Questions on proving, manipulating and applying inequalities.

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2
votes
1answer
34 views

Deriving a bounding $\delta$ of an interior point

This question is based on the Baby Rudin's 2.16: Regard $Q$, the set of all rational numbers, as a metric space, with $d(p,q)=\lvert p -q \rvert$. Let $E$ be the set of all $p \in Q$ such that $2 ...
2
votes
1answer
46 views

Strength of the inequalities between the means, comparing $Q_n-A_n, A_n-G_n, G_n-H_n$.

For $n$ real positive numbers $a_1, a_2, \ldots, a_n$, let $$Q_n = \sqrt{\frac {\sum_{k = 1}^n a_k^2}n},$$ $$A_n = \frac {\sum_{k = 1}^n a_k}n,$$ $$G_n = (\prod_{k = 1}^n a_k)^{\frac 1n},$$ $$H_n = ...
3
votes
1answer
80 views

Show that $(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6$ for $a^2 + b^2 + c^2 + d^2 = 1$.

For $a, b, c, d \in \Bbb R$ such that $a^2 + b^2 + c^2 + d^2 = 1$, show that $$(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4 \le 6.$$ The answer uses the mysterious identity ...
5
votes
3answers
120 views

Is the Riemann integral of a strictly smaller function strictly smaller?

We all know that if $f\leq{}g$ in $[a,b]$ then $$ \int_a^bf\,dx\leq\int_a^bg\,dx $$ now, imagine that we have $f<g$, is it true that $$ \int_a^bf\,dx<\int_a^bg\,dx $$
5
votes
4answers
244 views

Estimate the sum of alternating harmonic series between $7/12$ and $47/60$

How can I prove that: $$\frac{7}{12} < \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} < \frac{47}{60}$$ ? I don't even know how to start solving this...
0
votes
1answer
52 views

Function and inequality problem

Let $f$ an ascending and convex function on $(0,+\infty)$. I must to prove that: $$ f(\vert \sin(x) \vert +3) -f(\vert \sin(x) \vert)< f(x+3)-f(x) $$ I know that a solution of that is to ...
0
votes
1answer
26 views

Prove, for every $l \geq 3$ , the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds

I need to prove that for every $l \geq 3$, the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds. ($l$ is integer) This is what I tried so far. $$ \begin{align} x &= ...
5
votes
0answers
159 views
+50

Prove $\sum_{cyc}\left(\frac{a^4}{a^3+b^3}\right)^{\frac34} \geqslant \frac{a^{\frac34}+b^{\frac34}+c^{\frac34}}{2^{\frac34}}$

When $a,b,c > 0$, prove $$\left(\frac{a^4}{a^3+b^3}\right)^{\frac34}+\left(\frac{b^4}{b^3+c^3}\right)^{\frac34}+\left(\frac{c^4}{c^3+a^3}\right)^{\frac34} \geqslant ...
3
votes
1answer
19 views

“Reverse” of frobenius matrix norm inequality

Suppose that we have some $m \times n$ matrix $C$, and its full rank (skeleton) decomposition $$ C = AB^T, $$ where $A$ is $m\times r$ and $B$ is $n\times r$ for some $r$. We know that frobenius norm ...
0
votes
1answer
24 views

Ιnequality relationship

Let $a,b,c,d$ positive numbers. They are connected with the relations $$b<d,\quad a<c,\quad b<a,\quad d<c$$ Is it possible to prove that $a-b<c-d$?
3
votes
0answers
80 views

Prove $\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$

$a,b,c >0$ and $a+b+c=3$, prove $$\left(\frac{a+1}{a+b} \right)^{\frac25}+\left(\frac{b+1}{b+c} \right)^{\frac25}+\left(\frac{c+1}{c+a} \right)^{\frac25} \geqslant 3$$ I try to apply AM-GM ...
3
votes
0answers
99 views

Prove $(x^2y+y^2z+z^2x)\cdot \left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2} +\frac{1}{(z+x)^2}\right) \geqslant \frac94$

