Questions on proving, manipulating and applying inequalities.

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2
votes
3answers
94 views

How to prove that $(\frac{n}{k})^k\leq{{n}\choose{k}}\leq\frac{n^k}{k!}$?

How to prove that $(\frac{n}{k})^k\leq{{n}\choose{k}}\leq\frac{n^k}{k!}$? I can only manage to see the second inequality, could any one give a hint about the first one?
2
votes
1answer
22 views

Necessary likelihood ratios conditions for stochastic dominance

Suppose $X$ has CDF and PDF $F_X$ and $f_X$ with support $(-\infty,\infty)$ and $Y$ has CDF and PDF $F_Y$ and $f_Y$ also with support $(-\infty,\infty)$. Muller (2001) claims that if $X$ is ...
2
votes
3answers
71 views

a log inequality

Can anyone offer some guidance on proving the following inequality? Define $\Lambda_1(a)=-a\log a$ and $\Lambda_2(a,b)=-(a+b)\log(a+b)$. Then if $a$, $b$, $c$, and $d$ are non-negative numbers summing ...
2
votes
6answers
75 views

Prove that $xy+yz+zx \leq x^2+y^2+z^2$

Prove that $xy+yz+zx \leq x^2+y^2+z^2$ . Hint: Use $\frac{a+b}{2}\geq\sqrt{ab}$ First I tried using the hint by setting $a=x$ and $b=y+z$, however this results in the inequality: $$x^2+y^2+z^2 ...
1
vote
3answers
96 views

Solving a system of five polynomials

I am trying to solve the following system of equations for tuple $\left(a,b,c,d,t\right) \in \mathbb{R}^{4} \times [0,1]$, with parameter $\ell\in\mathbb{R}$. $$ \begin{eqnarray} a\frac{t^{2}}{2} - ...
-6
votes
3answers
38 views

Solving the inequality $2-(x/4-3)+1 \le 4-2/3x$ [closed]

I need to solve $$2-(x/4-3)+1 \le 4-2/3x$$
5
votes
1answer
105 views

Let $f : [0,1] \to \mathbb{R}$, prove that $2 \int_{0}^{1} f(x)dx \ge f\Big(\frac{1}{n}\Big) + \sum_{k=1}^{n-1}\frac{1}{k} f\Big(\frac{k}{n}\Big)$

Let $f : [0,1] \to \mathbb{R}$ be a differentiable function with a continuous derivative such that $f(x) \ge xf'(x), \forall x \in [0,1]$. Prove that: $$2 \int_{0}^{1} f(x)dx \ge ...
0
votes
1answer
164 views

Algebra question about inequalities [closed]

Let $n>0$ and let there be two positive integers $x,y$ such that $x^n+y^n=1$ Prove, $$\left(\sum_{k=1}^{n} \frac{1+x^{2k}}{1+x^{4k}}\right)\left(\sum_{k=1}^{n} ...
1
vote
1answer
75 views

Prove that $(n-1) \sum_1^n \cot(\theta_i) \leq \sum_1^n \tan(\theta_i) $

n is a positive integer and $\theta_i$ is such that $ 0^\circ \leq \theta_i \leq 90^\circ $ for all positive integers $i \leq n$ and $\sum_1^n \cos^2(\theta_i) = 1$. Prove that $(n-1) \sum_1^n ...
-1
votes
0answers
23 views

Want proof of inequalities with necessary conditions for parameters [closed]

(i) $\sum\limits_{n=0}^{\infty}\left(a_n t^n\right)^s \leq \left(\sum\limits_{n=0}^{\infty} a_n t^n\right)^s, \mathrm{where} \; s\in N,$ (ii) ...
1
vote
1answer
24 views

Determining the region given by $(x-ay)(x-by)(x-cy)\dots<0$

It's easy to sketch the region given by the inequality $(x-ay)(x-by)<0$ where $a,b\in\mathbb{R}$. A cubic one would take a bit more work but doable nevertheless. But I was trying to find a general ...
4
votes
0answers
59 views

If $x_1, x_2,…,x_{10}$ are such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} \cos(x_i)$ [duplicate]

Take $x_1, x_2,...,x_{10}$ such that $\sum_{i=1}^{10} \sin^2(x_i) = 1$ with $x_1, x_2,...,x_{10}$ on $\left[0,\frac{\pi}{2}\right]$, prove that $3 \sum_{i=1}^{10} \sin(x_i) \leq \sum_{i=1}^{10} ...
3
votes
0answers
35 views

