Questions on proving and manipulating inequalities.

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5
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0answers
24 views

Counterexamples to the Matrix norm AM-GM inequality?

I am new here and this my first question, I hope I am being as clear as possible and apologize in advance for any misunderstandings. I am researching the Arithmetic-Geometric Mean (AM-GM) inequality ...
1
vote
3answers
41 views

$a^2+b^2+c^2=(abc)^2-2\leq 6$. Proof or counter-example needed for $a,b,c\gt 0$

I was working on an inequality proof in which I need to use the following inequality to conclude. $$\forall~a,b,c\gt 0~,~a^2+b^2+c^2=(abc)^2-2\implies a^2+b^2+c^2\leq 6$$ I can't think of any ...
5
votes
1answer
73 views

Prove that: $2(ab+bc+ca)-a^2-b^2-c^2\le6$.

Let $a,b,c>0$ such that: $\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}=1$. Prove that: $2(ab+bc+ca)-a^2-b^2-c^2\le6$. I have no idea for solve this problem.
2
votes
3answers
41 views

Prove $a^\alpha b^\beta + c^\alpha d^\beta \leq (a+c)^\alpha (b+d)^\beta$

I'm trying to prove that the sum of two log-convex functions is log-convex. I've figured out that this can be done by proving: $a^\alpha b^\beta + c^\alpha d^\beta \leq (a+c)^\alpha (b+d)^\beta$ for ...
-1
votes
1answer
29 views

The area of the graph consisting of all the points (x,y) such that $x^2 + y^2 \le 1 \le \left|x\right| + \left|y\right|$? [on hold]

I tried plotting graph of both functions but i am not able to get answer in the form of options given . What is the area of the region in $\mathbb R^2$ such that $$x^2 + y^2 \le 1 \le ...
0
votes
2answers
57 views

How to determine that if $\frac{n}{\sqrt{n^3+2}}>\frac{n+1}{\sqrt{(n+1)^3+2}}$ for any n is natural number

How to prove that $\frac{n}{\sqrt{n^3+2}}>\frac{n+1}{\sqrt{(n+1)^3+2}}$ for any n is natural number I have that problem when tried to determine an alternating series conditional converge or not. ...
0
votes
2answers
25 views

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$

Prove that $-x^2 \leq x^n \leq x^2$ for $-1<x<1, n\in \mathbb N, n \geq3$ I have no idea how to do this, I don't even know how to begin. Please help!
12
votes
3answers
63 views

Prove that $4x^2-8xy+5y^2\geq0$ - is this a valid proof?

I need to prove that $4x^2-8xy+5y^2\geq0$ holds for every real numbers $x, y$. First I start with another inequality, i.e. $4x^2-8xy+4y^2\geq0$, which clearly holds as it can be factorized into ...
0
votes
1answer
19 views

Invalid range from inequality

We were given this function and asked to give Range. $$f(x)~=~\dfrac{x^2}{x^2+1}$$ Now I took 3 cases and deduced that $\text{Range} = \left[~0,\infty ~\right)$ Now it is obvious that if we divide ...
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1answer
54 views

Prove the inequality, $\root3\of4\sin^2(x/2)<3(\sin x+1-x)^{2/3}$

Prove that $$\left(\sin^2{\frac{x}{2}}\right) \cdot \frac{\sqrt[3]{4}}{3} \cdot \frac{1}{{(\sin x + 1 - x})^{\frac{2}{3}}} <1$$
2
votes
3answers
45 views

Power function of fixed numbers.

Prove that $3^x-4^x+2x4^{x-1}\le0$, where $x\in[-0.5,0]$.Here is it's plot. I tried to do it by first and second derivative test but it involves $log$ which make the expression more complicated.
3
votes
1answer
59 views

Estimate the integral of $(1+x^2)^{-\alpha}$, where $\alpha>1/2$

I'm reading a proof of a theorem, and there's one step I couldn't understand why. It said that for all $a>0$ and $\alpha>1/2$, $$ \int_{a}^{\infty}(1+x^2)^{-\alpha} \ \mathrm dx ...
1
vote
1answer
23 views

Is this inequality of real numbers true?