$x,y,z > 0$ and $x+y+z=3$, prove $$(x^2y+y^2z+z^2x)\cdot \left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2} +\frac{1}{(z+x)^2}\right) \geqslant \frac94$$ My immediate thought is that this inequality is ...
2
votes
0answers
29 views

Prove $\sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$ for positive $a,b,c$

For $a,b,c >0$, prove $$\sum_{cyc} \frac{(a+2b+3c)^2}{(2a+b)^2} \geqslant 5 + 7 \cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$$ My notation $$\sum_{cyc}a^2b= a^2b+b^2c+c^2a$$ What I try: 1. Using C-S ...
4
votes
2answers
252 views
+50

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
0
votes
1answer
58 views

Question using Young inequality

In fact i am looking for $\alpha_1,\alpha_2,\alpha_3$ such that the following inequality is true:$a^\frac{1}{80}\le \alpha_1\frac{a^{\frac{1}{4}}}{b^{\frac{3}{4}}c^6}+\alpha_2b^{\frac{1}{16}}+\alpha_3 ...
1
vote
1answer
16 views

How to bound a ratio of integer and a real number in floor function?

I have : $$n\le\left\lfloor \dfrac{m}{x}\right\rfloor,$$ where $n$, $m$ are positive integers and $x<1$ is a positive real number. I would like to bound the ratio $\frac{n}{m}$. So I will get: ...
0
votes
0answers
12 views

Existence of solutions for a scaled integer linear inequality

Assume that I know there exist non-negative integer solutions to a linear system of integer equations (with coefficients from $\{-1,0,1\}$ and non-negative constant terms in my case). Is there any ...
2
votes
0answers
100 views
+50

Inequality with analytic functions on the unit ball

Let $g(z) = \sum_{n\geqslant 0} a_nz^n$ be an analytic function where $a_n$ only take values in $\{0,1\}$ (not sure if it is a necessary condition, it is just the case I'm considering). Let ...
7
votes
3answers
796 views

relationship between sum of squares and sum

I have to admit I am not good at math since it's been a while since I did the last math problem. I am working on a project where there is a problem that can be summarized like this: if ...
0
votes
1answer
61 views

Find all real solutions of $ \frac{ae^x}{2e^x-1} < 1 $

Question: Find all real solutions of $ \frac{ae^x}{2e^x-1} < 1 $, where $a$ is a positive constant. This is what I have attempted: Consider $$ \frac{ae^x}{2e^x-1} < 1 $$ Case 1: ...
0
votes
1answer
29 views

Solving two compound inequalities simultaneously

Is it even possible to solve this? I'm trying to find the point where one ceiling function is less than another: $\lceil \frac {q}{m_1} \rceil < \lceil \frac {q}{m_2} \rceil$ $K = \lceil ...
4
votes
1answer
65 views

Tricky Multivariable Inequality

Given 100 positive real numbers $x_1, x_2, \cdots, x_n$ that satisfy $x_1^2+x_2^2+\cdots+x_n^2>10000$ and $x_1+x_2+\cdots x_n\le 300$, prove that there exist three numbers from this set such that ...
-2
votes
0answers
30 views

How to get the value a in a expression that have log function? [closed]

There is an inequality ln(2a+1)>a Could someone show me step by step, how to get the range of a? Thanks~
2
votes
3answers
52 views

Do you write plus/minus if a variable squares equals the square root of a number?

For example, if I have $x^2 = \sqrt{49}$. I know that $7$ is the number, but as my final answer, do I write that $x = +\sqrt{7}$ and $-\sqrt{7}$ or just $x = \sqrt{7}$?
0
votes
7answers
87 views

How to proof this inequality [closed]

I saw a topic yesterday, where they used the following inequality as a hint: $$\frac{1}{a+b}+\frac{1}{a+c}\geq \frac{4}{2a+b+c}$$ for some positive integers $$a,b,c$$ and $$a+b+c=1$$ What is the ...
1
vote
1answer
90 views
+100

Existence of operator

I want to show that for $ s> \frac{1}{2} $ there is a bounded linear operator $ T: H^s(\mathbb{R}^n) \to H^{s-\frac{1}{2}}(\mathbb{R}^{n-1})$ following the below steps: Consider that $ u \in ...
-1
votes
0answers
33 views

