Questions on proving and manipulating inequalities.

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4
votes
1answer
93 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
votes
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
87 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
111 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
124 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
174 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
229 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
25 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
115 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 ...
8
votes
1answer
518 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
121 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
34 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
100 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
65 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
75 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
57 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
290 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
15 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
35 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
19 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 ...
7
votes
3answers
191 views

Is $ln(x)$ ever greater than $x$

Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement? Just wondering for some convergence related thing
1
vote
1answer
71 views

Proof of an inequality involving factorials

How can the following inequality be proven? $$\left(n!\right)^{\frac{1}{n}}\left((n+1)!\right)^{-\frac{1}{n+1}}\gt\dfrac{n}{n+1}$$ I know this is a result obtained in 1964, but I don't know how to ...
1
vote
1answer
45 views

Algebraic inequality:$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac1a+\frac1b+ \frac1c$

Prove that : If a,b,c $\in \mathbb{R^+}$ $$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac1a+\frac1b+ \frac1c$$ My attempt : We know that the sequence {a,b,c} and ...
2
votes
3answers
141 views

Name the property $f(x) \ge x$

It's a really one of the simplest properties you could imagine for a function. But I haven't been able to find a name for it. What do you call a function $f$ with the following property: $$f(x) \ge ...
0
votes
0answers
50 views

Two conditions, necessary and sufficient conditions.

I am currently studying for a test for a scholarship, in such a test, the following question appears (The part that I'm having trouble with in is in bold): For each of $A$ ~ $D$ in the following ...
1
vote
1answer
53 views

How to use mathematical induction with inequality?

I am stuck with this question. Given that $n$ is a positive integer where $n≥2$, prove by the method of mathematical induction that (a) $$ \sum_{r=1}^{n-1} r^3 < \frac{n^4}{4} $$ (b) $$ ...
1
vote
0answers
19 views

An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
1
vote
2answers
60 views

How to establish this inequality: $(1-a)(1-b)(1-c) \geq 8abc$ for $a+b+c=1$?

Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c = 1$. Then how to establish the following inequality? $$ (1-a)(1-b)(1-c) \geq 8abc.$$ My effort: Since $a+b+c =1$, we can write $$ ...
1
vote
0answers
24 views

$\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
2
votes
1answer
44 views

What is the most elementary proof of these inequalities?

Let $p$ be a non-zero integer, and let $x_1$, $\ldots$, $x_n$ be $n$ positive real numbers. Then we define the $p$-th power mean $M_p$ of these numbers as $$ M_p \colon= (\frac{x_1^p + \ldots + ...
0
votes
1answer
36 views

How to get n from n-1

I know this question is probably trivial, but I'm having great difficulty with it for some reason. So, I want to solve for $p$: $n-1 \geq 2(n-p)$ I know that the answer is $n \leq 2p -1 ...
1
vote
2answers
69 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
6
votes
2answers
186 views

How prove this Stronger AM-GM inequality $\frac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le A_{n}-G_{n}$

let $a_{i}>0,i=1,2,\cdots,n,n\ge 3$,show that $$\dfrac{n^2-1}{6}\min_{1\le i<j\le n}\left(\sqrt{a_{i}}-\sqrt{a_{j}}\right)^2\le\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}-\sqrt[n]{a_{1}a_{2}\cdots ...
6
votes
4answers
204 views

Inequality for harmonic means

Prove that for real numbers $a_1 ,a_2 ,...,a_n >0$ the following inequality holds $$\frac{1}{a_1 } +\frac{2}{a_1 +a_2 } +...+\frac{n}{a_1 +a_2 +...+a_n }\leq 4\cdot \left(\frac{1}{a_1} ...
0
votes
2answers
49 views

What does | mean in this exercise? And how do I solve it?

I was doing a practice exam for my SATs and I stumbled across this problem in the inequality section of the Algebra part. And I don't know what that symbol means and how to solve the problem with that ...
2
votes
1answer
39 views

How to prove this integral inequality?

Here is a problem: Let $B_r=\{ (x_1,x_2,\cdots,x_n)\in \mathbb{R}^n: x_1^2+x_2^2+\cdots+x_n^2<r^2\}.$ Let $f$ be a $C^1$ real function on $B_2$. Prove that $$\inf_{a\in R}\int_{B_2} ...
0
votes
3answers
52 views

How to derive this inequality?

How to derive the following inequality for all positive integers $n \geq 2$? $$ \frac{n!}{n^n} \leq \left(\frac{1}{2}\right)^k,$$ where $k$ denotes the greatest integer less than or equal to $\dfrac ...
0
votes
2answers
40 views

How to derive these inequalities?

I can derive the inequalities $$ n^p < \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^p $$ for any positive integers $p$ and $n$. These actually follow from the identity $$b^p - a^p = (b-a)(b^{p-1} + ...
0
votes
0answers
25 views

How about integral version of Holder's inequality?

In light of the fact that Minkowski's inequality have integral version, I thought there might be one for Holder's as well. I cannot find any through searching (there is an infinite product version in ...