Questions on proving and manipulating inequalities.

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1
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1answer
33 views

How can I determine the bounds for this inequality?

I have the following inequality: $$ -40 < \bigg(\frac{a}{2^{52}}\bigg) (2^{b}) < 40 ,\ \text{with}\ a \in [-2^{52}, 2^{52}]\ \text{and}\ b \in [-1024, 1024].$$ How can I "thin" the range of ...
1
vote
2answers
88 views

Permutation of positive real numbers

Consider a set of positive real numbers $\{P_1,P_2,\dots,P_n\}$ and a permutation of this set $\{Q_1,Q_2,\dots,Q_n\}$. Is it possible to find a permutation such that ...
0
votes
2answers
18 views

Region given by these inequalities in XY Plane

Given region as $ 0\leq x \leq y $ , $ x+y \leq 1$ . I did this as Is this correct ?
2
votes
1answer
24 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
0
votes
1answer
14 views

Young's inequality implies $L^p$ convergence of convolution

I am reading a material which states: If $f_n \to f$ in $L^1(\mathbb{R})$, $g \in L^P(\mathbb{R})$. Then $f_n*g \to f*g$ in $L^p(\mathbb{R})$ by Young's inequality. But I cannot see why Young's ...
4
votes
1answer
123 views
+100

A Cauchy-Schwartz type inequality

Given positive integers $k<n$ and positive real numbers $x_1$, $x_2, \dots, x_n$. Denote $$ A={x_1\over x_2+x_3+\dots+x_{k+1}}+{x_2\over x_3+x_4+\dots+x_{k+2}}+\ldots+{x_n\over x_1+x_2+\dots+x_k}$$ ...
-1
votes
1answer
38 views

How can I rearrange $|a-b|<|b|/2$ to get $a^2>(b^2)/4$? [on hold]

How can I rearrange $|a-b|<|b|/2$ to get $a^2>(b^2)/4 $?
1
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0answers
30 views

Non-trivial inequality

Equation (1) on page 7 of http://arxiv.org/pdf/1312.7308v1.pdf claims that: $$\frac{1}{t}\log \left(\frac{\log ((1+\epsilon)t)}{\omega} \right) \geq c \Rightarrow t \leq \frac{1}{c} \log \left( ...
-5
votes
1answer
57 views

Inequality on matrix norm: $ \lVert A^n \rVert \leq \lVert A \rVert^n $ [on hold]

If $A$ is a $n \times n$ matrix and assume we have a matrix norm $\lVert \cdot \rVert$. In a proof I need the following property: $ \lVert A^n \rVert \leq \lVert A \rVert^n $. I don't know how to ...
0
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1answer
41 views

How to prove that $(a+b-ab)^n+(1-a^n)(1-b^n) \geq n $ for $a,b\in [0,1]$ and $n\in\mathbb{N}$?

Let $a,b\in [0,1]$ and $n\in\mathbb{N}$. Prove the following inequality: $$(a+b-ab)^n+(1-a^n)(1-b^n) \geq n $$ I thought on using M Induction: Assuming that the inequality holds for $n=k,$ ...
1
vote
1answer
26 views

If $X\ge 0$ and $a\ge E[X]$, then $P(X\gt a)\ge (E(X)-a)^2/ E(X^2)$ [on hold]

I need help with this problem. Prove that if $X\ge0$ and $E[X^2]<\infty$ then for all $a\neq0$, $E[X]\le a$, we have $$P(X\gt a)\ge\frac {(E(X)-a)^2}{E(X^2)}$$ Progress I have my doubts if ...
6
votes
2answers
176 views
+100

Find Minimum value of $P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$

Given: $x,y\in (-\sqrt2;\sqrt2)$ and $x^4+y^4+4=\dfrac{6}{xy}$ Find Minimum value Of $$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$$ Could someone help me ?
0
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7answers
124 views

Prove $a^{2}+2ab+3b^{2}+2a+6b+3 \geq 0$

If $a,b\in\mathbb{R}$ prove that the following inequality holds: $a^{2}+2ab+3b^{2}+2a+6b+3 \geq 0$
6
votes
4answers
236 views

To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.

Let $f$ be a compactly supported nonnegative $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2\leq C f(x) $ by showing that for every point ...
0
votes
0answers
34 views

What bounds can we put on the largest root of a polynomial?