An Inequality problem… [closed]

For $a,b,c,d$ real numbers $a\ge b\ge c\ge d\ge 0$ such that $b(b-a)+c(c-b)+d(d-c)\le 2 - \frac{a^2}{2}$. Find the minimum value of $ \dfrac{1}{b+2006c-2006d} + \dfrac{1}{a+2006b-2006c-d} + ...
5
votes
1answer
43 views

Prove : $2(x^3+y^3+z^3)+3xyz \ge 3(x^2y+y^2z+z^2x)$ with $x,y,z \gt 0$

At first I tried to divide both side by $xyz$, the inequality became: $$2\sum {\frac{x^2}{yz}}+3 \ge 3\sum{\frac xy}$$ Let $$\frac xy = a;\frac yz = b;\frac zx = c;$$ So all we have to prove is ...
1
vote
2answers
43 views

What is the motivation of this inequality?

Problem Let $$S_n = \sum_{k=1}^n \left(\sqrt{1+\frac{k}{n^2}} -1\right)$$ Show that $\lim_{n\rightarrow \infty} S_n = 1/4$. Solution We first observe that for all $x ...
-2
votes
3answers
92 views

Prove $\sqrt{x^{2} + y^{2}} \le |x| + |y|$ [closed]

How do I prove this? x and y are real numbers. Thanks for the help.
0
votes
1answer
27 views

$L^p$ Norm of product of two bounded functions

If $f$ and $g$ are bounded functions in $L^p[a,b]$, does the following inequality hold in $L_p$ spaces? $$\|fg\|_p\leq\|f\|_p\|g\|_p$$
2
votes
1answer
125 views

How do I prove this set of inequalities using Cauchy-Schwarz? [closed]

Hello all I am trying to understand inequalities and their real world applications, typically optimization techniques. I got into to college as a math major and this fall would be my first semester. ...
1
vote
5answers
125 views

Inequality problem $\frac 1x + \frac 1y + \frac 1z > 5$ prove

Prove that: $$\frac 1x + \frac 1y + \frac 1z > 5$$ where $x+y+z=1$, $x, y, z$ are real numbers not equal $0$ and $x\neq y \neq z $
0
votes
3answers
62 views

show that $3^n< n!$ if n is an integer greater than 6

This prove requires mathematical induction Basis step: $n=7$ which is indeed true since $3^7\lt 7!$ where $3^7=2187$, $7!=5040$, and $2187< 5040$ hence p(7) is true. Inductive step: Assume: ...
0
votes
0answers
16 views

Can I bound this $N((1+\delta)^{2N}+1)\leq N^j$? for any $j\in \mathbb{N}$, where $\delta<1$

For $N\in\mathbb{N}$ large enough. Can I bound this $N((1+\delta)^{2N}+1)\leq N^j$ for any $j\in \mathbb{N}$, where $\delta<1/2$? I tried using Matlab for j=5, but I'd like some ideas for a proof. ...
0
votes
0answers
21 views

Is $(1 + \tau / (3p))^k \ge e^{k\tau/(4p)}$ really true?

In the paper "Preserving Statistical Validity in Adaptive Data Analysis", it says that if $p \in (0,1], \tau \in [0, 1/3]$ then $$(1 + \tau / (3p))^k \ge e^{k\tau/(4p)}$$ I understand that $(1 + \tau ...
1
vote
1answer
18 views

What values are used for the counter $k$ in this proof involving a convergent subsequence?

Lemma Suppose that $(x_n)$ is a Cauchy sequence in a metric space $(X,d)$, and $x\in X$. Also suppose $(x_{n_k})$ is a subsequence of $(x_n)$ such that $x_{n_k}\to x$ as $k\to \infty$. Then $x_n\to ...
1
vote
1answer
37 views

Prove $\sum_{k=1}^n \sqrt{a_k^2 + b_k^2} \ge \sqrt{(\sum a_k )^2 + (\sum b_k)^2}$

