Questions on proving, manipulating and applying inequalities.

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0
votes
1answer
15 views

Solving inequalities using sign patterns

I don't want the solution, I'm just confused as to what to do with the $e^x$ in this instance. Solve for $x$ in $e^x(2\cos x-1)≤0$ where $0≤x≤4\pi$. Typically I would just reduce down to $\cos x ...
0
votes
5answers
63 views

help me find inequality

If $a+2b>3$ and $b+3c>5$ then $a+b+c> \hspace{.1cm} ?$
0
votes
0answers
6 views

Muirhead's inequality generalization

Muirhead's inequality is a well-known fact: for all permutations of finite expressions $x_1^{\alpha_1}x_2^{\alpha_2}...x_n^{\alpha_n}$ (if the sets of powers satisfy $(\alpha_1,\alpha_2...\alpha_n) ...
3
votes
3answers
48 views

Proving an Inequality using a Different Method

Is there another way to prove that: If $a,b\geq 0$ and $x,y>0$ $$\frac{a^2}{x} + \frac{b^2}{y} \ge \frac{(a+b)^2}{x+y}$$ using a different method than clearing denominators and reducing to ...
1
vote
3answers
50 views

Prove inequality $x^sy^{1-s} \leq sx + (1-s)y$

Given $s \in (0,1)$, prove $$x^sy^{1-s} \leq sx + (1-s)y$$ for $x,y > 0$ Tried some algebraic manipulations but I'm guessing I need to use some trick. Any suggestions, hints?
0
votes
0answers
16 views

Calculate an upper bound for $\left|\frac{e^{i\alpha-\beta}-e^{-(i\alpha-\beta)}}{i\alpha-\beta}\right|$

Let $\alpha,\beta\in\mathbb R$. Calculate an upper bound for $$\left|\frac{e^{i\alpha-\beta}-e^{-(i\alpha-\beta)}}{i\alpha-\beta}\right|$$ I think that $\cosh$ is involved in the answer, but I can't ...
5
votes
1answer
113 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in V\\ ...
1
vote
0answers
27 views

On the Chernoff bound

Recently, I saw the Chernoff bound written as follows. Let $X_1,X_2,\ldots,X_n$ be drawn i.i.d. on alphabet $\mathcal{X}$ and let $f:\mathcal{X}\to [0,1]$ be any function. Let $\mathbb{E}f(X_1) = ...
0
votes
2answers
24 views

Inequality involving floor

Let $x$ be randomly chosen from $\{1,...n\}$. Define $X_{p}$ such that \begin{equation} X_p= \begin{cases} 1, & \text{if}\ p|x, \\ 0, & \text{otherwise.} ...
0
votes
0answers
42 views

Solving 3 linear equations with 6 variables, 3 of which are the same across each equation.

I feel like I could hash out the answer on my own but am struggling to think of an elegant way to show it. The equations are as follows: $$3.5x-2.5y-3z=A$$ $$-7.5x+3.75y+5.25z=B$$ ...
2
votes
1answer
53 views

Inequality on integrals of continuous functions: $\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$

Let $f\colon [0, 1] \to \mathbb{R}$ be a continuous function. How to prove $$\int_0^1 f^2(x)\,dx \geq \left(\int_{0}^{1} f(x) \,dx\right)^2$$ (I'm not getting anything.. any hint is appreciated)
1
vote
1answer
48 views

Use stirlings approximation to prove inequality.

I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't ...
9
votes
0answers
45 views

Problem with inequality $\min (x_1,x_2,\ldots,x_n)$

let $0\le x_i$, $i=1,2,\ldots,n$, and $a_i=1+(i-1)d$, $d\in[0,2],\forall i\in\{1,2,3,\ldots,n\}$, show that $$(1+a_n)\left(x_1+x_2+\cdots+x_n\right)^2\ge 2n \min(x_1,x_2,\ldots,x_n) \left(\sum_{i=1}^n ...
1
vote
1answer
18 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
3
votes
3answers
177 views

Proving the inequality $\frac{a+b}{2} - \sqrt{ab} \geq \sqrt{\frac{a^2+b^2}{2}} - \frac{a+b}{2}$

Show that for any two positive real numbers $a$ and $b$, $\frac{a+b}{2} - \sqrt{ab} \geq \sqrt{\frac{a^2+b^2}{2}} - \frac{a+b}{2}$ My attempt: $(\sqrt a-\sqrt b)^2\geq0\\\frac{a+b}{2}\geq ...
2
votes
1answer
32 views

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ ...
0
votes
0answers
21 views

Finding upper critical value with Chebyshev's inequality

Consider $X$ is a Poisson random variable with distribution $X$~$Pois(\theta)$. I define the mean in my hypothesis as $\lambda$ and nominal significance level $\alpha$. Null hypothesis $H_0 : ...
7
votes
2answers
97 views

Name of $|x|^p+|y|^p\le (|x|+|y|)^p$ ($p\ge 1$)?

