Questions on proving and manipulating inequalities.

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1
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4answers
116 views

Solving the logarithimic inequality $\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$

I tried solving the logarithmic inequality: $$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$ several times but keeping getting wrong answers.
0
votes
1answer
63 views

inequality funny question

I'm not sure what they want here: solve the inequality in realtion to $x$ for various values of $a$ : $\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$
4
votes
2answers
84 views

Prove that $\sqrt[3]{\frac{a^{2}}{(b+c)^{2}}}+\sqrt[3]{\frac{b^{2}}{(a+c)^{2}}}+\sqrt[3]{\frac{c^{2}}{(b+a)^{2}}}\geq \frac{3}{\sqrt[3]{4}}$ [closed]

Let $a;b;c \in \mathbb{R^+}$ such that $a+b+c=3$. Prove : $\sqrt[3]{\frac{a^{2}}{(b+c)^{2}}}+\sqrt[3]{\frac{b^{2}}{(a+c)^{2}}}+\sqrt[3]{\frac{c^{2}}{(b+a)^{2}}}\geq \frac{3}{\sqrt[3]{4}}$
0
votes
2answers
65 views

Using mean value theorem to show that $\cos (x)>1-x^2/2$

I have a question, by applying the mean value theorem to $f(x)=\frac{x^2}{2}+\cos (x)$, on the interval [0,x], show that $\cos (x)>1-\frac{x^2}{2}$. We know that ...
6
votes
2answers
232 views
+50

If $a_1a_2\cdots a_n=1$, then the sum $\sum_k a_k\prod_{j\le k} (1+a_j)^{-1}$ is bounded below by $1-2^{-n}$

I am having trouble with an inequality. Let $a_1,a_2,\ldots, a_n$ be positive real numbers whose product is $1$. Show that the sum $$ ...
2
votes
2answers
61 views

Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$ \left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E) $$ So far I have : For $x=0$ this inequality if not ...
3
votes
0answers
32 views

How to estimate $\displaystyle\sum_{|\alpha|\leq p} \binom{p}{|\alpha|}$?

I'm trying to find some estimates for some PDE I'm working on. I'd like to estimate the sum $$\sum_{|\alpha|\leq p} \binom{p}{|\alpha|},$$ where $\alpha\in\mathbb N_0^n$ and $\mathbb N_0=\mathbb ...
0
votes
0answers
32 views

Help with a simple function inequality

Let $R\geq 1$, $f(x)$ a function and $1_A$ the indicator function of the set $A$; is this inequality true? $$\frac{\vert f(x)\vert}{R}1_{R\leq\vert x\vert\leq 2R}\leq\frac{\vert f(x)\vert}{1+\vert ...
0
votes
0answers
13 views

Is there an effective bound known for the coefficients of half integer weight cusp forms?

If $f(z)=\sum a_n q^n$ is a cusp form (of integer weight) normalized so that $a_1=1$, we have the inequality $$\vert a_n \vert \leq d(n) n^{(k-1)/2},$$ known as the Deligne bound (in which $d(n)$ ...
1
vote
1answer
41 views

show an integral is bounded by a constant independent of a parameter

This is a question in Treves. Suppose $a>1$ and $\tau \in \mathbb R $, (i) show that for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge(\tau ^2+|\xi|^2+a^2)^{1/2}$ (ii) ...
0
votes
0answers
40 views

how to prove this inequality(ask for help ) [duplicate]

assume $$a>0,b>0,c>0,d>0$$ I want to prove: $$ \frac{1}{4}(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a})\geq (\frac{a^4+b^4+c^4+d^4}{4})^{1/4} $$ How to prove this? Thanks ...
5
votes
1answer
38 views

Inequality in four variables which sum up to 4

The positive real numbers $x,y,z,t$ satisfy $x+y+z+t=4$. Is the inequality $$x\sqrt{y}+y\sqrt{z}+z\sqrt{t}+t\sqrt{x}\leq4$$ true for all $x,y,z,t>0$?
2
votes
1answer
30 views

If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers. Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$. I don't understand why that is. Any help would be greatly appreciated.
0
votes
0answers
19 views

$ || \lambda(A) - \lambda(B) ||_p \prec_k || \lambda(A -B) ||_p$?