I know this may be a primary problem but I cannot solve it. $$\sum_{k=1}^n \sqrt{a_k^2 + b_k^2} \ge \sqrt{(\sum a_k )^2 + (\sum b_k)^2}$$ with $a_k,b_k$ are positive reals and n is an integer. I ...
7
votes
2answers
92 views

show that $\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } \ge \frac { 2 }{ 1+a } +\frac { 2 }{ 1+b } +\frac { 2 }{ 1+c } $

Let $a,b,c$ are positive numbers,if $$a+b+c=1$$ show that $$\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } \ge \frac { 2 }{ 1+a } +\frac { 2 }{ 1+b } +\frac { 2 }{ 1+c } $$ I am ...
2
votes
3answers
115 views

Show that $|A+A|\geq (2n-1)$

Consider a set $A$ consisting of $n$ natural numbers $\{a_i\}_{i=1}^n$ such that $a_1<a_2 < \cdots <a_{n-1} < a_n$. Define the set $A+A$ such that it contains $a_i + a_j \ ; \ i \leq j$ ...
3
votes
2answers
35 views

Bound for a series containing $2^k$ and $k!$

Because I needed to evaluate the series $$S=\displaystyle\sum_{k=1}^{+\infty}\dfrac{1}{2^k+k!}$$ using the Milne inequality, I found for it, the bound: $$S\lt1-\dfrac{1}{e}.$$ Is it possible to have ...
1
vote
1answer
48 views

How is this inequality called? (And how to improve this process)

I am reading a book and it mentions the following: Let $u \in H^1_0(G)$; then $$\lVert u\rVert ^2_{L^\infty(G)} \le C \lVert u \rVert_{L^2(G)}\lVert u'\rVert_{L^2(G)}$$ Note: Here $G = (a,b) \subset ...
0
votes
0answers
16 views

Bound difference between weighted mean and mean of mean

Given a real finite series $\{x_N\}$ of data (eg. a year of daily mean of a measurements). If I compute monthly means from this data, and I get two series $\{\bar{x}_1,\dots,\bar{x}_m\}$ which stands ...
1
vote
1answer
28 views

Tighten a mean inequality

For a given real finite series, say $\{x_n\}$, and a subset of this series $\{y_m\}$ ($m < n$), that is a subsequent serie where some terms of $\{x_n\}$ have been withdrawn. If we would to compare ...
0
votes
1answer
57 views

Product of the differences of two pd matrices and their respective inverses is pd

Given two $\textbf{positive definite (pd), Hermitian}$ matrices X and Y, I am trying to determine whether $(X-Y)(Y^{-1}-X^{-1})$ will always be pd as well, and how to prove this. This formulation ...
1
vote
1answer
27 views

Prove the following inequality .

While I was going through matrix norms, I came across the fact that Operator norm of a matrix is less than equal to Frobenius Norm of the matrix. While trying to prove this fact I came across the ...
3
votes
3answers
48 views

Two real numbers, $x$ and $y$, satisfy the condition $x + y = 2 $. Show $xy(x^2+y^2) \leq 2$

Question: Two real numbers, $x$ and $y$, satisfy the condition $x + y = 2 $. Show $xy(x^2+y^2) \leq 2$ What I have attempted: Consider $$x+y=2$$ $$ \Leftrightarrow (x+y)^2 = 2^2 ...
0
votes
1answer
43 views

Prove this inequality $x^x\cdot y^y\cdot z^z\ge\frac{27}{(6-x-y-z)^3}$

Let $x,y,z\in (0.1)$ show that $$x^x\cdot y^y\cdot z^z\ge\dfrac{27}{(6-x-y-z)^3}\tag{1}$$ or $$x\ln{x}+y\ln{y}+z\ln{z}+3\ln{(6-x-y-z)}\ge 3\ln{3}$$ since $f(x)=\ln{x}$,By Jenson inequality ...
1
vote
2answers
40 views

If $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$

I am trying to find a simple way to show the fact that if $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$ for any number $d$. Is there a way to do this besides breaking it up into the cases 1) ...
0
votes
0answers
21 views

Blow up inequality

How to prove that there exist a $ t^*$ such that the solution blows up in finite time for this inequality $$ y'(t) \geq (y(t))^p- C, \quad p>1, c >0 .$$