Proving those hard cyclic inequalities

From time to time some user asks for help to prove a cyclic inequality, that is, something like $$f(x,y,z)\le k$$ where $x,y,z$ are usually real positive numbers and $f$ is a 'cyclic function' (I ...
3
votes
4answers
146 views

Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+…+\frac1{2^n - 1} \le n$

$\dfrac{n}{2} \le \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n - 1} \le n $ I've Tried for hours but didn't got any striking idea. I don't have any efforts to show rather than induction. ...
2
votes
2answers
148 views

Derive one inequality from another

One of the steps to the solution of my problem is to apply the following inequality: $$\frac{y}{x} + \frac{x}{y} \ge 2$$ which is true for $x > 0$ and $y > 0$. My textbook suggests to use ...
1
vote
1answer
74 views

Inequality problem: $\tanh(\pi x)\sin(\pi x)\geq x^2(1-x^3)$ on $[0,1]$.

How to show that $$\tanh(\pi x)\sin(\pi x)\geq x^2(1-x^3)$$ on $[0,1]$? I tried to expand $\tanh(\pi x)\sin(\pi x)$ in a Taylor expansion: $$\tanh(\pi x)\sin(\pi x)=\pi^2 x^2 - \frac{\pi^4 ...
2
votes
1answer
87 views

Is this a valid proof for $\left | a+b \right | \leq \left |a \right| + \left | b \right | $?

$\left | a+b \right | \leq \left |a \right| + \left | b \right | \Rightarrow$ $\sqrt{{(a+b)}^2} \leq \sqrt{{a}^2} + \sqrt{{b}^2}$ ${(\sqrt{{(a+b)}^2})}^2 \leq ({\sqrt{{a}^2} + \sqrt{{b}^2}})^2 ...
2
votes
1answer
32 views

Is the inequality solution below legal?

First of all, I know this is very basic, but I've taken a big break from University for personal reasons and I've forgotten a lot of basic math. I've been doing the practice problems in the Calculus ...
0
votes
1answer
41 views

$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}}$ , where $X \sim N(0,1)$

Prove that: $$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}},$$ where $X \sim N(0,1)$ and $c>0$, $t \in\mathbb R$. The problem should be solved easily by using the equality: $$P(X \geq c) = P(e^{Xt} ...
0
votes
0answers
34 views

Succinct notation for specifying that eigenvalues must have negative real part?

Is there a succinct way to denote that all eigenvalues of a matrix $A$ have negative real parts? If the eigenvalues were real, I could simply write this as $$-1 < A < +1$$ since we have the ...
2
votes
0answers
62 views

Prove this inequality $a^4+b^4+c^4+9\ge 4(a^2b+b^2c+c^2a)$

let $a,b,c>0$ and such $abc=1$, show that $$a^4+b^4+c^4+9\ge 4(a^2b+b^2c+c^2a)$$ Schur inequality $$a^4+b^4+c^4\ge \sum_{cyc}ab(a^2+b^2)-abc(a+b+c)$$ It suffices to show that ...
7
votes
5answers
95 views

Show that $\left(1 + \frac{x}{n}\right)^{-n} \le 2^{-x}$, when $x,n \ge 0$, $x \le n$

Show that $\left(1 + \frac{x}{n}\right)^{-n} \le 2^{-x}$, when $x,n \ge 0$, $x \le n$. This is driving me crazy... I have plotted the graphs to be sure that the inequality is true, and it is, but I ...
-1
votes
3answers
93 views

How to prove this inequality $(\frac{n+1}{e})^{n} < n! < e(\frac{n+1}{e})^{n+1}$? [closed]

$\Bigl(\frac{n+1}{e}\Bigr)^{n} < n! < e\Bigl(\cfrac{n+1}{e}\Bigr)^{n+1}$
0
votes
3answers
78 views

Does $\Pr(\{X\leq x\})\geq\Pr(\{Y\leq x\})$ imply $\Pr(\{X\leq Y\})=1$?

Suppose that $(\Omega,\mathcal{F},P)$ is a probability space and $X,Y:\Omega\to\mathbb{R}$ are random variables satisfying $$ P(\{X\leq x\})\geq P(\{Y\leq x\}),\quad\forall x\in\mathbb{R}. $$ ...
4
votes
1answer
60 views

Approximation of irrational numbers?

