Questions on proving and manipulating inequalities.

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1
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2answers
41 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 $n$?
-2
votes
1answer
20 views

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

Is $\ln (\ln n) < \frac{\sum n}n$ ? Or is this grossly innaccurate or too vague? (The comments about blocking my questions are totally unfair. I haven't asked that many questions. I have never ...
0
votes
0answers
20 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} + ...
3
votes
5answers
123 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 ...
1
vote
4answers
31 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 ...
-3
votes
0answers
41 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
9 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 ...
1
vote
0answers
31 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 ...
-6
votes
0answers
36 views

Need help solving $\frac{x+2}{4} - \frac{x-2}2 \lt 3$

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.
-1
votes
1answer
31 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 ...
3
votes
6answers
57 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! ...
10
votes
0answers
81 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 $$\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}\leq\frac{3\sqrt{3}}{4}$$
1
vote
2answers
40 views

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}$.
1
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4answers
26 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 ...
-1
votes
2answers
30 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
votes
1answer
26 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 ...
1
vote
0answers
19 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
19 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
56 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
251 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 ...
1
vote
1answer
45 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
88 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)! = ...
6
votes
0answers
25 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
70 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 [on hold]

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 ...
4
votes
0answers
64 views

Conditions for Trace Inequality Tr( ( A² - B² ) Z) >= 0

Consider the $M \times M$ complex matrices ${\bf A}, {\bf B}, {\bf Z} \succeq {\bf 0}$. We have the relation ${\bf A} \succeq {\bf B} \succ {\bf 0}$, i.e. both are nonsingular. Further assume $\bf Z = ...
3
votes
1answer
51 views

Find $\lim_{n\to\ \infty}\sqrt[n]{\frac{\sum_{i=1}^p a_i^n}{p}}$

I was trying to solve a question of an entrance exam. I am having trouble in the following problem. Please help me. For positive real numbers $a_1, a_2, \ldots, a_p$ find the value of ...
7
votes
1answer
86 views

when index is irrational number with inequality

Let $x>0$, show that $$x^{\sqrt{3}}+x^{\frac{\sqrt{3}}{2}}+1\ge 3\left(\dfrac{1+x}{2}\right)^{\sqrt{3}}$$ we consider $$f(x)=2^{\sqrt{3}}(x^{\sqrt{3}}+x^{\dfrac{\sqrt{3}}{2}}+1)- ...
1
vote
3answers
30 views

an inequality with noninteger order

I want to Show that the following inequality is true or not: For $0<q<1$ and $a\geq b\geq0$, $$a^q-b^q\leq 2(a-b)^q.$$ Could you please help me in showing this inequality is true or not? ...
1
vote
3answers
66 views

Positivity of power function.

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a\le0$. I tried to do it by first derivative test but derivative become more complicated (same with 2nd derivative for ...
1
vote
1answer
123 views

Are these inequalities true?

The following is a question regarding an inequality direction. With $a,b,c $ being real and non-negative. Can I say that following is ALWAYS true. $$1+(a+b+c)^2 \le 2(a+b+c)^2$$ What about the ...
0
votes
1answer
27 views

Question regarding an inequality relationship [closed]

If $a,b,c >0$, is true that the following is always satisfied? $$(a+b+c)^2 > {(a+\sqrt{b^2+c^2})^2} $$
5
votes
1answer
68 views

Proving $\frac {2n}{(a+b)^n} \le \frac {1}{a^n} + \frac {1}{b^n}$ for $a,b>0, n\in\mathbb{N}$ by induction

prove using induction: $$ \frac {2n}{(a+b)^n} \le \frac {1}{a^n} + \frac {1}{b^n} $$ $$a,b \gt 0 , n \in N$$ my attempt: base $n=1$: $$ \frac {2}{(a+b)} \le \frac {1}{a} + \frac {1}{b}$$ ...
1
vote
1answer
31 views

Precalculus equation solving with inequalities

For which a $\in \mathbb{R}$ does the equation $$|x-1| + 2|x-2| = a$$ have two solutions? I divided this into three cases: Case 1 $x \geq 2$: $$x-1 +2(x-2) = a$$ $$x-1+2x-4 = a$$ $$3x-5 = a$$ ...
0
votes
1answer
23 views

How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality?

Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u ...
0
votes
1answer
35 views

How to bound this difference between two logarithmic expression

I want to bound the difference between two logarithmic expression shown below with a constant number i.e not function of $x,y,z$ where $x,y,z \in \mathbb{C}$. The difference is $$ ...
4
votes
1answer
55 views

Let $l$ be a natural number. Prove that $n\lt\sqrt{n ^ 2 + l}\lt n+1$ for almost every $n$.

In my assignment I have to prove the following statement: Let $l$ be a natural number. Prove that for almost every $n$ the following inequality is true: $$n\lt\sqrt{n ^ 2 + l}\lt n+1$$ I chose ...
0
votes
2answers
22 views

Inequality Solution Set

Find the solution set for the inequality: $|x-2| > |x+6|$ I'm just not sure how to work this out? I've found the answer is $(-\infty,-2)$ using wolframAlpha, however there isn't a step by step. ...
2
votes
0answers
42 views

On the second part of solution of a question due to Erdos

Problem. Let $a_1<a_2<\dotsb<a_n\le 2n$ be a sequence of positive integers. Then $$ \min [a_i,a_j]\le 6\left(\Big[\frac n2\Big]+1\right), $$ where $[a_i,a_j]$ denotes the least ...
3
votes
3answers
125 views

Prove that $a^2+b^2+c^2\geq [2(a-b)^2(b-c)^2(a-c)^2]^{1/3}$

Mathematica seems to know that this statement is true, yet I am struggling to prove it. Possible useful inequalities are Minkowski and the geometric mean. Using the geometric mean inequality I can ...