Let $\alpha\in (0,1/2)$ be a parameter. Is it true thet for every $x>y>0$ real numbers we have $$y^{-\alpha} - x^{-\alpha} \leq C y^{-\alpha -\frac{1}{2}} (x-y)^{1/2}$$ for some constant ...
2
votes
0answers
21 views

Obtain an inequality of real numbers

Let $x,y> 0$ be real numbers such that $x>y$. Let $\alpha \in (0,1/2)$ be a parameter then I obtained the following inequality: $$y^{-\alpha} - x^{-\alpha} \leq C ...
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2answers
55 views

About primes and Euler's totient function.

Is the number of primes $< n$ itself less than the number of positive integers that are less than $n$ and relatively prime to $n$?
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2answers
34 views

About an inequality using 'lnlnn' related to Robin's inequality.

Is $\ln\ln n < \sigma(n)/n$ ? Or is this grossly innaccurate or too vague? (The comments about blocking my questions is totally unfair. I haven't asked that many questions. I have never been ...
0
votes
0answers
25 views

A Integral inequality.

For any positive integer $n \in {\mathbb{N}^ + }$, prove inequality $$\int_{ - \pi }^\pi {\left| {\cos \left( {\frac{{2n + 1}}{2}t} \right)} \right| \cdot \left| {\frac{1}{{\sin \left( {\frac{t}{2} + ...
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votes
5answers
129 views

Inequality in Algebra: $1 \leq x_1 x_2 \cdots x_n$ implies that $2^{n} \leq (1 + x_1)(1+x_2) \cdots (1 + x_n).$

How do I prove that if $x_1, \ldots, x_n$ are positive real numbers, then $$1 \leq x_1 x_2 \cdots x_n \text{ implies that } 2^{n} \leq (1 + x_1)(1+x_2) \cdots (1 + x_n).$$ I attempted a proof by ...
2
votes
4answers
43 views

Domain of the function $f(x) = \sqrt{\frac{3^x-4^x}{x^2-4x-4}}$ will be?

I tried solving this question by $1.$ $-1$ and $4$ will not be in domain because denominator can not be zero . $2.$ Either both denominator and numerator will be positive or negative so that whole ...
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votes
0answers
47 views

How does this inequality follow from the given conditions? [on hold]

$a, b, c>0, a^2 + b^2 + c^2 \le3$ then $ab + bc +ca - abc \le2$
2
votes
0answers
17 views

Determining asymptotics of a function given a series of difference-like inequalities

I have a function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ and I know it satisfies the following properties. $f(x) \leq \frac{\log{\sqrt{2}}}{2x}$ and for all $A \geq 1$ and $B \geq ...
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vote
0answers
39 views

Measure of set satisfying acute version of AM/GM

Let $N>2$ be integer. Take a $N$ real numbers $x[k] \in [0,1]$. By the AMGM inequality we have that their geometric mean is less or equal to their sum divided by $N$. My question is: what is the ...
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votes
0answers
41 views

Need help solving $\frac{x+2}{4} - \frac{x-2}2 \lt 3$ [on hold]

Could some one please explain how to solve the inequality, step by step in the most simple as possible way? $$\frac{x+2}{4} - \frac{x-2}2 \lt 3$$ Your help would be much appreciated.
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votes
1answer
33 views

Is $\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$? [on hold]

Is it true that for some fixed $k\ge2$ and for all sufficiently large $x$ and $y$ with $y\ge x$ we have, $$\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$$ where $\pi(x)$ is the prime counting function. I am ...
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votes
0answers
19 views

When n > 1, which of the following expressions must increase in value as n increases? [on hold]

When $n > 1$, which of the following expressions must increase in value as n increases? $1- {1\over n}$ $\sqrt{n}$ ${1\over n^2}$
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6answers
61 views

Prove that $2^n(n!)^2 \leq (2n)!$

Prove that $2^n(n!)^2 \leq (2n)!$ One can also use the following result to prove the above: $2 · 6 · 10 · 14 · · · · · (4n − 2) = \frac{(2n)!}{ n!}$. The above relation gives, $(2n)!=2^n n! ...
12
votes
1answer
196 views

How to prove that $\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$?