Given two Hermitian matrices $\mathbf{A}$ and $\mathbf{B}$ and eigenvalue function $\lambda(\cdot)$ which returns eigenvalues of a matrix in non-increasing order. I found the following is true from ...
1
vote
1answer
101 views

Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$ [closed]

I have a beautiful inequality, but I can only prove part of cases. Given any positive real numbers $a,b,c$, we have $$(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$$ How can we prove this ...
1
vote
2answers
45 views

Show that for all $(\tau, \xi) \in \mathbb R^{n+1}$ we have $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$

Show that, for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$ This is the exercise 7.4 in the book by Francois Treves. It is just a fundamental ...
1
vote
1answer
33 views

Clarification: how to get the following asymptotics

I'm having some trouble justifying some steps in a paper. Let $a_n$ be an increasing sequence of integers satisfying $n! \le a_n \le 2(n!)$, and let $f:\mathbb{N} \to \mathbb{N}$ be a function ...
3
votes
1answer
138 views

let $a,b,c >0 $ and $abc=1$,prove that $\sqrt{1+8a^2}+ \sqrt{1+8b^2}+ \sqrt{1+8c^2}\leq 3(a+b+c )$

let $a,b,c >0 $ and $abc=1$,prove that $\sqrt{1+8a^2}+ \sqrt{1+8b^2}+ \sqrt{1+8c^2}\leq 3(a+b+c )$ can anyone help me with this question. i've tried to assume that $a\geq b \geq c $ as my teacher ...
17
votes
1answer
402 views

How prove this inequality with n variables

Question: Suppose that $x_{1},x_{2},\cdots,x_{n}$ are real numbers, such that $$x_{1}x_{2}\cdots x_{n}\neq 0$$ and $$\dfrac{x_{1}}{x_{2}}+\dfrac{x_{2}}{x_{3}}+\cdots+\dfrac{x_{n}}{x_{1}}=0$$ ...
3
votes
3answers
292 views

Sufficient conditions for bound

Let $m\leq n$ be nonnegative integers and $x > 0$. I would like to find sufficient conditions on $m,n,x$ (as tight as possible) s.t. $$\frac{ \binom{n}{m} \sum_{j=0}^m j\binom{n}{m-j}x^j }{ x ...
1
vote
2answers
109 views

How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$

There is convex body $T$ (with the area is $1$), show that there is a triangle $\Delta ABC$, such $A,B,C\in T$, and $$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$ This problem is from China ...
0
votes
1answer
49 views

How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$?

How prove $\frac{x^{3}+y}{y^{3}+x}-1\geq \ln \frac{(x^{2}+1)^{2}}{x}-\ln \frac{(y^{2}+1)^{2}}{y}$ where $x, y\geq 1$?
0
votes
0answers
61 views

equivalent LMIs

page 63 of LMI book of Stephen boyd: why and how LMI conditions 5.14 and 5.12 are equivalent? how to get from 5.12 to 5.14? [ A'P+PA+Lambda*C'*C PB+lambda*C'*D; (PB+lambda*C'D)' -lambda(I-D'*D)] ...
9
votes
11answers
409 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
1
vote
2answers
27 views

A question about an inequality involving integers $\ge 0$ [closed]

$y^2 \le x < (y + 1)^2$ is true for a unique $y$ where $y \ge 0$ and $x \ge 0$ . Suppose $x > 9$, then why is $y \ge3$? Suppose $x = 10$. Then $y = \sqrt{10}$ or $y < \sqrt{10}$. So, ...
0
votes
1answer
26 views

Chong inequalites about permutations

I read about two inequalities called Chong's inequalities. They state: $$\sum_{k=1}^N\dfrac{a_k}{a_{\pi(k)}}\ge N$$ and $$\displaystyle\prod_{k=1}^Na_k^{a_k}\ge\prod_{k=1}^N a_k^{a_{\pi(k)}}$$ I ...
0
votes
2answers
30 views

Product across inequalities

Suppose I have a sequence of numbers $\{a_i \}_{i=1}^{N}$ and $\{b_i \}_{i=1}^{N}$ with $0\leq a_i\leq 1$ and $0 \leq b_i \leq 1$ for $i=1,...,N$. Is stating If $a_i \leq b_i$ $\forall i$ then ...
1
vote
2answers
67 views

$a,b,c \geq 0$,prove that $a^2+b^2+c^2+abc+5 \geq3(a+b+c) $

$a,b,c \geq 0$, prove that $a^2+b^2+c^2+abc+5 \geq3(a+b+c)$ I'm certain that this problem could be solved by using dirchlet's theory.but I do not know how to apply it exactly.
4
votes
1answer
70 views

Prove $a+b+c \geq ab+bc+ca$

If $a,b,c$ are positive real numbers that $$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\geq 1$$ is true,Prove:$$a+b+c \geq ab+bc+ca$$ Additional info:Additional info: We should only ...
1
vote
1answer
60 views

Proof of an inequality involving $(N-1)!$

How is it possible to prove the following inequality? ...
3
votes
2answers
87 views

