Questions on proving and manipulating inequalities.

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0
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2answers
42 views

proof of triangular inequality modified $|x+y|=|x|+|y|$ iff $|xy|>0$

$$|x+y|=|x|+|y| \iff |xy|>0$$ I tried to prove the above inequality but i cant find a way. I tried assuming the first condition is true and tried to derive the second part of it but it seems i ...
0
votes
0answers
26 views

An inequality with $\omega(n)$ [duplicate]

Prove: For any positive integer $k, N$, $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}\left(\omega (n)\right)^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q}$$ Where $\sum\limits_{q\leq ...
1
vote
3answers
68 views

Proof of $\sin2x+x\sin^2x \lt\dfrac{1}{4}x^2+2$

How can be proven the following inequality? $$\forall{x\in\mathbb{R}},\left[\sin(2x)+x\sin(x)^2\right]\lt\dfrac{1}{4}x^2+2$$ Thanks
1
vote
1answer
68 views

Ratio of 2 Gammas, approximation with power

Find all value of $\alpha$ such that $\lim\limits_{x\rightarrow +\infty}\left(\frac{\Gamma(x+\alpha)}{\Gamma(x)}-x^{\alpha}\right)=0$. (note: $\alpha$ is a constant with respect to $x$) By ...
0
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1answer
42 views

An inequality used to prove minkowski's inequality [closed]

How to prove the following : $$ |x+y| \leq 2(|x|^p + |y|^p)^{\frac{1}{p}}, x,y \in \Bbb R, p \geq 1$$
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1answer
50 views

Linear Programming?

An agriculture company has 80 tons of fertilizer Alpha and 120 tons of fertilizer Bravo. The company mixes these fertilizer into two products. Product Super requires 2 parts of fertilizer A and 1 part ...
2
votes
1answer
41 views

Solve: $\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$

Given $a_1,a_2,\ldots,a_n \in\mathbb{R}$. Solve the following equation on $\mathbb{R}$: $$\sum_{i=1}^n \max\left\{x-a_i,0 \right\}=1.$$ I am not sure that a closed-form solution exists, so iterative ...
0
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0answers
44 views

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
3
votes
1answer
31 views

Is the norm of the average $\le$ the norm of the max?

Given $\pmb X \in \mathcal{R}^p$, denote the elements of $\pmb X$ as $\pmb x_i$ for $i= 1, \dots, n$. Denote the $t(\pmb X)$ as the average of $\pmb X$ \begin{equation} \pmb t(\pmb X) = \frac 1 n ...
0
votes
1answer
48 views

How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?

I would appreciate some help proving the inequality $$\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}.$$ Thanks in advance!
1
vote
3answers
36 views

Greatest value of the binomial coefficient. [duplicate]

How should I prove the greatest value of the binomial coefficient $C(n,r)$ occurs for $r=\left[\cfrac{(n+1)}{2}\right]$ ?
0
votes
1answer
51 views

Use of AM/GM Inequality

I am currently attempting to prove the following inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$ for all $ a,b,c>0$ My instinctive plan of attack is to use the AM/GM ...
2
votes
2answers
111 views

Proving $\displaystyle \frac{\sin^3x}{x}\lt 0.69$ for any $x\gt 0$

Question : How can we prove strictly that the following inequality holds for any $x\gt0$?$$\frac{\sin^3x}{x}\lt 0.69$$ This seems difficult though it doesn't look so. Can anyone help?
2
votes
1answer
31 views

Inequations on holomorphic functions

Let $f : \mathbb{C}\setminus\{0\} \to \mathbb{C}$ an holomorphic function such as $$\exists C_0 > 0 / \forall z \in \mathbb{C}\setminus\{0\} \qquad |f(z)| \leqslant C_0\left( |z| + ...
2
votes
1answer
66 views

Proving a sharp inequality

After experimenting, I've come to the conclusion that if $x\geq y\geq z\geq 0:$ $$\sum_{x,y,z}\frac{x}{\sqrt{x+y}}\geq \sum_{x,y,z}\frac{y}{\sqrt{x+y}}$$ (the sums are cyclic) Does anyone know how ...
2
votes
1answer
55 views

Proof of $\left|\sum_{k=1}^N\dfrac{k!}{N^k}-\sum_{k=1}^N\frac{1}{kN}\right|\le\dfrac{1}{4}$

Is it possible to prove the following inequality? $$\left|\sum_{k=1}^N\dfrac{k!}{N^k}-\sum_{k=1}^N\frac{1}{kN}\right|\le\dfrac{1}{4}$$ Thanks
1
vote
1answer
30 views

Help to prove the condition that a right half-open interval is not empty

The right half-open interval is defined as: $[a,b) = \{x \in \mathbb{R}|a \le x \lt b\}$ I need to prove: $[a,b) \ne \emptyset \iff a<b$ My attempt: For $\Rightarrow$: $$\begin{align} ...
0
votes
1answer
40 views