Problem Suppose $\theta>1$ is an irrational algebraic integer, i.e. $\theta\not\in\mathbb Z$ but satisfies a monic polynomial with integer coefficients, and $\{a_n\}_{n\ge0}$ is a sequence of ...
0
votes
3answers
32 views

Solution for inequality: $ \frac{\log p}{e^{wp}} \leq z^2 $

I am currently looking at some concentration bounds (Law of iterated logarithm) and trying to find the number of samples/steps required so that the deviation from mean is within some small $\epsilon$. ...
0
votes
3answers
45 views

Solving rational inequalities

I am having difficulty with solving: $$\frac{(2x-1)}{(x-5)} > \frac{(x+1)}{(x+5)}.$$ I tried to move the one side over so that there was zero opposite the equation but I ended up making some ...
1
vote
3answers
39 views

(Discriminant) For which values of k will the equation g(x) = x + k have two real roots that are of opposite signs?

I am currently in Grade 12 and came across the following question in a past paper: $$g(x) = \frac{2}{x+1}+1$$ The question asks: For which values of k will the equation $g(x) = x + k$ have two real ...
1
vote
1answer
30 views

Polynomial inequalities vs rational inequalities

A question from one of the comprehension questions I have is: How would the intervals of the solution set differ between a polynomial inequality and a rational inequality? I have tried to research ...
0
votes
1answer
9 views

How to use differential inequality to establish a bound on variable?

Given $\frac{dN}{dt} \leq \Lambda-\mu N,$ how can it be shown that $N(t) \leq N(0)e^{-\mu t} + \frac{\Lambda}{\mu}(1-e^{-\mu t})?$ I guess a more fundamental question (I've never worked with ...
0
votes
1answer
54 views

Proof of Inequality involving exponentials

I would to be able to show that $$ \exp\Bigl(\frac{1}{1+x}\Bigr)-1 \gt \frac{1-\exp(-y/x)}{y}~~~\text{for}~~~1\le y \le x $$ My Attempt: I start by fixing $x$ and letting $$ f(y) = ...
0
votes
1answer
32 views

Solving logarthmic inequality

Question Find integral solutions of this inequality$$\left (\frac{1}{10}\right )^{\log_{x-3}^{x^2-4x+3}} \ge 1$$ My try : I took log on both sides and got $\log_{x-3}^{x-1} \le-1$ but ...
1
vote
2answers
61 views

Solving following irrational inequaltiy

Here is the inequality: $$x-3<(x^2+4x-5)^{1/2}.$$ My try: First of all I know for RHS to be defined $x^2+4x-5$ should be greater than $0$,so I got the intervals $[-\infty,-5]\cup[1,\infty]$, then ...
1
vote
3answers
28 views

Proving one function is greater than another in a certain interval

Given $f(x,y)=y~(1-x)$ and $g(x,y)=(1 - x^y)$, how can I prove that $f(x,y) \geq g(x,y)$ for $0 \leq x \leq 1$ and for all $y \geq 1$? From my tries, it seems that there are two possibilities: If ...
7
votes
0answers
70 views

How to prove this inverse of Holder inequality?

How to prove inverse of HÖlder inequality let $p,q>0,a,b,x,y>0$, and such $$\dfrac{1}{p}+\dfrac{1}{q}=1$$ show that ...
0
votes
0answers
16 views

Set of 3 inequations involving 3 unknowns with a maximum

I am capable of finding a relation between unknowns x, y and z involved in this set of 3 inequations: $\begin{cases} ax - y - z \leq x \\ -x + by - z \leq y \\ - x - y + cz \leq z\end{cases}$ This ...
4
votes
0answers
49 views

Upper bounding a definite integral

So I have the following problem. Let $F$ be the set of functions for which $|f(x)| \le 2$ for all $x$ and $\int_{0}^{5} [f(x)]^2dx \le 16$. Over all the functions in $F$, compute the maximum ...
0
votes
1answer
16 views

Complex inequality question

I am trying to understand why the following holds: \begin{align*} \Re((1-\imath)(A+B)) \geq \Re((1-\imath)A) - \sqrt{2}|B|, \end{align*} where, \begin{align*} A:= \sum_{x=1}^{[\sqrt{k}]} ...
8
votes
6answers
145 views

