Questions on proving, manipulating and applying inequalities.

learn more… | top users | synonyms (1)

0
votes
3answers
47 views

proving an inequality related to $AM\ge GM$

$$a^2+ab+b^2\ge 3(a+b-1)$$ $a,b$ are real numbers using $AM\ge GM$ I proved that $$a^2+b^2+ab\ge 3ab$$ $$(a^2+b^2+ab)/3\ge 3ab$$ how do I prove that $$3ab\ge 3(a+b-1)$$ if I'm able to prove the ...
1
vote
3answers
43 views

Prove $\ln x \ge \frac{x-1}{x}$

Prove that for every $x>0$: $$\ln x \ge \frac{x-1}{x}$$ What I did: $$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$ $$f(1) = g(1) = 0 $$ So it's enough to prove that $$ f'(x) \ge g'(x)$$ ...
0
votes
1answer
30 views

Name of the inequality $|x|+|y| \geq |x+y|$?

What is the name of the inequality $|x|+|y| \geq |x+y|$? I remember seeing this inequality and thinking it was the triangle inequality, but that only holds if $x,y,z$ are the side lengths of a ...
1
vote
1answer
43 views

Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far: induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates ...
3
votes
0answers
62 views

Prove that $a+b^2+c^3+d^4 \ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$

If $0 < a \le b \le c \le d$ and $abcd = 1$ prove that $$a+b^2+c^3+d^4 \ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$$ I first thought of multiplying both sides with ...
2
votes
0answers
18 views

Prove that $(n-1)!S_m\geq (n-m)!m!P_m.$

If $a_1, a_2,\cdots a_n$ be all positive rationals such that $S_n=a_1^m+a_2^m+\cdots +a_n^m$, $P_m=\sum a_1^ma_2^m\cdots a_m^m$ (the sum of products m taken m at a time). Prove that ...
0
votes
0answers
24 views

On even almost perfect numbers other than the powers of two

(Note: There is a related question in MSE as well as in MO.) Let $\sigma(x)$ be the sum of the divisors of $x$. We say that $X$ is almost perfect if $\sigma(X) = 2X - 1$. Antalan and Tagle (in a ...
-3
votes
1answer
115 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
2
votes
6answers
81 views

Prove $\frac{1}{3}(x+y+z)^2 \geq xy + yz + xz.$

Prove that for nonnegative $x,y,z$ that $\frac{1}{3}(x+y+z)^2 \geq xy + yz + xz.$ I saw this result in a problem but didn't know how to prove it. I tried expanding and collecting to get the ...
0
votes
1answer
31 views

Estimate $|f(x)| \le \frac C{|x|^3}$

Let $$f(x) = \frac{\sin x}x+\frac{\sin(x-1)}{2(x-1)}+\frac{\sin(x+1)}{2(x+1)}.$$ Find the common denominator and use common trigonometric identities to establish that $$|f(x)| \le \frac ...
0
votes
0answers
5 views

Norm and Inner Product Inequality in Hilbert spaces

Let $H$ be a Hilbert space, and suppose that $C \subset H$ is closed, convex and nonempty. Then, for $y_{j}=P_{C}(x_{j})$, $j=1,2$ where $P_{C}$ is the metric projection onto $C$ and $x_{1},x_{2} \in ...
1
vote
0answers
23 views

Criteria for inequality

I am working with an inequality and I need to prove something of the shape $$c\cdot a+d\cdot b \leq a\cdot b$$ The numbers $a$ and $b$ have a specific form, but for the $c$ and $d$ I only know that ...
3
votes
0answers
15 views

Sum of Complex Numbers and Modulus Inequality

Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that: $ \left| \displaystyle\sum_{j \in S}z_{j} \right| \geq ...
0
votes
0answers
4 views

Lagarias and Robin theorems versus multiplicative property

If I use for example Robin's theorem, see here in the section Growth of arithmetic functions, or Lagarias equivalence, see (5) here has sense ask us what is the more sharp inequality for ...
3
votes
1answer
51 views

Prove that $a^ab^bc^c\geq (\frac{a+b}{2})^{\frac{a+b}{2}} (\frac{b+c}{2})^{\frac{b+c}{2}}(\frac{c+a}{2})^{\frac{c+a}{2}}$

Prove that $$a^ab^bc^c\geq \left(\frac{a+b}{2}\right)^{\frac{a+b}{2}} \left(\frac{b+c}{2}\right)^{\frac{b+c}{2}}\left(\frac{c+a}{2}\right)^{\frac{c+a}{2}}\geq \left(\frac{a+b+c}{3}\right)^{a+b+c}$$ ...
0
votes
0answers
15 views

Inequality in the proof of Weak Harnack Inequality

Let $\Omega \subset \mathbb{R}^{n}$ a bounded domain s.t $B_{1} \subset \Omega$ , $u \in H^{1}(\Omega)$ a nonnegative supersolution in the weak sense of the equation $Lu=-D_{i}(a_{ij}(x)D_{j}u)$ ...
0
votes
1answer
21 views

Why is the following reverse triangle inequality true for given series?