An Inequality Involving Prime Numbers

Let $p_i$ be the $i^{th}$ prime number. It seems as though the following inequality is true for all positive integers $m$ and real numbers $x>1$: ...
0
votes
2answers
41 views

How to apply the transitive law when there is a $\le$

The transitive law states that: For real numbers $a$, $b$ and $c$: $a<b \text{ and } b<c \Rightarrow a<c$ I am not sure how to apply it in the following cases ($x \in \mathbb{R}$): $a ...
4
votes
1answer
95 views

On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
0
votes
2answers
32 views

Quick floor function

This isn't true, right? $$k\left\lfloor\frac n {2k}\right\rfloor\leq \left\lfloor\frac n k\right\rfloor$$ $2<k\leq \left\lfloor\dfrac {n-1} 2\right\rfloor$, $n>4$, $k,n$ are coprime.
0
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2answers
25 views

Floor Function Bound?

I am trying to prove or disprove the following bound: $2+\left(n-\left\lfloor\dfrac n k\right\rfloor k\right)\ge \left\lfloor\dfrac n k\right\rfloor$, where $2<k\le \left\lfloor\dfrac {n-1} ...
2
votes
2answers
88 views

Proof $(\frac{n+1}{n})^n>2$ for positive $n$ [closed]

I would like to see some proofs that $(\frac{n+1}{n})^n>2$ for $n\in\mathbb R^+$ have some experience with inequalities, but I don't know too much theory. Regards
11
votes
0answers
113 views

Stronger version of AMM problem 11145 (April 2005)?

How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$: ...
12
votes
2answers
129 views

Is there a probabilistic proof of the inequality $4p(1-p) \leq 1$ for a probability $p$?

Let $p\in(0,1)$. The inequality $4p(1-p)\leq 1$ is very easy and elementary, but I wonder if there is a probabilistic proof of it. By that, I mean constructing a “natural” probability space and an ...
3
votes
0answers
177 views

How prove this stronger Cauchy-Schwarz inequality for traces of compression matrices

Question: Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
11
votes
2answers
231 views

Why is Volume^2 at most product of the 3 projections?

Is there a simple proof for $$ \text{Vol}^2(P)\le \prod_{i=x,y,z} \text{Area}(\text{Proj}_i(P)), $$ where $P\subset \mathbb R^3$ and $\text{Proj}_z(P)$ denotes the projection of $P$ to the $z=0$ ...
0
votes
1answer
26 views

Inequality in intergration

i saw this in solution of some exercise they said that (the real exercise i already post it here ) $$\dfrac{e^{-xt}}{1+t^5}\leq e^{-xt} \Longrightarrow ...
3
votes
6answers
118 views

If $ x^2+y^2+z^2 =1$ for $x,y,z \in \mathbb{R}$, then find maximum value of $ x^3+y^3+z^3-3xyz $.

If $ x^2+y^2+z^2 =1$, for $x,y,z \in \mathbb{R}$, what is the maximum of $ x^3+y^3+z^3-3xyz $ ? I factorize it... Then put the maximum values of $x+y+z$ and min value of $xy+yz+zx$... But it is ...
9
votes
1answer
525 views

How to prove $\frac{1}{4}(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a})\ge \sqrt[4]{\frac{a^4+b^4+c^4+d^4}{4}}$

Let $a,b,c,d>0$, show that $$\dfrac{1}{4}\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)\ge \sqrt[4]{\dfrac{a^4+b^4+c^4+d^4}{4}}$$ I know this is interesting ...
1
vote
0answers
94 views

Inequality with size of sets

Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1}+p_2+\cdots+p_k= 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ ...
1
vote
1answer
19 views

Inequality involving two altitudes of an isosceles triangle and its base

I am trying to solve the following multiple choice problem: $ABC$ is a triangle such that $AB=AC$. Let $D$ be the foot of the perpendicular from $C$ to $AB$ and $E$ the foot of the perpendicular from ...
1
vote
2answers
40 views

Relation between the GM of two sides of a triangle and the bisector of angle between them

I am trying to solve the following multiple choice problem : Let the bisector of the angle $C$ of a triangle $ABC$ intersect the side $AB$ at a point $D$. Then the geometric mean of $CA$ and $CB$ ...
2
votes
1answer
122 views

Algebra question : Prove the inequality.

Let $a , b \ \& \ c$ be positive real numbers satisfying : $$\cfrac{a}{1+b+c} + \cfrac{b}{1+c+a} + \cfrac{c}{1+a+b} \ge \cfrac{ab}{1+a+b} + \cfrac{bc}{1+b+c}+ \cfrac{ca}{1+a+c} $$ Prove that ...
0
votes
1answer
35 views

Minkowski inequality for $l_p$ norm.

