Questions on proving and manipulating inequalities.

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1
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1answer
65 views

I can't understand how can prove

I don't understand how we can prove that inequality, without integration $$\frac{1}{x}\int_{x}^{2x}(2-\frac{1}{y+2})\,dy \geq 2 - \frac{1}{x+2}.$$ P.S: Here is what I try... if can someone help me to ...
0
votes
0answers
34 views

Prove a lower bound for a/b+b/c+c/a

Given $a,b,c>0$, prove the inequality $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq \frac{(1+a^2)}{(1+a\dot\,b)} + \frac{(1+b^2)}{(1+b\dot\,c)} + \frac{(1+c^2)}{(1+c\dot \,a)}$$ I have tried ...
1
vote
1answer
33 views

Decide whether set is convex, connect and bounded.

Let $A=\{ \left(x,y,z \right)\in \mathbb{R}^3 : x^2+y^2-z^2+1<0\}$. Decide whether set A is: a) convex (definition i know: Set $A\in \mathbb{R}^k$ is convex set if for all $x,y \in A$ line segment ...
0
votes
0answers
22 views

estimate an equality on sine function

I want to prove the following: For any given $\epsilon>0$, there exists a $\delta>0$ such that for any fixed $0<\theta<\frac{\pi}{2}$ with $\frac{\pi}{2}-\theta<\delta$, there exists ...
-2
votes
1answer
37 views

Use the Mean Value Theorem to prove an inequality [on hold]

Using MVT, prove the following equation to be true: $$\sqrt{x} - \frac{x-y}{2\sqrt{y}} < \sqrt{y} < \sqrt{x} - \frac{x-y}{2\sqrt{x}},$$ given that $y>x>0$
2
votes
5answers
38 views

Complicated inequality, how to prove?

If $-\pi/2 <\theta<\pi/2$, how do I prove that $\left|\sin (\theta) - \sum_{k=0}^n (-1)^k \frac{\theta^{2k+1}}{(2k+1)!}\right| \leq \frac{|\theta|^{2n+2}}{(2n+2)!}.$
8
votes
6answers
141 views

Prove that $(n+1)^{n-1}<n^n$

How would one prove that $$(n+1)^{n-1}<n^n \ \forall n>1$$ I have tried several methods such as induction.
1
vote
2answers
51 views

If $x,y \in \mathbb{R}$ where $x\leq y$ and $y\leq x$. Does $x=y$?

I'm trying to complete this problem: Let $A$ be a nonempty set and suppose $\alpha$ and $\beta$ are both suprema of $A$. Prove that $\alpha = \beta$. The first thing i did was try to find an ...
4
votes
4answers
83 views

Inequality $|e^z -1| \le 2 |z|$ for complex $z$ with $|z|\le1$

I am trying to prove that for $z \in \mathbb C, |z|\le 1$: $$|e^z -1| \le 2 |z|$$ But I'm stuck and I need help. I showed that for all $z$: $|e^z -1| \le |z|e^{|z|}$ but it does not seem useful. ...
2
votes
1answer
20 views

Complex modulus Inequality using $|exp(z)-1|$

I think I am almost there: Prove $\left|z\right|/4 < \left|\exp(z)-1\right|<7\left|z\right|/4$ for all $0<|z|<1$. MY ADVANCES First we note that $$ \left|\exp(z)-1\right| = ...
0
votes
1answer
32 views

How to apply Cauchy-Schwarz inequality to show an infinite series is bounded?