Let $a,b,c>0: (a+b)(b+c)(c+a)=ab+bc+ca$. How to prove that $$\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$$
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vote
2answers
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Complex Number inqualities

Although the inequalities are not defined on complex numbers. But does the inequality $x < 4 + 5i$ be said to possess any solutions ? Where $ i = \sqrt{-1}$.
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4answers
28 views

Solution set of modulus inequations

$$|2x+5|\leq\dfrac{1}{2}$$ What will be the solution set? My attempt: For $x \in \left(-\infty,-\dfrac{5}{2}\right)$ $x\geq -\dfrac{11}{4}$ For $x \in \left[-\dfrac{5}{2},\infty\right)$ $x\leq ...
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votes
2answers
36 views

Analysis Inequality with all positive numbers [on hold]

Prove: $$ab\leq \varepsilon a^2+{b^2\over4\varepsilon}$$ with $\varepsilon\gt0,\ a>0,\ b>0$
0
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1answer
27 views

Solving a linear inequality without choosing points to check

Consider an inequality like this: (2x-1)/(x+5) > 0 If we start by multiplying both sides by (x+5), we get 2x-1>0, which has the solution x > 1/2. However, the ...
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0answers
20 views

A trace inquality for the product of symmetric PSD matrices

I'm estimating the expectation of a quadratic form, using two different estimators, and would like to compare the variances. The first is a MC estimator, and the other is the Hutchinson estimator. I ...
1
vote
5answers
67 views

Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$

This is easy to prove for negative $x$ but what about positive $x$? Should I use MVT?
2
votes
1answer
18 views

First hit of a martingale

I came across this result somewhere and I don't grasp its proof in its entirety. Let $M$ be a continuous martingale such that $M_0 = 0$. Define $\tau_x = \inf\{t\geq 0: M_t =x \}$. Then, $$P\{\tau_a ...
1
vote
1answer
20 views

An integral inequality involving increasing function

Let $0\leq a< b \leq \pi/2$ Let $f:[a,b]\to\mathbb R$ be a positive, increasing function. Prove that $\left|\int_a^b f(t)\cos(t)dt\right|\leq f(b)(b+\sin(b))-f(a)(b+\sin(a))$ I ...
0
votes
2answers
59 views

Prove for integers $a,b\ge 3$ that $a^2+b^2+1>(a+1)(b+1)$

Prove for integers $a,b\ge 3$ that $a^2+b^2+1>(a+1)(b+1)$. Original question asked for positive real solutions, but I've changed it to integers. It's question I've come up with. AM-GM ...
2
votes
4answers
254 views

Explain this inequality, related to logarithms

I am trying to understand a proof of Stirling's formula. One part of the proof states that, 'Since the log function is increasing on the interval $(0,\infty)$, we get $$\int_{n-1}^{n} \log(x) dx ...
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1answer
48 views

Show that $ \sqrt{8a+b^3}+\sqrt{8b+c^3}+\sqrt{8c+a^3}\geq 9 $ [on hold]

Let $ a, b, c $ be non-negative numbers such that $ a+b+c=3 $. Show that $ \sqrt{8a+b^3}+\sqrt{8b+c^3}+\sqrt{8c+a^3} \geq 9 $ When does equality hold?
5
votes
5answers
96 views

Is it possible to prove this? $\ln(\frac{x}{x-1}) < \frac{100}{x} $ for $ x > 1$

$-\ln(1-(\frac{1}{x})) < \frac{100}{x} $ for $ x > 1$ is what I want to prove. I pulled a negative sign out and I got $\ln(\frac{x}{(x-1)}) < \frac{100}{x} $ for $ x > 1$. How do I ...
3
votes
1answer
16 views

Under what conditions is $ \max_{a \in A} (f(a) - g(a)) \geq \max_{a \in A} f(a) - \max_{a \in A} g(a) $ true?