I checked these What is the difference between square of sum and sum of square? Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$. It is easy to see $p$-th power ...
3
votes
2answers
36 views

How to show the inequality is strict?

This is an exercise from Rudin's Principles of Mathematical Analysis, Chapter $6$. Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) = 0$, and $$\int_a^b ...
5
votes
4answers
269 views

Minimum value of reciprocal squares

I am bit stuck at a question. The question is : given: $x + y = 1$, $x$ and $y$ both are positive numbers. What will be the minimum value of: $$\left(x + \frac{1}{x}\right)^2 + ...
-1
votes
0answers
27 views

Could someone give a detailed (yet elementary) proof for Jensen's inequality?

I want to prove that Suppose there is a function $f:[a,b] \to \mathbb R$, and there are $x_i \in [a,b], w_i \gt 0 $ for $i=1,\dots,n$ such that $\sum_{i=1}^nw_i=1$, then if the function is convex, ...
1
vote
2answers
24 views

A question on inequality and differentiation of logarithms

Show by differentiating that $\ln x$ is a concave function of $x$. Deduce that if $p,q,x,y$ are positive real numbers with ${1\over p}+{1\over q}=1$, then $$xy \lt {x^p\over p}+{y^q\over q}$$ I ...
-1
votes
2answers
28 views

What is the Solution? x ≥8lgx

x ≥ 8lgx I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.
3
votes
3answers
91 views

Does $\frac{x+y}{2}>\frac{a+b}{2}$ hold?

$a$ and $b$ are two real positive numbers. Given that $x=\sqrt{ab}$ and $y=\sqrt{\frac{a^2+b^2}{2}}$, which one has a higher value, $\frac{x+y}{2}$ or $\frac{a+b}{2}$? We know that ...
3
votes
1answer
56 views

Rewriting $|x-10|+|y-5|\leq 7$ so that absolute values disappear - Algebra

Equation 1: $|x-10|+|y-5|\leq 7$ I want to rewrite this equation into equations that do not have the absolute value. $|A|\leq b$ can be written as $A \leq b$ $A \geq -b$ I want to apply the ...
1
vote
2answers
42 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
6
votes
4answers
119 views

Show that $2^{\sqrt{2}}>1+\sqrt{2}$

Given that $\sqrt{2}>1.4$ and $(1+\sqrt{2})^5<99$, I need to show that $2^{\sqrt{2}}>1+\sqrt{2}$ From the given inequalities, I deduce that $(1+\sqrt{2})<\sqrt[5]{99}$ and ...
1
vote
0answers
21 views

Tighter upper bounds with ratios of powers of norms

This question arises in concentration or sparsity measures for finite sequences. Given $x\in \mathbb{R}^K$ and $1 \le r < s$, i try to find a tight upper bound for $$\psi_{r,s}(x) = \frac{\sum_1^K ...
3
votes
0answers
62 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
2
votes
2answers
53 views

O.I.M. polygon inequality

I am trying to prove an inequality which was used to prepare the Romanian O.I.M. team. I seem to lack ideas on how to tackle this problem. We take a convex polygon $P_1\ldots P_{n+2}$ and consider ...
0
votes
1answer
21 views

Is there an upper bound for expectation of product of two measurable function on a random variable?

I wonder if there is an useful upper bound for $\mathbb{E}_{x\sim p(x)}[f(x)g(x)]$ in the following form: $$ \mathbb{E}_{x\sim p(x)}[f(x)g(x)] \leq \mathbb{E}_{x\sim p(x)}[f(x)]\times xxxxxx $$ The ...
3
votes
0answers
22 views

The Hardy-Littlewood-Sobolev Inequality

Let $f:\mathbb R^n \to \mathbb C$, $n\ge 2$. I saw the line that the inequality $$ \left\| |x|^{-1} * |f|^2 \right\|_{L^\infty} \le C\|f\|_{L^{\frac{2n}{n-1},2}}^2 $$ with some constant $C>0$. Here ...
1
vote
2answers
94 views

how to solve the a,b,c inequality?