For a polynomial $p(x)=x^{n+1}+a_{n} x^{n} + \cdots + a_1$ with roots $|x_1| < \cdots < |x_n|$ can we find relatively simple function $M(a_1, \dots, a_n)$ such that $$|x_i| \leq M(a_1, \dots, ...
7
votes
5answers
586 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 ...
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3answers
47 views

How to solve this inequality? $2\cos(x+1)>0$ [on hold]

Please help me answer this question. How can I solve the following inequality? Solve the following inequality: $2\cos(x+1)>0$. Thank you.
3
votes
2answers
66 views

What is the largest $k$ such that $ \frac { k(abc) }{ a+b+c } \le \left( a+b \right) ^{ 2 }+\left( a+b+4c \right) ^{ 2 } $?

Find the largest value of $ k $ such that $ \frac { k(abc) }{ a+b+c } \le \left( a+b \right) ^{ 2 }+\left( a+b+4c \right) ^{ 2 } $
5
votes
1answer
80 views

Why it is true for rapid decreasing function $g$ that: $\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\leq A_{l,k}(1+|y|)^{l}$

If $g$ is of rapid decrease, that is $\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x)|<\infty$, then we have: $$\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq ...
18
votes
2answers
480 views

Maximizing curious symmetric function from simple combinatorics

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (This question has been posted at ...
-1
votes
0answers
18 views

Trace of matrix with orthogonal matrix

Let $H$ be an orthogonal matrix. Let $\Phi$ be given matrix. What is $Tr(H\Phi)$? Any lower bounds that will be useful? What if $H$ is also symmetric?
72
votes
0answers
2k views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
3
votes
0answers
70 views

How prove this complex inequality with same as (2014 china CMO) Cauchy-Schwarz inequality

let $r$ is give numbers,let $z_{1},z_{2},\cdots,z_{n}$ such $|z_{i}-1|\le r,i=1,2,\cdots,n,r\in(0,1)$ show that ...
0
votes
0answers
97 views

Prove the Schwarz inequality using $ 2xy \leq x^2 + y^2 $

Im really bad at analysis and this problem was recommend to me to help me grasp some basics of $\epsilon $ $\delta $ So im doing a problem ( though its like 12 pieces ) this is i guess the fourth ...
1
vote
2answers
67 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
0
votes
0answers
28 views

Variational Inequalities and how they are used?

I am doing undergrad research in this field next semester and I have never heard of this topic before. I tried wikipedia and reddit for help but nothing seems to help. I just want to know what I'm up ...
5
votes
1answer
60 views

What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications. What I want is a collection of questions/problems which look very complex but get ...
5
votes
3answers
178 views

Let a,b $\in$ $R$ . Show that $a^4+b^4+8\ge 8ab$

Let a,b $\in$ $R$. Show that $a^4+b^4+8\ge 8ab$. The question is from the inequalities section of An Excursion in mathematics by Bhaskaraycharya Pratisthanan. My heuristics include using the AM-GM ...
3
votes
2answers
155 views

let $a,b,c >0 $ and $abc=1$,prove that $\sqrt{1+8a^2}+ \sqrt{1+8b^2}+ \sqrt{1+8c^2}\leq 3(a+b+c )$

let $a,b,c >0 $ and $abc=1$,prove that $\sqrt{1+8a^2}+ \sqrt{1+8b^2}+ \sqrt{1+8c^2}\leq 3(a+b+c )$ can anyone help me with this question. i've tried to assume that $a\geq b \geq c $ as my teacher ...
4
votes
1answer
66 views

The equality case of the Schwartz inequality

Question: The fact that $a^2 \geq 0$ $ \forall a \in \mathbb{R}$; elementary as it may seem, is nevertheless the fundamental idea upon which most important inequalities are ultimately based. The ...
1
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2answers
39 views

Proving that this function is negligible

Let $f(n) =\frac{1}{2^{\sqrt n}}$, where $n \in \mathbb{N}$. I want to prove that $\forall a \in N, a \ge1\; \exists k: f(n) \le n^{-a}, \forall n \ge k$ I attempted to solve the inequality ...
2
votes
1answer
89 views

How prove $\frac{2}{3}<\frac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\frac{3}{2}$

Let $x\in(0,1]$, show that $$\dfrac{2}{3}<\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\dfrac{3}{2}$$ My try: since $$\begin{align}\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3} ...
2
votes
1answer
64 views

How find this inequality minimum $\sum_{cyc}\sqrt{a^2+b^2+ab-2a-b+1}$

Let $0<a,b,c<1$, find this follow minimum $$\sqrt{(a+b)^2-(a+1)(b+2)+3}+\sqrt{(b+c)^2-(b+1)(c+2)+3}+\sqrt{(c+a)^2-(c+1)(a+2)+3}$$ My try: since ...
4
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2answers
211 views

How prove this matrix inequality $\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$

