{n} |z_0 - a_k | \geq 1 $ for $a_1, \dots , a_n \in U_{1} [0] $?

Let $a_1 , \dots , a_n $ be points in the unit circle/ball in $\mathbb{C}$ around $(0,0)$ (also known as $U_{1} [0]$), which do not necessarily differ from one another. How to prove that there exists ...
2
votes
3answers
83 views

Easy Inequality Problem

Solve the given inequality by interpreting it as a statement about distances on the real line. $$\lvert x+1\rvert > \lvert x-3\rvert$$ I am confused on what this question is asking. Can anyone ...
6
votes
3answers
146 views

How to prove $(\frac{n+1}{e})^n<n!<e(\frac{n+1}{e})^{n+1}$ without integrating method?

How to prove $$\left(\frac{n+1}{e}\right)^n<n!<e\left(\frac{n+1}{e}\right)^{n+1}$$ without integrating method? In fact we could prove this by noticing that $$i<x<i+1\Rightarrow \ln ...
5
votes
2answers
119 views

Prove $x^ny(x - y) + y^nz(y - z) + z^nx(z - x) \ge 0$

Prove the inequality with $x, y,z$ is the sides of a triangle and $n\in \mathbb Z \land n\ge2$ $${x^n}y(x - y) + {y^n}z(y - z) + {z^n}x(z - x) \ge 0 \tag 1$$ I can prove the inequality with ...
0
votes
0answers
244 views

Using the Simplex algorithm to solve systems of linear inequalities?

I am trying to understand how I could use the first phase of the Simplex method (i.e. constructing a tableaux corresponding to an initial feasible solution) in order to solve systems of linear ...
0
votes
1answer
51 views

Looking for suggestions on how to proceed with showing that:

for $x \ge 2863:$ $$\ln\left(\left\lfloor\frac{x}{6}\right\rfloor!\right) < \sum_{k=5}^{\infty}-\mu(k)\ln\left(\left\lfloor\frac{x}{k}\right\rfloor!\right)$$ I've written a java application which ...
0
votes
2answers
70 views

Some inequality with complex variables and a concavity of a complex function.

I am doing some project. I have to calculate the estimates of an operator. But I was stuck on a part. I need to show the following form of inequality to derive a conclusion what I want to show. ...
12
votes
1answer
225 views

A series: $1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\cdots$

Denote $$b_1=1,b_{n}=b_{n-1}-\dfrac{S(b_{n-1})}{n},(n>1 )\tag1$$ where $S(x)=1$ if $x>0,S(x)=-1$ if $x<0$, and $S(0)=0.$ So ...
12
votes
1answer
182 views

Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$

How am I supposed to prove this inequality? $$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$ Jordan inequality might be an option but led me ...
2
votes
0answers
53 views

Inequality with $\|\cdot\|_p$ norm

Let $x_1, \ldots, x_{2m}$ be $\{0,1\}$ Bernoulli random variables, i.e. variables which takes values $0$ and $1$ with equal probability. Let $S_m$ be group of all permutations $\pi$ on $\{1, \ldots, ...
4
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1answer
375 views

Conjecture regarding trapping rational numbers in some special intervals

Conjecture: Let $b\in\mathbb{N}_{\geq3}$ and $\{x_i\}$ be a collection of $b−2$ rational numbers greater than $1$. Does there always exist a natural number $a$ such that for all $i$ there exists some ...
2
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1answer
158 views

A problem involving Schwarz lemma (from Gamelin)

I have a problem that I cannot solve; it was homework this past semester, I didn't get it then, and now I'm going over past problems and am stuck on it again. It reads-- suppose $f(z)$ is analytic ...
2
votes
1answer
180 views

$\log_{2}{3} > \log_{3}{5}$?

Which one is larger $\log_{2}{3}$ or $ \log_{3}{5}$? Edit : Without use of numerical calculations, just use properties of logarithm, exponentials. we cant use $\log 3 = 0.477$ and $\log 2 = 0. 301$ , ...
6
votes
9answers
964 views

Why is $9<\sqrt{89}<10$?

Explain why $9<\sqrt{89}<10$. How do you explain this? I'm doing revision and we haven't been taught it yet but it will be on the test. $\sqrt{389}$ is also between two consecutive whole ...
2
votes
1answer
74 views

Explain why the solution to the inequality is wrong

Problem: $$x/2 - 4/x - 1 > 0$$ Simplified to: $$(x^2-8)/2x > 1$$ Right solution (put all at one side, bring to one fraction) is union of: $$x^2-8-2x>0, x > 0$$ and $$x^2-8-2x<0, x ...
1
vote
1answer
286 views

Chernoff bound proof using Markov

Does anyone familiar with the following format of Chernoff bound: $$ Pr\left(\frac{1}{n}\sum\limits_{i=1}^n X_i \gt T\right ) \le \inf_{\gamma \gt 0}{\left ( \frac{E[e^{\gamma X_i}]}{e^{\gamma T}} ...
7
votes
2answers
2k views

Showing equality in Cauchy-Schwarz inequality

With $\mathbf{u,v}$ being vectors in $\mathbb{R}^n$ euclidean space, the Cauchy–Schwarz inequality is $$ {\left(\sum_{i=1}^{n} u_i v_i\right)}^2 \leq \left(\sum_{i=1}^{n} ...
2
votes
2answers
3k views

How do I prove the arithmetic-geometric mean inequality?

I am following along with this bare-bones proof of the arithmetic-geometric mean inequality with two real numbers. I'm having difficulty understanding the logic behind this step: $$ ...
7
votes
5answers
722 views

Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)

If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$. As a first step to prove this ...
1
vote
0answers
42 views

Estimation of a scalar product

I encountered the following, which shouldn't be that hard, but I can't get my head around it. The problem is the following estimate (part of a bigger equation, but here's just the difficult part): ...
17
votes
1answer
325 views

$x^3-3x-3=0$, prove that $10^x<127$

$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
3
votes
1answer
426 views

Jensen's inequality and $L^p$ norms

Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
0
votes
3answers
45 views

Let $a,b \in \Re$. If 0 < $\epsilon$ < min{|a|, |b|}. Show this inequality

Let $a,b \in \Re$. If 0 < $\epsilon$ < min{|a|, |b|}. $ {\frac{|a+\epsilon|}{|b+\epsilon|}} \leq {\frac{|a|+\epsilon}{|b|-\epsilon}}$ I tried to use triangular inequality. But have no idea of ...
3
votes
2answers
686 views

Do inequalities hold under square-root (or exponentiation in general)?

This has been bothering me lately. My proof-skills are rusty (and were never great to begin with). I dimly recall having seen this (or something related to it) in a math course I took a while ago, but ...