I would like to know why: $\left| \sum\limits_{t=0}^\infty \delta^tr(s_t,a_t)\right|\le \sum\limits_{t=0}^\infty \delta^t|r(s_t,a_t)|$ where: $\delta \in (0,1)$ and is a $r(s_t,a_t)$ is a real ...
1
vote
0answers
30 views

Solve this inequality for B

I am working on a program that is supposed to qualify a value as "in range" and I have come up with the expression: $$\lvert a-b\rvert \leq c$$ to determine the value. Plugging in test numbers ...
5
votes
3answers
697 views

Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$

I'm asked to used induction to prove Bernoulli's Inequality: If $1+x>0$, then $(1+x)^n\geq 1+nx$ for all $n\in\mathbb{N}$. This what I have so far: Let $n=1$. Then $1+x\geq 1+x$. This is true. Now ...
4
votes
1answer
71 views

Integral Inequality $\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$

Let $f:[0,1]\to\mathbb{R}$ be a continuous function such that $\int\limits_0^1f(x)dx=0$. Prove that $$\int\limits_0^1f^2(x)dx\geq12\left( \int\limits_0^1xf(x)dx\right)^2.$$ My approach as follow Let ...
0
votes
0answers
32 views

Finding solutions to a system of equations: $b+c=46_a b\times c=545_a$

Whenever i try to solve this system of equation i fail and don't even know where to get started: $b+c=46_a$ $b\times c=545_a$ $1\le a < b < c, (a,b,c) \in \mathbb{N}$ Edit: I tried to solve ...
0
votes
0answers
8 views

interval boundaries for covariance of two variables, given their covariance with a third variable

I am thinking about three random variables $X, Y, Z$. If the correlation coefficient (alternatively, covariance, assuming $Var(X,Y,Z)=1$) between variables $X,Y$ is given, as is correlation between ...
0
votes
1answer
24 views

Inequality involving probabilities

While working on stochastics processes, I have found the following inequality, which I have not been able to proof: Let $h>l>1$ and $0\leqslant p\leqslant 1$ (probability). Then ...
2
votes
4answers
152 views

Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b\in \mathbb R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
0
votes
0answers
73 views

Prove a lower bound for $a/b+b/c+c/a$ [closed]

Given $a,b,c>0$, prove the inequality $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge \frac{1+a^2}{1+ab} + \frac{1+b^2}{1+cb} + \frac{1+c^2}{1+ac}$$ Do you have idea for solve this?
0
votes
0answers
25 views

Show $x - \sin(x) <\frac{x^3}{6}$ if $x>0$ [duplicate]

I wish to show that if $x>0$, then $x -\sin(x) < \frac{x^3}{6}$. I'm not really sure where to start with this. Does anyone have ideas?
0
votes
1answer
26 views

Convex optimization problem: linear equality and inequality constraints

When linear equality constraints can be converted in an inequality constraints for a strongly convex optimization problem? I mean, I got the same solution for both the following problem: 1) $\min_x ...
2
votes
2answers
80 views

How to prove this inequality? $(a+b+c=1)$

Show that if $a,b,c$ are positive reals and $a+b+c=1$, then the following must hold: $$\frac{2(a^3+b^3+c^3)}{abc}+3 \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ What I have tried is using $abc \leq ...
0
votes
1answer
315 views

Square root's inequality [closed]

Could anyone help me solve this inequality? I would be really grateful if you showed how to calculate it with steps. Thank you in advance. $\sqrt{x+2} + \sqrt{x-5} \ge 5 - x$
1
vote
1answer
34 views

Condition on Inequalities

Here $$X=\frac{p}{b+q}+\frac{b^2 p (1+q)+2 b p \left(a p+q^2\right)+q \left(a (-1+p) p+q+p q^2\right)}{\left(b+a p+b q+q^2\right)^2}$$ $$Y=\frac{p^2}{b+a p+b q+q^2}$$ Where $a$, $b$, $p$ and $q$ are ...
0
votes
1answer
26 views

Simple inequality with exponential

I have bounded $A$ by $$ e^{-\epsilon c}(\cosh c)^n $$ for any $c>0$, and if I'm correct the minimum occurs when $\tanh c=\epsilon/n$. By the right choice of $c$, I want to show that $$ A\le ...
1
vote
0answers
24 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
5
votes
1answer
38 views

How to prove this inequality $(\sum_i x_i y_i)^2 - \sum_i x_i^2y_i^2 \leq 1-1/n$?