Consider: $$ \max_{a \in A} (f(a) - g(a)) \geq \max_{a \in A} f(a) - \max_{a \in A} g(a). $$ Intuitively, it seems obvious it should be true, but I was having a hard time coming up with a rigorous ...
2
votes
3answers
56 views

For which values of $N$ is $x^N \ge \ln x$ for all $0 < x < \infty$?

Find, with proof, the smallest value of $N$ such that $$x^N \ge \ln x$$ for all $0 < x < \infty$. I thought of adding the natural logarithm to both sides and taking derivative. This gave ...
2
votes
2answers
24 views

Help with Factorial Inequality Induction proof

So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step. What I have is: (n=1) $3^{(1)} + (1)! = 4$ and $((1) + 3)! = ...
7
votes
0answers
26 views

If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?

I wanted to ask a separate question to focus on an elementary issue from my question Does the inverse of a polynomial matrix have polynomial growth?. Let $p : \mathbb{R}^n \to \mathbb{R}$ be a ...
1
vote
1answer
20 views

Prove that $\frac{1}{\left|1-\frac{|x|}{|y|}\right|}\leq \frac{1}{\left|1-\frac{\delta_x}{\delta_y}\right|}$.

Let $|x|\leq \delta_x$ and $|y|\geq \delta_y$ where $\delta_x, \delta_y >0$ and $\delta_x< \delta_y $. Prove that $$\frac{1}{\left|1-\frac{|x|}{|y|}\right|}\leq ...
2
votes
2answers
71 views

Minimise $ab+bc+ac$

Let $a,b,c \in \mathbb R$, and $a^2+b^2+c^2=1$ How can I calculate the minimum value of $ab+bc+ac$? (i.e. most negative) I've tried using the fact that $(a-b)^2+(b-c)^2+(a-c)^2 \ge 0$ but this gives ...
1
vote
1answer
34 views

Find Min: $A= \frac{bc}{a(b+2c)} +2 [ \frac{ac}{b(c+a)} + \frac {ab}{c(2a+b)}]$

Given $a,b,c>0$ such that: $ \frac{4a}{b} (1+ \frac{2c}{b}) + \frac{b}{a} (1+ \frac{c}{a})=6$ Find Min: $A= \frac{bc}{a(b+2c)} +2 [ \frac{ac}{b(c+a)} + \frac {ab}{c(2a+b)}]$ My try: Let: ...
1
vote
5answers
67 views

Any tips for solving $\frac{|4x-2|}{|2x+1|} \le 1$ as succinctly as possible?

$\frac{|4x-2|}{|2x+1|} \le 1$ So as I currently see it, I have two choices: 1) Attempt to solve algebraically but that has led me down some long paths when I believe the question should be solvable ...
0
votes
2answers
55 views

Triangle Inequality Like Equation [closed]

If we are in $R^2$ and define $d(a,b)$ as the set of points between $a$ and $b$ we can create an equation like this: $$d(x,z) \subseteq d(x,y) \times d(y,z)$$ where the $\subseteq$ is the subset ...
0
votes
2answers
35 views

looking for the first n that satisfies this equation

I have been trying to find the first $ n \in \Bbb N $ that will satisfy the following equation: $$ \sqrt[n]{2} - 1 < 10^{-8} $$ So far I have tried something like $$ \sqrt[n]{2} - 1 < 10^{-8} ...
3
votes
3answers
77 views

Prove $x^2 - x + 1 $ is always positive.

While solving a question, I came up with an inequality : $(1+x)(1-x+x^2)>0$ The book stated - where $(1-x+x^2)$ is always positive as $D<0$ and $a>0$ I'm not that sure how did it ...