I wish to show that for $(a_k)$ a sequence of numbers, $a_k \in \mathbb{R}$ then claim : $|\sum\limits_{k = n+1}^m a_k | \leq ||\sum\limits_{k = n+1}^\infty a_k| - |\sum\limits_{k = ...
1
vote
1answer
10 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
0
votes
2answers
26 views

How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...
1
vote
3answers
37 views

Proving the convergnce of a sequence

So, I have to prove that the sequence defined as $a_{n+1}=\frac{6(1+a_n)}{7+a_n}$ converges and then find the limit. I have few questions; Do i have to assume that $a_n \geq 0$ or $a_n \leq0$. ...
1
vote
0answers
56 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
-1
votes
0answers
45 views

Proving $n^n \cdot \left(\frac{n+1}{2}\right)^{2n}\geq \left(\frac{n+1}{2}\right)^3$

Proving $$n^n \cdot \left(\frac{n+1}{2}\right)^{2n}\geq \left(\frac{n+1}{2}\right)^3$$ , Where $n\in \mathbb{N}$ $\bf{My\; Try::}$ Using $\bf{A.M\geq G.M}\;,$ We get ...
2
votes
5answers
93 views

How do I prove $\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}…+\frac{1}{n+n}$

How do I prove the following inequality- $$\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$ without the help of induction? Thanks for any help!!
1
vote
2answers
66 views

Prove that $\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}…+\frac{1}{n+n}$

How do I prove $$\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$ without using induction? Note that clearly $n\neq 0$ Thanks for any help!!
4
votes
5answers
87 views

Prove that $\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$

Without using Mathematical Induction, prove that $$\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$$ I am unable to solve this problem and don't know where to start. Please help me ...
5
votes
1answer
128 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
3
votes
1answer
40 views

Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
0
votes
0answers
21 views

Understanding ADMM: how is it applied to this particular problem?

While reading Boyd's paper on ADMM I encountered an issue. Consider the following problem: Problem. Minimize $f(u) + g(v)$ subject to $Au + Bv = c$, where $f$ and $g$ are closed, proper, convex and ...
0
votes
2answers
28 views

Triangle Inequality?

I'm having trouble proving the following claim: $\forall a, b, c \in \mathbb{R}_+: T(a, b, c) \Rightarrow [|a − b| < c$ and $|b − c| < a$ and $|a − c| < b]$ Where $T(a, b, c)$ is a ...
2
votes
1answer
29 views

Prove that $|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$

Prove for any $a, b \in [-1, 1]$ that $$|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$$ I'm sure there is a solution using the Cauchy-Swartz inequality. Thus i tried to ...
1
vote
1answer
31 views

Prove Jensen Inequality holds for a function

Given function $$f:\mathbb{R}^n_{+} \rightarrow \mathbb{R}, \ f(x) = \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}}$$ Show that for any $x, y \in \text{dom} \ f, \theta \in [0,1]$, ...
0
votes
0answers
17 views

How to prove the following equivalence

$T$ is a positive real number $\lambda_k$ is a family of positive reals number that $\to \infty$ with $k$ $f(s)$ is a function that is a $o(s)$ and $0<f(s)<s$ I want to prove that $\liminf_k ...
1
vote
4answers
73 views

Proving that the exponential inequality $e^x \ge x^e$ holds for all $x \ge 0$ [duplicate]

How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$? I tried to do this by setting $f(x)=e^x-x^e$ Plotting this function shows this easily, as seen here. However, when I tried to prove ...
2
votes
3answers
70 views

Trying to prove for all integers: $n \ge 1 \implies \frac{2n+1}{2n+2} \ge \frac{\sqrt{n}}{\sqrt{n+1}}$

Been racking my brain on this one.. I've tried some things but not sure if it flows logically: $\forall x \in \mathbb{Z}: n \ge 1$ $n+2 \ge 1$ $2n+2 \ge n+1$ $\frac{2n+1}{2n+2} \ge ...
1
vote
2answers
67 views

how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$

Prove that if $a,b,c$ are non-negative real numbers, then $(ab + bc + ca)^2 \geq 3abc(a+b+c)$. I tried to compute from $(a-b)^2 + (b-c)^2 + (c-a)^2 \geq 0$.
1
vote
2answers
44 views

Is this function bounded above?

