Questions on proving and manipulating inequalities.

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2
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1answer
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8
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1answer
120 views

A generalization of arithmetic and geometric means using elementary symmetric polynomials

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. A while ago I noticed that if you form the polynomial $$ P(x) = (x - a_1)(x-a_2) \cdots (x-a_n) $$ then: The arithmetic mean of $a_1, \ldots, ...
0
votes
2answers
17 views

Integration inequality question help: Sketch the curve y=1/u for u > 0…

Sketch the curve $y=\frac{1}{u}$ for $u > 0$. From the diagram, show that $\int\limits_1^{\sqrt{x}}\frac{du}{u}< \sqrt x-1$, for x > 1. Use this result to show that $0 < \ln(x) < ...
2
votes
1answer
31 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...
3
votes
1answer
19 views

Using Conditional Jensen inequality proof the following

$X_1,X_2,\ldots,X_n$ are i.i.d. random variables, $X_1>0$, $E[X_1]=\mu$, $E[X_1^k]<\infty$ for $1<k \leq2$. Proof: $$ E\left[\left(\frac{1}{n}\sum_{i=1}^nX_i\right)^k\right]\leq ...
1
vote
1answer
323 views

Tricks to solve inequalities

I am wondering if there are some tricks to solve inequalities which are not manageable analytically. For example consider the inequality (say we restrict on positive $x$): $\displaystyle \frac {\text ...
1
vote
3answers
67 views

If $a,b,c\in\mathbb{R^+}$ such that $ abc = 1 $ and $ ab + bc + ca = 5 $. Prove that $ 17/4 \leq (a+b+c)\leq 1+ \sqrt{32}. $

If $a,b,c\in\mathbb{R^+}$ such that $ abc = 1 $ and $ ab + bc + ca = 5 $. Prove that $$ \frac{17}{4} \leq (a+b+c)\leq 1+ \sqrt{32}. $$ My attempt Tried using Vieta but it didn't work. Also I used ...
3
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2answers
331 views

Inequalities involving arithmetic, geometric and harmonic means

Let $A$, $G$ and $H$ denote the arithmetic, geometric and harmonic means of a set of $n$ values. It is well-known that $A$, $G$, and $H$ satisfy $$ A \ge G \ge H$$ regardless of the value $n$. ...
3
votes
1answer
154 views

Solve an inequality using Cauchy-Schwarz Inequality

Le $a,b,c,d \in \mathbb{R^{+}}$. Using Cauchy-Schwarz Inequality prove that the following inequality holds: $$\frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} \ge \frac{1}{\frac 1a + \frac 1b} + ...
-1
votes
0answers
14 views

Is it true that they really chose dashed lines for > and < and solid lines for $\ge$ and $\le$ in linear inequalities for graphing? [on hold]

I've noticed that they use > and < for dashed lines and $\ge$ and $\le$ for solid lines in linear inequalities for graphing also above, below, to the left, or to the right. I know that the dashed ...
2
votes
1answer
54 views

Prove that $ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2\ge\left( \frac{x_1+\dots+x_n}{n} \right)\left(\frac{y_1+\dots+y_n}{n} \right). $

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that ...
3
votes
0answers
43 views

If a positive decreasing function satisfies $f(0)=1$ and $f''\le f$ on $[0,1)$, then $f'(0)\ge-\sqrt{2}$

Let, $f(x)$ be a twice differentiable function defined on $(-1, 1)$ and $f(0) = 1$. Let, $f(x) ≥ 0$, $f'(x) ≤ 0$ and $f''(x) ≤ f(x)$ for all $x ≥ 0$. Show that, $f'(0) ≥ -√2$. I am telling you ...
1
vote
1answer
74 views

Evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$

I Have to evaluate $\pi$ using $\arctan(\frac{\sqrt{3}}{3})$ with an error with no more than $10^{-10}$ using taylor approximation $ p_{2n-1}(x) \approx\arctan(x)$ . So, After manipulation, I get: ...
4
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1answer
137 views

Does every record of the arithmetic derivative of natural numbers occur at a practical number?

