Questions on proving and manipulating inequalities.
4
votes
1answer
43 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 ...
8
votes
3answers
192 views
Funny integral inequality
Assume $f(x) \in C^1([0,1])$,and $\int_0^{\frac{1}{2}}f(x)\text{d}x=0$,show that:
$$\left(\int_0^1f(x)\text{d}x\right)^2 \leq \frac{1}{12}\int_0^1[f'(x)]^2\text{d}x$$
and how to find the smallest ...
1
vote
3answers
90 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:
$$\|f*g\|_q\leq ...
5
votes
1answer
76 views
How prove this $\left(\sqrt{a^2+b^4}-a\right)\left(\sqrt{b^2+a^4}-b\right)\le a^2b^2$
let $a,b\in R$,and such that
$$\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$$
I think this is very beatifull problem, have you nice methods? Thank ...
5
votes
1answer
335 views
Help understanding proof of Schwarz Inequality
I'm working through Spivak's Calculus over the summer, and I'm currently on problem 19 of Chapter 1, which involves proving the Schwarz inequality. The first two parts of the proof are fairly ...
1
vote
2answers
35 views
Spivak problem on Schwarz inequality
I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality:
$$
x_1y_1 + x_2y_2 \leq ...
7
votes
1answer
139 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
2answers
94 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
4
votes
1answer
87 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
63 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
votes
1answer
47 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
2answers
130 views
Unfamiliar inequality rules in proofs. (learning proofs)
Without going into too much detail, I don't think we actually went over inequalities in much depth in any crappy high school or college math class I've ever taken. Now that I am studying proofs, I ...
1
vote
0answers
32 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, ...
1
vote
1answer
32 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
$C_{n}^{m}=\dfrac{n!}{m!(n-m)!}$
4
votes
2answers
73 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
2answers
29 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.
...
6
votes
8answers
669 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 ...
3
votes
2answers
52 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 ...
1
vote
3answers
44 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 ...
4
votes
3answers
69 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
$$i<x<i+1\Rightarrow \ln ...
0
votes
0answers
72 views
Simplifying $\frac{1}{n}\sum_{k=1}^n f(\frac{1}{k})$
Suppose that $$\displaystyle \forall x\in \mathbb{R}_+^* \quad f(x)=\frac{x^2-1}{4}-\frac{\ln(x)}{2}.$$
How can I simplify this: $$I(n)=\frac{1}{n}\sum_{k=1}^n f\left(\frac{1}{k}\right)$$
and prove ...
0
votes
0answers
20 views
Using the Simplex algorithm to solve systems of linear inequalities?
I am trying to understand how I could use the first phase of the Simplex method (i.e. constructing a tableaux corresponding to an initial feasible solution) in order to solve systems of linear ...
3
votes
1answer
120 views
Dimension of solution space for system of linear inequalities
Let's say I have a system of inequalities: $Ax \leq g$ for some $A \in \mathbb{R}^{4\times4}$, $x \in \mathbb{R}^4$, $g \in \mathbb{R}^4$, and $A$ is full rank. Here, the $\leq$ denotes element-wise ...
0
votes
1answer
38 views
Looking for suggestions on how to proceed with showing that:
for $x \ge 2863:$
$$\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)$$
I've written a java application which ...
8
votes
0answers
121 views
A series: $1-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\cdots$
Denote $$b_1=1,b_{n}=b_{n-1}-\dfrac{S(b_{n-1})}{n},(n>1 )\tag1$$
where $S(x)=1$ if $x>0,S(x)=-1$ if $x<0$, and $S(0)=0.$
So ...
17
votes
8answers
725 views
Comparing $2013!$ and $1007^{2013}$
I have to compare the following two numbers:
$$2013! \text{ and } 1007^{2013}$$
where $n! = 1 \times 2 \times \cdots \times (n-1) \times n$.
I tried in different ways to group the $1 \times 2 ...
12
votes
2answers
119 views
Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$
How am I supposed to prove this inequality?
$$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$
Jordan inequality might be an option but led me ...
1
vote
2answers
35 views
Inequality- Absolute Value general powers
Iam trying to understand the following inequality:$p>0$
Let $$T_{m}:=\sum_{i=1}^{m}\left(\left|\int_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}\right|^{p}-\left|g ...