How to prove this inequality: $$(x_1y_1+x_2y_2+ \cdots + x_ny_n)^2 - (x_1^2y_1^2+x_2^2y_2^2+\cdots+x_n^2y_n^2)\leq 1-\frac{1}{n},$$ where $x_i,y_i \geq 0,i=1,2,\ldots,n$, and ...
0
votes
3answers
193 views

Proving inequalities: $\sum_{k=1}^n a_k \leq \sum_{k=1}^n ka_k \leq n \sum_{k=1}^n a_k$

I'm really bad when it comes to proving inequalities. I have prove this: these are all positive $\sum\limits_{k=1}^n a_k \leq \sum\limits_{k=1}^n ka_k \leq n \sum\limits_{k=1}^n a_k$ Where would i ...
2
votes
0answers
63 views

Can Waring problem be solved with triangle inequality?

When we calculate the difference $\frac{3^k-1}{2^k-1}-\left(\frac{3}{2}\right)^k,$ we get $\frac{3^k-2^k}{4^k-2^k}.$ Then solving: $1-\frac{3^k-2^k}{4^k-2^k}>\frac{3^k-2^k}{4^k-2^k},$ we get ...
0
votes
1answer
56 views

Prove that $1+\sum_{m=1}^n \frac{1}{m!}\geqslant 2\ $ for $n=2,3,4,5,\ldots$

I was given Prove that $1+\sum_{m=1}^n \frac{1}{m!}\geqslant 2\ $ for $n=2,3,4,5,\ldots$ I understand that $1+\sum_{m=1}^n \frac{1}{m!}>2$, but I don't see how $1+\sum_{m=1}^n ...
4
votes
1answer
69 views

How to understand/remember Holder's inequality

If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}=1$ and if $f \in L^p$ and $g \in L^q$, then $f\cdot g \in L^1$ and $$\int |fg| \leqslant ||f||_p \cdot ||g||_q$$ I think ...
1
vote
1answer
15 views

$g(x)=\frac{x}{m}-1+\log m$, $g(x) \ge log x$

Let $m \in \mathbb N$ and $g(x)=\frac{x}{m}-1+\log m$, if $m-\frac{1}{2} \le x<m+\frac{1}{2}$. How could I show that $g(x) \ge \log x$?
3
votes
1answer
25 views

Inequality involving sums and products (related to multi-stage rockets)

Let $a_i$ and $b_i$ be strictly positive real numbers for $i=1,\ldots, n$. I wonder whether the following inequality holds in general or does not:$$\frac{\sum_{i=1}^n a_i+\sum_{i=1}^n ...
2
votes
1answer
54 views

$x\over(1-x)$ $y\over(1-y)$ $z\over(1-z)$ >= 8 when $x ,y ,z $ are positive proper fractions and $x+y+z = 2$

Q. Prove that $x\over(1-x)$ $y\over(1-y)$ $z\over(1-z)$ $\geq$ 8 when $x ,y ,z $ are positive proper fractions and $x+y+z = 2$ What I did: From A.M. G.M. inequality, $(x+y+z)\over3$ $\geq$ ...
2
votes
2answers
35 views

Solving an inequality $B<n!$ without a calculator or gamma function?

Is there a way to solve $B<n!$ where $B$ is some very large real number (suppose for example $B=10^{17}$) without a calculator or gamma function? At the very least, to find the nearest integer for ...
-3
votes
1answer
46 views

Is this property with inequality true? [closed]

Is the following implication true ? If $5(a+b+c+d) \geq 10a + b + c + d\;$ and $\;5(a+b+c+d) \geq a + 10b + 10c + 10d$, then $5(a+b+c+d) \geq 10 (a+b+c+d)$
-1
votes
1answer
28 views