Consider nonconstant functions $f(x), g(x) \neq x$. Suppose there exist positive constants $k_1$ and $k_2$ such that $k_{1} x \leq f(x) \leq k_{2} x$ and $\frac{1}{2}k_{1} x \leq g(x) \leq k_{2} x$. ...
4
votes
2answers
133 views

Prove that $7<e^2<8$

I was asked by my teacher to prove that $7<e^2<8$ using only algebraic methods and knowing that $2<e<3$. I don't know how to do this, where to start from, but I guess that I would need ...
0
votes
0answers
9 views

What function gives this inequality?

Let $i<j<k<l$ be positive integers. I want to find a "nice" function $f(x, y)$ such that $f(i, k)+f(j, l)>\max(f(i, j)+f(k, l), f(i, l)+f(j, k))$. This seems a bit tricky because the ...
0
votes
2answers
25 views

Question on double inequality with radicals

A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$? Thank you very much.
2
votes
1answer
79 views

What is the intuition behind the Cauchy-Schwarz inequality in the real numbers?

The Cauchy-Schwarz inequality states that $$\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$$ The proof, with the discriminant argument, is ...
3
votes
2answers
46 views

Sketch the set of points satysfing an inequality $|z+1|+|z-1|\leq2$

The inequality is $$|z-1|+|z+1|\leq2$$ I used a triangle inequality to show that Since triangle inequality states: $$|z+w|\leq|z|+|w|$$ Then $$|z-1+z+1|\leq|z-1|+|z+1|\leq2$$ So $$|2z|\leq2$$ From ...
0
votes
0answers
10 views

Inequality problem for Markov Process

Is there any upper bound available for the following quantity $$E[\max_{1 \leq k \leq n} X_k]$$ where $\{X_n\}$ is a Markov chain.
0
votes
1answer
28 views

Question on statement of Cauchy-Schwarz inequality: $\vert\langle x,y \rangle \vert \leq \Vert x \Vert \cdot \Vert y \Vert$

Denoting the Cauchy-Schwarz inequality as Wikipedia does, $$\vert\langle x,y \rangle \vert \leq \Vert x \Vert \cdot \Vert y \Vert$$ and noting that $$\vert\langle x,y \rangle \vert = \Vert x\cdot y ...
0
votes
2answers
61 views

An inequality on an arbitrary function

I'm trying to find the complexity of a program and reduced the question to the following one: Let $g$ be a function from natural numbers (including $0$) to natural numbers. Assume that for every $n ...
1
vote
1answer
22 views

Solving inequality to state function f>1

I have the following function, $$f = \frac{x-a}{y-a} $$ I want to specify the condition for $f>1$ I wrote it as, $$f>1$$ when $$\frac{x-a}{y-a}>1$$ so, I rewrote it as, $$x-a>y-a$$ ...
2
votes
3answers
41 views

Why are some solutions excluded if we simply multiply the denominator in an fraction inequality?

I have the next inequality for which I have to find the solutions : $$\frac{2x-5}{3x-1}\geq 1, \text{ where } x\in\mathbb{R}$$ I know I have to subtract $1$ and then I have to analyse the sign for ...
0
votes
1answer
31 views

How to solve the “$\min$” function coming with absolute value in an inequality

Prove that if $|x-a|<\min(k/(2|1+b|),1)$ and $|y-b|<(k/2(1+|a|))$, then $|xy-ab|<k.$ In this question I don't know how to deal with "$\min$" part in the first inequality.
1
vote
1answer
26 views

How to show that $(X-a)^+\le X^++|a|$

How to show that $(X-a)^+\le X^++|a|$, where $X, a$ are real Is the following OK; $(X-a)^+ +(a-X)^+=|X-a|$ and If the claim (in the yellow box) is not true then also; $(a-X)^+> a^++|X|$ but ...
3
votes
2answers
43 views

Proof of $a^x ≥ x+1 \; \forall x \in \Bbb R \implies a=e$

I'm trying to prove the following : Let $a>0$ a real number. Then : $\quad a^x ≥ x+1 \;\; \forall x \in \Bbb R \iff a=e$ I managed to prove the '$\Longleftarrow$' part : $x≥0$ then ...
2
votes
3answers
43 views

Solving inequlity with $e^x$

I'm studying differential calculus, but one of the questions involves solving an inequality: $$(x-2)e^x < 0$$ I intend to go deeper in solving inequalities later, but I just want to understand ...