To show that $e^x > 1+x$ for any $x\ne 0$ [duplicate]

$$e^x>1+x$$ is what I want to show. So let's define a function: $$h\left(x\right)=e^x-x-1$$ and investigate its derivative: $$h'\left(x\right)=e^x-1$$. Easy to see that at $x=0$ it has a ...
0
votes
2answers
41 views

Does this inequality hold for all n>=1?

$$\ln ( \ln ( ( n+1 ) ^ {1/2} ) ) - \ln ( \ln ( n ^ {1/2} ) ) < \frac1 { (n / \ln (n ) ) ^ 2 + 1}$$ It seems that this is true for all $n\geq1$. I tried proving that by induction but I ...
2
votes
1answer
31 views

What is the best time complexity of checking the inequality $a_1x_1 + \cdots + a_mx_m \le K$ to have a non-negative integer solution?

We know that all the coefficients $a_1, a_2, \ldots , a_m$ are integer. Also, $K$ is an integer number. I only need to know if the inequality has a integer solution or not. It means that there is no ...
1
vote
0answers
55 views

Where does this inequality in a paper come from?

It's probably simple but I'm not sure why I'm not seeing it. The inequality is from a paper: $$\begin{align*} \sum_{i=1}^4 \rho_i (x_i-1)(1-\sum_{j=1}^4 \alpha_{ij}x_j) &\leq\begin{split} ...
1
vote
3answers
64 views

Help with proof of $(n+1)^n > n! 2^n$

I have already managed to prove it using induction and Bernoulli's inequality but I wonder if there is another way. My proof goes like this: (This is my first time using MathJax, so I apologize for ...
0
votes
2answers
53 views

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$ options $a.)\ 1<x<6\\ b.)\ 1 \leq x<6\\ c.)\ 1<x\leq 6\\ \color{green}{d.)\ 1\leq x \leq 6}$ I ...
2
votes
2answers
59 views

$x_1,x_2,…,x_n$ are positive real numbers, $\sum_{i=1}^n x_i^2 = 1$, to find the minimum value of:

$$\sum_{i=1}^n \frac{x_i^5}{s - x_i}$$ with $$s = \sum_{i=1}^{n}x_i$$ I used Cauchy Schwarz inequality: $$(\sum_{i=1}^n \frac{x_i^5}{s - x_i})(\sum_{i=1}^{n}\frac{s-x_i}{x_i}) \geq 1$$ ...
4
votes
3answers
47 views

$a_1,a_2,…,a_n$ are positive real numbers, their product is equal to $1$, show: $\sum_{i=1}^n a_i^{\frac 1 i} \geq \frac{n+1}2$

it says to use the weighted AM-GM to solve it, because the inequality is not homogenous I've tried to use $$\lambda _ i = \frac{a_i^{\frac1i -1}}{\sum_{k=1}^n a_k^{\frac1k -1}}$$ this $\lambda$ is ...
2
votes
1answer
30 views

$a_1,a_2,a_3,b_1,b_2,b_3$ are positive real numbers, show: $\sqrt[3]{(a_1+b_1)(a_2+b_2)(a_3+b_3)} \geq \sqrt[3]{a_1a_2a_3} + \sqrt[3]{b_1b_2b_3}$

The question says one only needs the AM-GM inequality, I've been stuck here for more than one hour. $$(a_i + b_i) \gt a_i$$ and $$a_i + b_i \gt b_i$$ therefore, $$ ...
4
votes
3answers
112 views

How to show $ \Big\vert \frac{\sin(x)}{x} \Big\vert $ is bounded by $1$?

This may be a silly question, but I cannot figure it out. I want to prove that $ \Big\vert \frac{\sin(x)}{x} \Big\vert \leq 1 $ for $x\in[-1,0)\cup(0,1]$, but I don't even know where to start.
0
votes
0answers
33 views

Solving differential equation and the inequality

I have stuck in a small step were I need to solve for t: $$ e^{t(\lambda_3-\lambda_1)} \geq 1 \Rightarrow t \geq\frac{1}{\lambda_3 - \lambda_1}$$ I don´t understand how the solution is required. ...
12
votes
3answers
220 views

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

let $0<a\le b\le c\le d$, and such $abcd=1$,show that $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$ it seems harder than This inequality $a+b^2+c^3\ge ...