Approximation using $1-x \le e^{-x}$

Suppose I want to approximate a number $p_k = (1-\frac{1}{365}) \cdot ... \cdot (1-\frac{k-1}{365})$. $k$ is a natural number. The book I'm reading says it can be done using the fact that $1-x \le ...
2
votes
2answers
627 views

Proof of Frechet-Hoeffding Copula bounds

How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$. I'm struggling to ...
1
vote
3answers
78 views

Solve $\frac{(x-1)^{204}(x+3)^5(x-4)^{2015}}{(x+5)^{102}}\ge 0$

Solve $\frac{(x-1)^{204}(x+3)^5(x-4)^{2015}}{(x+5)^{102}}\ge 0$ Just wanted to share a nice and quick technique i learnt for such problems.
2
votes
1answer
62 views

Proving the inequality $\frac{1}{k!}+\frac{1}{(k + 1)!}+\frac{ 1}{ (k + 2)! }+…\leq {(\frac{e}{k})}^k$

In the first part of the question we showed that $P(X \geq k)\leq E(e^{tX}e^{-kt})$ for all $t \geq 0$ and real $k$ by the use of Markov's inequality. This wasn't too bad. Now, in the second part, ...
0
votes
2answers
45 views

Solving inequality equation involving sum of binomial coefficients

I have a function $f(k,\,i)$ involving binomial coefficients: $$f(k,\,i)\,=\left(\begin{matrix}k+i \\ k\end{matrix}\right)=\frac{(k+i)!}{k!\,i!}$$ And the following sum over this function (expansion ...
-2
votes
1answer
55 views

Estimating $\left|\frac{1}{y} - \frac{1}{y_0}\right|$ when $|y-y_0|$ is small [closed]

If $|y-y_0| < \min\left(\frac{|y_0|}{2}, \frac{\varepsilon |y_0|^2}{2}\right)$ where $y\neq 0$ and $y_0 \neq 0$, Prove that $\left|\frac{1}{y} - \frac{1}{y_0}\right| < \varepsilon$. I was ...
2
votes
0answers
81 views

Find the minimum value of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x+3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x+3}}$ [duplicate]

Let $x$ be a real number. Find the minimum value of $$P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x+3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x+3}}$$ This is a problem from 2015 ...
0
votes
0answers
28 views

Polynomial optimization and AM-GM inequality

I want to maximize the function $f(\mathbf{x},\mathbf{y}) = \sum \limits_{k=1}^{K}p_k(\mathbf{x})q_k(\mathbf{y})$, where $0 < p_k(\mathbf{x}) \leq \delta_k$ and $0 < q_k(\mathbf{y}) \leq ...
1
vote
3answers
80 views

Easy inequality going wrong

Question to solve: $$\frac{3}{x+1} + \frac{7}{x+2} \leq \frac{6}{x-1}$$ My method: $$\implies \frac{10x + 13}{(x+1)(x+2)} - \frac{6}{x-1} \leq 0$$ $$\implies \frac{4x^2 -15x-25}{(x-1)(x+1)(x+2)} ...
5
votes
3answers
82 views

Prove that $\det(AA^T+I)\ge 1$

If $A$ is a matrix with real entries, prove that $$\det(AA^T+I)\ge 1.$$ I tried using the eigenvalues. One thing came into my mind: maybe $AA^T$ is positive definite (I don't know whether this is ...
3
votes
0answers
62 views

How prove $\sigma(4^p-1)<(2^{p+1}-1)^2$

If $p$ is an odd prime numbers, show that $$\sigma(4^p-1)<(2^{p+1}-1)^2$$ where $\sigma(n)$ stands for the sum of divisors. I thought of using the formula for $\sigma(n)$: If ...
0
votes
1answer
31 views

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close ...
2
votes
1answer
23 views

Prove the set is closed with respect to its norm…

Let $V$ be a normed vector space over R. Let $W$ be a proper closed subspace of $V$. We say $w^*$ is a best approximation in $W$ to $v^* \in V$ if $\|v^*-w^*\| \leq \|v^*-w\|$ for all $w \in W$. ...