Consider the arithmetic derivative of natural numbers, as defined here. By this definition, for every integer $n>1$, with canonical prime factorization ...
0
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1answer
30 views

does this inequality imply the Cauchy Schwarz inequliaty?

Let $z_i,w_i \in \mathbb{C}$. My teacher in class proved the following inequality: $$ \Big( \sum_{j=1}^n \mathrm{Re}(z_jw_j) \Big)^2 \leq \sum_j |z_j|^2 \cdot \sum_j |w_j|^2$$ Question: Does this ...
0
votes
0answers
13 views

Upper-bounds to $B_z(a,b)$

Is there any standard technique to produce nice upper-bounds to the incomplete beta function $$B_z(a,b)=\int_0^z t^{a-1} (1-t)^{b-1} dt \,?$$ Disclaimer: this question is intentionally not too ...
1
vote
2answers
58 views

Absolute value inequality - Please guide further

Prove that if the numbers $x$, $y$ are of one sign, then $\left|\frac{x+y}{2}-\sqrt{xy}\right|+\left|\frac{x+y}{2}+\sqrt{xy}\right|=|x|+|y|$. Expanding the LHS, ...
5
votes
0answers
144 views

Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...
2
votes
5answers
86 views

Proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$

What is the proof for $\forall x \in [0, \frac{\pi}{2}]\quad \sin(x) \ge \frac{x}{2}$ ? Assuming it is true.
1
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1answer
30 views

A problem of inequality and request for help

Problem If $n$ be a positive integer, prove that $$\begin{align} \frac{1}{2\sqrt{n+1}} < \frac{1.3.5 \dots (2n-1)}{2.4.6 \dots 2n}< \frac{1}{\sqrt{2n+1}} \end{align}$$ Following is my ...
3
votes
3answers
46 views

Prove this inequality.

Let $S=a_1+...+a_n<1$ where $a_i>0$. Prove that $1+S<(1+a_1)\cdot ... \cdot (1+a_n)<{1\over 1-S}$. I started with the right inequality but I am not sure it iss plausible (I did something ...
2
votes
2answers
69 views

Show $\lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + \lambda_na_n\rvert < 1$ when $\lvert a_i\rvert < 1$ and $\lambda_i\geq 0$

If $\lvert a_i\rvert < 1$, $\lambda_i\geq 0$ for $i = 1,\ldots,n$ and $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$, show that $$ \lvert\lambda_1a_1 + \lambda_2a_2 + \cdots + ...
3
votes
4answers
62 views

How to show $ \sin x \geq \frac{2x}{\pi}, x \in [0, \frac{\pi}{2}]$?

I have tried the following: $$ f(x) = \sin x-\frac{2x}{\pi} \\ f'(x)= \cos x-\frac{2}{\pi} \\ f''(x) = -\sin x \leq 0 $$ But this doesn't seem to be heading in the right direction as it would appear ...
4
votes
1answer
62 views

Proving $\sqrt{100,001}-\sqrt{100,000} < \frac{1}{2\sqrt{100,000}}$

Proving $\sqrt{100,001}-\sqrt{100,000} < \frac{1}{2\sqrt{100,000}}$ I squared both sides of the equation to get $100,001 + 100,000+-200\sqrt{10}\sqrt{100,001} < \frac{1}{400,000}$. I am just ...
1
vote
1answer
31 views

Bounding the increments of $\sin(e^{-x})$

Show that $|\sin(e^{-b}) -\sin(e^{-a})| \leq {b-a \over e^{-a}}$ for all $a \leq b$ This is part of a basic calculus class so i would appreciate answers suitable for my knowledge.
0
votes
1answer
27 views

Simple inequality proof in analysis

Just need verification on whether my proof is valid. I couldn't find a straightforward way to prove this inequality directly, so I tried a proof by contradiction instead. The question: Let $a, b \in ...
1
vote
2answers
139 views