How find the maximum of the value $x^2_{1}+x^2_{2}+\cdots+x^2_{2014}$

Question: let $x_{i}\in[-11,5],i=1,2,\cdots,2014$,and such $$x_{1}+x_{2}+\cdots+x_{2014}=0$$ find the maximum of the value $$x^2_{1}+x^2_{2}+\cdots+x^2_{2014}$$ since ...
5
votes
2answers
127 views

Inequality with $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}$

Inspired by this recent question, I suggest this. Let $n=2,3,4, \ldots .$ Then $$ \frac{7}{12} < \cfrac 1 {1 + \cfrac {1^2} {1 + \cfrac {2^2} {\ddots + \cfrac \vdots { 1 + \, {n^2} \,}}}} \leq ...
1
vote
1answer
31 views

Computational complexity and the big $\mathcal{O}$

I have a question about this Big $\mathcal{O}$ problem. I have the question down $90\%$, but the other $10\%$ isn't getting to me. I will write out the entire question and I'll point out the step, ...
5
votes
4answers
361 views

Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
5
votes
3answers
249 views

Is symmetry a valid option in inequalities?

Consider two questions: Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of 'e'? My answer: Since when e is maximum when all other variables are equal ...
4
votes
8answers
189 views

Show that the inequality holds $\frac{1}{n}+\frac{1}{n+1}+…+\frac{1}{2n}\ge\frac{7}{12}$

We have to show that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{7}{12}$ To be honest I don't have idea how to deal with it. I only suspect there will be need to consider two ...
-2
votes
3answers
77 views

$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=1$ prove that $a+b+bc+ac-abc \leq 2$

$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4$ prove that $ab+bc+ac-abc \leq 2$ can any one help me with this problem,I believe Dirichlet's theorem is the key for this sorry for making mistake over and over ...
1
vote
2answers
67 views

Induction inequality on sum of reciprocals

I have to prove that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{1}{2}$ for natural $n$ Checking for $n=1$ we have $\displaystyle 1+\frac{1}{2}=\frac{3}{2}\ge \frac{1}{2}$ ...
2
votes
1answer
100 views

Prove $\sum \limits_{cyc} \frac{1+b^2+c^4}{a^+b^2+c^3}\geq 3$

If $a,b,c$ are positive real numbers, prove $$\sum \limits_{cyc} \frac{1+b^2+c^4}{a+b^2+c^3}\geq 3$$ Additional info: We should only use Cauchy (preferred to used at least once and more than ...
1
vote
1answer
58 views

To control first derivative with the function itself.

Let $f$ be a compactly supported positive $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2< C f(x) $ by showing that for every point ...
1
vote
3answers
28 views

Inequality involving floor function and fractions

I have little to no experience working with floor inequalities so I am kind of stuck on this one. It seems pretty intuitive though. So basically I want to show that ...
4
votes
2answers
97 views

prove $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq 3(a^2+b^2+c^2)$

If $a,b,c$ are positive real numbers and $a+b+c=1$,Prove: $$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq 3(a^2+b^2+c^2)$$ Additional info:We can use AM-GM and Cauchy inequalities mostly.We are ...
4
votes
2answers
60 views

prove $\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$

If $a,b,c$ are positive real numbers,prove:prove $$\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$$ Additional info:We can use AM-GM and Cauchy ...
0
votes
0answers
37 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
0
votes
1answer
43 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
-1
votes
1answer
53 views

Finding two sided bounds on $(x+y)/(xy)$ given inequalities for $x$ and $y$

Given $\dfrac{1}{6} < x < \dfrac{1}{2}$ and $\dfrac{1}{7} < y < \dfrac{1}{3}$, can we determine bounds for $\dfrac{x+y}{xy}$?
1
vote
0answers
52 views

Arithmetic Mean and Geometric Mean Question, Guidance Needed

I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff ...
10
votes
1answer
138 views

prove that : $\frac{a^2+b^2}{a+b} + \frac{b^2+c^2}{b+c}+ \frac{c^2+a^2}{c+a} \geq 3$

For $a^2+b^2+c^2 =3$, with $a,b,c$, positive real numbers, prove that $$\frac{a^2+b^2}{a+b} + \frac{b^2+c^2}{b+c}+ \frac{c^2+a^2}{c+a} \geq 3.$$ can any one help me with this problem.
1
vote
1answer
35 views

Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
2
votes
1answer
53 views

prove $\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$

If $x$,$y$,$z$ are positive real numbers,Prove:$$\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$$ Using this two inequality: $\sum ^n_{i=1} \sqrt{a_ib_i}\leq\sqrt {ab} $ (we call it ...