0
votes
1answer
146 views
Proving inequality using Cauchy-Schwarz
How to prove $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y}\leqslant\frac{2(x^2+y^2+z^2)}{x+y+z}$ where $x,y,z$ are all positive real numbers? The hint was to use the Schwarz ...
17
votes
26answers
3k views
2
votes
1answer
48 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
117 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 $\log 2 = 0.
301$ , ...
3
votes
2answers
275 views
$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$
May I know the standard proof technique to prove such kind of inequalities.
$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$
Thanks!
1
vote
1answer
79 views
$(\log_23)^x-(\log_53)^x\geq(\log_23)^{-y}-(\log_53)^{-y}$
$(\log_23)^x-(\log_53)^x\geq(\log_23)^{-y}-(\log_53)^{-y}$
I guess the function $f(x)=(\log_23)^x-(\log_53)^x$ monotonically increasing, so I get the answer $x\geq-y$，but how to prove it not using ...
3
votes
3answers
143 views
Inequalities of expectations
Given two random variables $X$ and $Y$, integrable and everything. Is it true that,
$E(|XY|^2) \leq E(|X|^2) E(|Y|^2) $ ?
If not, can I use something so that I get,
$E|XY|^2 \leq (E|X|^p)^{1/p} ...
2
votes
1answer
55 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
$$x^2-8-2x<0, x ...
1
vote
1answer
28 views
Chernoff bound proof using Markov
Does anyone familiar with the following format of Chernoff bound:
$$
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}} ...
6
votes
2answers
72 views
Showing equality in Cauchy-Schwarz inequality
With $\mathbf{u,v}$ being vectors in $\mathbb{R}^n$ euclidean space, the Cauchy–Schwarz inequality is
$$
{\left(\sum_{i=1}^{n} u_i v_i\right)}^2 \leq \left(\sum_{i=1}^{n} ...
1
vote
1answer
64 views
Solution to a functional equation
Let $n,i$ be positive integers and $C$ a strictly positive real value.
Consider the equation for $f$ :
$$1*\ln(f(n)) = C * \sum_{3<i* \ln(i) < \sqrt{n}} \left(\ln[f( i* \ln(i) )-1)] - \ln[(f( i* ...
2
votes
2answers
57 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:
$$
...
0
votes
1answer
78 views
Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
Let $a_1, a_2, a_3, \dots , a_n$ be positive integers whose sum is 1. Prove that:
$\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$.
I thought maybe the Cauchy ...
5
votes
4answers
126 views
Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)
If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$.
As a first step to prove this ...
15
votes
1answer
206 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
1
vote
0answers
24 views
Estimation of a scalar product
I encountered the following, which shouldn't be that hard, but I can't get my head around it.
The problem is the following estimate (part of a bigger equation, but here's just the difficult part):
...
2
votes
1answer
60 views
Jensen's inequality and $L^p$ norms
Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
0
votes
3answers
35 views
Let $a,b \in \Re$. If 0 < $\epsilon$ < min{|a|, |b|}. Show this inequality
Let $a,b \in \Re$. If 0 < $\epsilon$ < min{|a|, |b|}.
$ {\frac{|a+\epsilon|}{|b+\epsilon|}} \leq {\frac{|a|+\epsilon}{|b|-\epsilon}}$
I tried to use triangular inequality.
But have no idea of ...
9
votes
2answers
208 views
How prove this integral inequality $\int_{0}^{s}f(x)\,dx\le\int_{s}^{1}f(x)\,dx\le\dfrac{s}{1-s}\int_{0}^{s}f(x)\,dx$
let $f(x)>0$ is continuous and is increasing on $[0,1]$,and
$s=\dfrac{\int_{0}^{1}xf(x)dx}{\int_{0}^{1}f(x)\,dx}$
show that
...
1
vote
2answers
31 views
Do inequalities hold under square-root (or exponentiation in general)?
This has been bothering me lately. My proof-skills are rusty (and were never great to begin with). I dimly recall having seen this (or something related to it) in a math course I took a while ago, but ...
7
votes
2answers
88 views
Help with an inequality problem
I came to ask this because I am really stuck at this problem. I have tried everything from arithmetic mean, geometric mean and harmonic mean. Also, I have tried playing with the variables and such, ...
9
votes
2answers
173 views
Inequality $\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0$
Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$,
$$
\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0.
$$
I have this question a very long time ago from a book or magazine but I ...