$a,b,c>0,a+b+c=3,$ prove that: $$\frac{a^3}{a^2+2b+c}+\frac{b^3}{b^2+2c+a}+\frac{c^3}{c^2+2a+b}\geq\frac{3}{4}$$
1
vote
2answers
51 views

$2^{m+1}-2^n\geq(m-n)^2$

So basically, I'm trying to see if this equality holds for large enough $n$ and such that $m\geq n$. However, Wolfram Alpha won't show me much, so I'm here to see if anyone has a solution for this. ...
4
votes
3answers
45 views

Multiplicating inequalities

I have two inequalities: $|x|\leq\sqrt{x^2+y^2}$ and $|y|\leq\sqrt{x^2+y^2}, \forall x,y \in \Bbb R$, can I multiply these inequalities to get $|xy|\leq x^2+y^2$? If yes, what is the justification? ...
2
votes
1answer
35 views

Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
0
votes
1answer
32 views

Supremum vs Integral [closed]

Let $h$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}h(t)\leq\int_{r}^{\infty}h(t)\frac{dt}{t} $$
1
vote
0answers
19 views
+200

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. $\tilde{\theta}$ can take on values from $0$ to ...
2
votes
1answer
47 views

Find a constant $C$ such that $ \Bigg| \frac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n$

Consider the following: $$ \Bigg| \dfrac{\prod_{i=0}^{k-1} (n-i) }{n^k} - 1 \Bigg| \leq \frac{C}{n}, \forall k \leq n $$ How to find an expression for $C$ independent of $k$ and thus $n$? It arises ...
1
vote
2answers
88 views

Prove that Euclidean distance in $\mathbb{R}^n$ is a distance

I'm trying to show that: $$\forall x,y\in\mathbb{R}^n, d(x,y)=\left(\sum_{i=1}^n(x_i-y_i)^2\right)^{1/2}$$ is a distance. However I have not proved Cauchy-Schwarz yet and I'm pretty sure I wouldn't ...
3
votes
1answer
80 views

Solving the trigonometric equation $\tan^2x+\cot^2x=2-\cos^{2014}(2x)$

I was solving the trigonometric equation $$\tan^2x+\cot^2x=2-\cos^{2014}(2x) $$ I solve it by inequality $|a|+\frac{1}{|a| }\geq 2$. $$ L.H.S=\tan^2x+\cot^2x =\tan^2x+\frac{1}{\tan^2x} ...
6
votes
1answer
341 views

Application of Jensen's inequality to $x^x+y^y+z^z$

Claim: If $x, y, z >0$ and $x+y+z = 3\pi, $ then $x^x + y^y + z^z > 81.$ My attempt: Let $f(w) = w^w$, so $f$ is convex on $(0, \infty).$ By Jensen's inequality, $f(x\frac{x}{3\pi}+ ...
5
votes
2answers
247 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.
1
vote
0answers
18 views

Martingale and quadratic variation inequality

I have the following inequality $$\mathbb{E}(\mid[M^{\Pi^m},M^{\Pi^m}]_T^{1/2}-[M^{\Pi^n},M^{\Pi^n}]_T^{1/2}\mid^p)\leq \mathbb{E}([M^{\Pi^m}-M^{\Pi^n},M^{\Pi^m}-M^{\Pi^n}]_T^{p/2}),$$ where $M$ is a ...
0
votes
2answers
60 views

For which $x, y\in\mathbb{R ^+}$ do we have $|xy-\frac{1}{xy}|\le|x-\frac{1}{x}|+|y-\frac{1}{y}|$?

I need to find all $x, y\in\mathbb{R^+}$ such that the following inequality holds. $$\Big| xy-\dfrac{1}{xy}\Big|\le\Big|x-\dfrac{1}{x}\Big|+\Big|y-\dfrac{1}{y}\Big|$$ If I substitute $x=2$ and $y=3$ ...
2
votes
0answers
16 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
10
votes
3answers
124 views

Struggling with an inequality: $ \frac{1}{\sqrt[n]{1+m}} + \frac{1}{\sqrt[m]{1+n}} \ge 1 $

Prove that for every natural numbers, $m$ and $n$, this inequality holds: $$ \frac{1}{\sqrt[n]{1+m}} + \frac{1}{\sqrt[m]{1+n}} \ge 1 $$ I tried to use Bernoulli's inequality, but I can't figure it ...
1
vote
1answer
20 views

Property of an almost additive sequence of functions

We say that a sequene of functions $\Phi=(\phi_n)_n$ is almost additive if there exists a constant $C > 0$ such that for every $n,m \in \mathbb{N}$ and $x\in \Lambda$ we have \begin{equation*} -C + ...
-2
votes
3answers
74 views

inequality question? [closed]

$x= \sqrt{25}$ $y^2= 49$ What is the relationship between $x$ and $y$ ? $x>y$ $x<y $ $x\ge y $ $x\le y$ No relation can be established.
2
votes
1answer
35 views

A form of Nash's inequality, $\|f\|_2\le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta$

For $f\in \mathcal{S}(\mathbb{R})$ can anyone help me prove the following Nash inequality, $$\|f\|_2 \le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta.$$ I believe $\alpha$ and $\beta$ should be $2/3$ and ...