Two Problem: find $\max, \min$; number theory: find $x, y$

Find $x, y \in \mathbb{N}$ such that $$\left.\frac{x^2+y^2}{x-y}~\right|~ 2010$$ Find max and min of $\sqrt{x+1}+\sqrt{5-4x}$ (I know $\max = \frac{3\sqrt{5}}2,\, \min = \frac 3 2$)
4
votes
3answers
87 views

Let a,b,c be positive real numbers numbers such that $ a^2 + b^2 + c^2 = 3 $

Let $a,b,c\in\mathbb{R^+}$ such that $ a^2 + b^2 + c^2 = 3 $. Prove that $$ (a+b+c)(a/b + b/c + c/a) \geq 9. $$ My Attempt I tried AM-GM on the symmetric expression so the $a+b+c \geq 3$, but I ...
1
vote
4answers
54 views

What are all values of $x$ in $\mathbb{R}$ that satisfy $4 < |x+2| + |x-1| < 5$?

I am having some problems getting started with this problem, as I never had to deal with an inequality that was between two values with absolute values. Any help is appreciated. The problem is find ...
4
votes
2answers
105 views

Need to prove that “If $x+y \ge 1$ then $x \ge \frac 12$ or $y \ge \frac 12$”

So I have this one homework assignment where I have to prove the following clause "If $x+y \ge 1$ then $x \ge \frac 12$ or $y \ge \frac 12$". I thought that if I assign $x=y$ and put it like "$2x \ge ...
1
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4answers
3k views

Is this correct method to prove that $a^2 + b^2 + c^2 ≥ ab + bc + ac$, when $a,b,c \geq 0$?

Can I prove it like this: Let's say that $a=b=c$ so we get "If $a \geq 0$ then $3a^2 ≥ 3a^2$" Now I take the negation of that statement and get "If $a \geq 0$ then $3a^2 < 3a^2$" The anti-thesis is ...
2
votes
1answer
37 views

inequality involving power sums

Let $x_1, x_2, ... ,x_n$ be positive real numbers and define $S(k)$ to be the power sum $S(k) = x_1^k + x_2^k +... + x_n^k$ . It is given that $S(3) = 3$ and that $S(5) = 5 $. Find the best lower ...
2
votes
2answers
57 views

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$ [duplicate]

Show that $a^2 + b^2 + c^2 \geq ab + bc + ca$ for all positive integers $a$, $b$, and $c$. I am not sure how to approach this problem. Should I divide this problem into multiple cases based on ...
1
vote
1answer
20 views

Prove that for all $m$, there exist some $k$, such that $(m-n)^2 > m^2$ for all $n>k$

I have a problem where I need to prove: $\forall m \in \mathbb{N}:\exists m \in \mathbb{N} ∋(m−n)^2>m^2~∀n>k$ My thought was since it is only "there exists some k.." can I not say: if $k = ...
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vote
2answers
337 views

Find $\limsup$ and $\liminf$ of a sequence and prove $\liminf a_n \leq \limsup a_n$.

I have a question of finding lim sup and lim inf of $a_n=\frac{1}{n} + (-1)^n$ and prove $\liminf a_n \leq \limsup a_n.$ So the work below is what I did for the first part. $a_{odd\ n} = ...
1
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1answer
220 views

limit inf /sup if $x_n\leq y_n$

I have just 2 problems : 1) Find the $\limsup x_n$ and $\liminf x_n$ where $x_n= e^{-n}$. 2) Let $x_n\leq y_n$ for every $n\in\mathbb{N}$ . Show that $\liminf x_n\leq\liminf y_n$ and $\limsup ...
2
votes
2answers
2k views

Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$

I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences. NOTE: subsequential limits are limits of subsequences Prove: $a_n$ is bounded $\implies ...
3
votes
1answer
46 views

How to prove/dispute the following log inequality?