Question: Let $A_{n\times n}$ and $B_{n\times n}$ be positive Hermitian matrices. Show that $$\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}.$$ I know this $$\det(A+B)\ge \det(A)+\det(B)$$ But My ...
2
votes
3answers
255 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 ...
12
votes
2answers
207 views
+50

Bernoulli's inequality and an unexpected limit

This question is inspired by What would happen to Bernoulli's inequality if $x<-1$?. Let $x_n=\min\{x\in{\bf R}:(1+x)^n\geq 1+nx\}$, where $n$ is natural and odd (my mistake in the first ...
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votes
0answers
28 views

Solving the inequality: $0.275k_1^2k_2(1-k_2)+0.05k_1k_2-0.5k_1 +1 <0$? [on hold]

where $k_1>0$, $0<k_2<1$ and $k_1k_2<2$ This question come from a stability analysis by Jury test. I would like to get an answer to identify k_1 could you help me to solve this ...
1
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0answers
17 views

Inequality to bound $\sum_i a_i b_i - \sum_i c_i d_i$ (harmonic eigenfunction/graph) type sum with constraints

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
4
votes
1answer
55 views

Inequality $\frac{x^3+y^3}{x-y}>4$

Let $x>y>0$ and $xy\geq 1$. Prove that $$\frac{x^3+y^3}{x-y}>4.$$ Of course we can factor $(x^3+y^3)=(x+y)(x^2-xy+y^2)$, but it is not very useful. For fixed $x-y$, we can try to find the ...
2
votes
1answer
43 views

Is this an upper bound or lower bound?

I came across a probability distribution function in my work, it is however difficult to find in closed form, therefore I am looking to either upper bound or lower bound it. Assuming $a,b,T$ are ...
1
vote
1answer
24 views

Trouble Understanding Proof Of Invariant Relationship

In part of a proof I am reading this is stated: $2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ) + (a_n + c_n )^2 + (b_n + d_n )^2 ≥ 2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ).$ (1) From this invariant inequality ...
-4
votes
0answers
45 views

is this systems of equations solvable

i saw this equation posted yesterday and i wanted to ask if i could have help helping this kid solve it. he is in pre-algebra and i am in algebra so not much of a difference. i have tried to solve for ...
0
votes
2answers
50 views

Verifying Property of Stochastic Integral

I am trying to verify this simple property for a stochastic integral. Given that f(t,w) is a bounded, nonanticipating function for a given Wiener process $W_t$ show that $E((\int_{0}^{T} f(s,w) ...
0
votes
2answers
15 views

Finding possible values for a function

After diving into simple inequalities, I've come across a particular exercise that requires me to find all possible value for $x$ for a given function. After searching about it, it seems to be simply ...
-1
votes
1answer
19 views

Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
0
votes
0answers
30 views

How prove this the number of ordered $n$-tuples $(\varepsilon_{1},\cdots,\varepsilon_{n})$such this following inequality is $2^{n-100}$

Interesting Question: for any complex numbers $z_{1},z_{2},\cdots,z_{n}$ such $$\begin{cases} |z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2=1\\ |z_{i}|\le\dfrac{1}{10},i=1,2,\cdots,n \end{cases}$$ ...
0
votes
1answer
36 views

Why $\left\|\sum_{i=1}^nx_i\right\|^2\leq n\sum_{i=1}^n\|x_i\|^2$

Why $$\left\|\sum_{i=1}^nx_i\right\|^2\leq n\sum_{i=1}^n\|x_i\|^2$$ for arbitrary norm on the inner product space over the real field? My attempt $$\left\|\sum_{i=1}^nx_i\right\|^2 = ...
3
votes
1answer
58 views

Prove that $a^ab^bc^c\ge (abc)^{(a+b+c)/{3}}$ [duplicate]

Prove that $$a^ab^bc^c\ge (abc)^{(a+b+c)/3}$$ My attempt: $$a^ab^bc^c\ge (abc)^{(a+b+c)/{3}}\implies ...
4
votes
3answers
395 views

Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.

Prove that the following inequality holds $$a^a b^b c^c\ge (abc)^{\frac{a+b+c}{3}}$$ if $a,b,c$ are positive. I'm not sure how to handle these kinds of powers. Are there any "famous" but not ...
3
votes
3answers
869 views

Proving Holder's inequality using Jensen's inequality

Let $p$ and $q$ be positive reals such that $\frac{1}{p}+\frac{1}{q} = 1$, so that $p,q$ in $(1,\infty)$. For $\vec a$ and $\vec b \in \mathbb{R}^2$ prove that $|\vec a \cdot \vec b | \leq ||\vec ...