I'm trying to prove the Minkowski inequality for the $l_p$ norm: $$ \| f + g\|_p \le \|f\|_p + \|g\|_p $$ where $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are Lebesgue measurable functions and $p ...
5
votes
1answer
101 views

How to prove the inequality?

Given $0<x<1$, $0<a<b<1$, and $a+b<1$, how to prove $a^x(1-ax)<b^x(1-bx)$? I've tried using $f(x)=x^t(1-xt)$ to do some manipulations (including derivations), but failed.
5
votes
1answer
67 views

A cyclic three variable inequality

I have a prove of the following inequality that depends upon somewhat messy algebra. I would like to learn how to prove it in a more elegant way. For positive numbers: $$\frac{x}{4x+4y+z} + ...
0
votes
1answer
34 views

$f(x)=sec(x)$ inequality inconsistency\trouble

I'm currently attempting to find the range of $f(x)=\sec(x)$ by considering $\cos(x)$ in the intervals of $0<\cos(x)\leqslant 1$ and $-1\leqslant \cos(x)<0$ (as $\sec(x)$ is undefined for ...
3
votes
2answers
77 views

there exist some real $a >0$ such that $\tan{a} = a$

How can i prove that there exist some real $a >0$ such that $\tan{a} = a$ ? I tried compute $$\lim_{x\to\frac{\pi}{2}^{+}}\tan x=\lim_{x\to\frac{\pi}{2}^{+}}\frac{\sin x}{\cos x}$$ We have the ...
2
votes
1answer
58 views

Figuring out when $f(x) = \sin(x^2)$ is increasing and decreasing

Regarding the function $f(x) = \sin(x^2)$, I'm supposed to figure out when it is increasing/decreasing. So far, I've found the derivative to be $f'(x) = 2x\cos(x^2)$. So long as I can solve the ...
1
vote
0answers
20 views

Do these inequalities make sense?

I have two sets of inequalities and i just want to know if they are correct. The parameters $\mu, K, d_1, \sigma_1,\sigma_2$ and dependent variables $H,F$are positive. Also $\sigma_2>\sigma_1$. ...
0
votes
1answer
60 views

$|x|^p+|y|^p\geq |x+y|^p$ for $0<p\leq 1$ [closed]

How to prove such inequality: $|x|^p+|y|^p\geq |x+y|^p$ for $0<p\leq 1$ and $x,y \in \mathbb{R}$?
4
votes
2answers
78 views

Prove the inequality for all $N$

Show that the following inequality holds for all integers $N\geq 1$ $$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$$ where $c_1,c_2$ are some constants. I have ...
0
votes
3answers
54 views

In proof by induction, what does it mean when condition for inductive step is lesser than the propsition itself?

My question is regarding the question posed at the end of the proof. My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false. ...
0
votes
0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
11
votes
5answers
291 views

How find this maximum of the $\sin^2{\theta_{1}}+\sin^2{\theta_{2}}+\cdots+\sin^2{\theta_{n}}$

Question: let $\theta_{1},\theta_{2},\cdots,\theta_{n}\ge 0$,and such $$\theta_{1}+\theta_{2}+\theta_{3}+\cdots+\theta_{n}=\pi$$ find the $P$ the maximum of value $P(n)$ ...
0
votes
0answers
16 views

Vysochanskij Petunin vs. Cantelli inequality for random variables

The well known Cantelli inequality states: $$Pr(|X-\mu|\ge\alpha)\le\frac{2\sigma^2}{\sigma^2+\alpha^2}$$ where $X$ is a real valued random variable, $\mu$ the mean value and $\sigma^2$ the variance ...
0
votes
3answers
38 views

Meaningful lower-bound of $\sqrt{a^2+b}-a$ when $a \gg b > 0$.

I know that, for $|x|\leq 1$, $e^x$ can be bounded as follows: \begin{equation*} 1+x \leq e^x \leq 1+x+x^2 \end{equation*} Likewise, I want some meaningful lower-bound of $\sqrt{a^2+b}-a$ when $a ...
0
votes
0answers
20 views

Application of mean value theorem to function $x \to (x-y)^{a-1-d/2}$

How can I show the following inequality by mean value theorem, for a constant $C>0$ $2|(x+h-y)^{a-1-d/2} - (x-y)^{a-1-d/2}|^p \leq C (x-y)^{(a-2-d/2)p}h^p$ Proof: Let $f(b) =(x+h-y)^{a-1-d/2}$, ...
4
votes
2answers
120 views

How prove this inequality?

show that $$\dfrac{\sqrt{2}}{2}<f(n)=\dfrac{\sqrt{2n+1}-1}{1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{n}}}<\dfrac{\sqrt{3}}{2}$$ I know this ...