I was wondering if the following inequality is true: $$\forall x,N\in \mathbb N^+: \lceil \log_2\left(\lfloor\frac{N}{x}+1\rfloor\right)\rceil\leq \lceil\log_2 (N+1)\rceil - \lfloor\log_2 (x)\rfloor ...
2
votes
1answer
55 views

Proof Involving Generalized Mean

Let $x=(x_1,...,x_n) \in \mathbb R^n$ and $$g(p)=\sqrt[p]{\frac{1}{n}\sum_{k=1}^{n} |x_k|^p)}$$ Using Hölder's inequality, show that $g(p)$ is increasing on $(0,\infty)$. For a sequence with ...
0
votes
1answer
19 views

Writting an inequality to represent a situation (Help needed) [on hold]

In the course of the day, Annalise will drive her car to work and later use the city rail system to get around her different branch offices. At one rail stop, she needs to ride a bus. Will she stay ...
1
vote
3answers
78 views

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then $\lim \inf (x_n) \leq \lim \inf(y_n)$ [duplicate]

If $X=\{x_n\}, Y = \{y_n\}$ be bounded sequences of real numbers. Then, if $x_n \leq y_n~\forall~n$, then show that $\lim \inf (x_n) \leq \lim \inf(y_n)$ and $\lim \sup (x_n) \leq \lim \sup (y_n)$ ...
1
vote
1answer
21 views

How to prove the inequality? [on hold]

Set $f(x)=1-(1-\lambda)^x$, where $\lambda \in (0,1)$, show that $f(x)/x \ge f(x)-f(x-1)$ holds for any $x\ge 1$.
3
votes
2answers
69 views

lim sup, lim inf, and inequalities for $a_n \le b_n$ [duplicate]

Suppose we have two sequences ${a_n}$ and ${b_n}$, which satisfies $ a_n \le b_n$ for $n=1,2,3,\ldots$. Do we have the following inequalities to be true? $$\limsup_{n \to \infty} a_n \le \limsup_{n ...
7
votes
4answers
450 views

$X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$

Can anyone prove this question? I tried but I didn't get any I idea, so I hope someone can solve it. Let $X_n\leq Y_n$ for each $n\in \Bbb N$. Show that $\liminf X_n \leq \liminf Y_n$ and $\limsup ...
3
votes
4answers
482 views

Given that a,b,c are distinct positive real numbers, prove that (a + b +c)( 1/a + 1/b + 1/c)>9

Given that $a,b,c$ are distinct positive real numbers, prove that $(a + b +c)\big( \frac1{a}+ \frac1{b} + \frac1{c}\big)>9$ This is how I tried doing it: Let $p= a + b + c,$ and $q=\frac1{a}+ ...
1
vote
0answers
54 views

A problem of inequality

Let $a_1, a_2, a_3$; $b_1, b_2, b_3$; $c_1, c_2, c_3$; $d_1, d_2, d_3$ be all real numbers. We need to show that $$\begin{align}(a_1b_1c_1d_1 + a_2b_2c_2d_2 &+ a_3b_3c_3d_3)^4\\ &\leq ...
0
votes
1answer
35 views

Find $\min x^TAy+b^Tx+c^Ty$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

The problem seems to be easy but I can't find a solution :( Problem: Given $A\in\mathbb{R}^{m\times n}, A\ge 0, b\in\mathbb{R}^{m}, c\in\mathbb{R}^{n}$. Minimize $f(x,y) = x^TAy+b^Tx+c^Ty$ subject to ...
1
vote
0answers
14 views

Existence of solution for linear matrix inequality?

Suppose $x$ is a $n\times1$ column vector. How to know whether the following matrix inequality has solution or not? $$Ax\leq B$$ where $A$ is a $m\times n$ matrix and $B$ is a $n\times 1$ column ...
1
vote
1answer
35 views

Proving that a system of equalities and inequalities is inconsistent

Prove that the system $a,b,d,e,f,g,h,i>0$ $ae+ai−bd+ei−fh=0$ $aei−hfa-bdi−gbf=0$ is inconsistent. I tried using some standard techniques such as factoring, or multiplying an equality and ...
2
votes
1answer
45 views

Integral inequality $\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$

Let $f(0) = 0$ and $0<f'(x)\leq1$ for all $x \geq0$, then prove: $$\int_0^x{f(t)^3 dt \leq \left( \int_0^x f(t) dt\right)^2} :\forall x>0$$ The hint I was given was "differentiate, factor and ...