Questions on proving, manipulating and applying inequalities.

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2
votes
1answer
69 views

How prove $-\sqrt{2}\log(\cos x)\leq\sqrt{x\tan x-\sin^{2}x}$?

How prove that $-\sqrt{2}\log(\cos x)\leq\sqrt{x\tan x-\sin^{2}x}$ for all $x\in\left [ 0,\frac{\pi}{2}\right)$?
7
votes
1answer
164 views

$a,b,c \geq 0$ and $a+b+c=3$ prove that $\frac{a^2+bc}{b+ac} + \frac{b^2+ac}{c+ab} + \frac{c^2+ab}{a+bc} \geq 3$

$a,b,c \geq 0$ and $a+b+c=3$ prove that $\frac{a^2+bc}{b+ac} + \frac{b^2+ac}{c+ab} + \frac{c^2+ab}{a+bc} \geq 3$ can anyone help me solve this problem,i've tried to use C-S and also AM-GM for couple ...
5
votes
1answer
218 views

Inequality involving traces and matrix inversions

The following question kept me wondering for some weeks: Given the symmetric matrices $A,B,C\in\mathbb{R}^{n\times n}$ where $A$ and $C$ are positive definite (hence invertible), and $B$ is positive ...
0
votes
1answer
53 views

Prove ${\sum \limits_{cyc}\frac{a^4+a^2+1}{a^6+a^3+1}\leq\sum \limits_{cyc}\frac{3}{a^2+a+1}}$

If $a,b,c$ are positive real numbers,Prove:$${\sum \limits_{cyc}\frac{a^4+a^2+1}{a^6+a^3+1}\leq\sum \limits_{cyc}\frac{3}{a^2+a+1}}$$ Additional info: We should only use Cauchy and AM-GM. ...
2
votes
1answer
25 views

Estimate of line integral of O(x^n) function

Let $f$ be an analytic function in some sector in the complex plane behaving as $$f(z)=\mathcal O(z^n)$$ for some $n$ as $z\to\infty$. Can one prove in general that line integrals of $f$ (in this ...
1
vote
1answer
94 views

Don't understand inequality in order to prove Algebraic Limit Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 2.3.3 on page 45, i.e., the Algebraic Limit Theorem. In particular, letting $\lim a_n = a$ and $\lim ...
7
votes
1answer
289 views

An interesting inequality about the cdf of the normal distribution

When approaching this other question I came out with the inequality: $$\frac{1}{4+x^2}e^{-x^2/2} \leq\Phi(x)\Phi(-x)\leq \frac{1}{4}e^{-x^2/2},\tag{1}$$ where $\Phi(x)$ is the cdf of the standard ...
0
votes
1answer
42 views

Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
0
votes
1answer
33 views

How to prove $|q|\ge 1 \Rightarrow |a|\ge |d|$?

Let $a,d,q \in \mathbb{Z}$ and $a=dq$ How do I show that $|q| \ge 1 \Rightarrow |a| \ge |d|$? I've tried: $|q|\ge 1 \Rightarrow (q>1 \text{, if } q>0) \text { or } (-q>1 \text{, if } ...
1
vote
4answers
159 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
77 views

inequality funny question

I'm not sure what they want here: solve the inequality in realtion to $x$ for various values of $a$ : $\frac{(a+2)x}{a-1} - \frac{2}{3} < 2x-1$
0
votes
2answers
118 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
288 views

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
67 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
33 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
50 views

Is there a standard symbol that denotes the set of relational operators

I am writing a research paper, and I would like to somehow denote the set of relational operators $$ \left\{ =,>,<,\leq,\geq,\neq\right\} $$ Before I use some random symbol for that purpose, ...
1
vote
0answers
49 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 $d(n)$ ...
0
votes
1answer
97 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
46 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 ...
4
votes
4answers
145 views

How to prove this $(n+1)^n < n^{n+1}$ for $\space n \ge 3$

I'm having some more trouble with induction I know how to prove this using $\ln$, but I need to use induction only. prove that: $(n+1)^n < n^{n+1}$ for any $ n\ge 3$
4
votes
1answer
49 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
36 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
29 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
134 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 ...
0
votes
2answers
52 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
39 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
2answers
187 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 ...
18
votes
1answer
477 views

If $\prod x_k\neq0$ and $\sum\frac{x_k}{x_{k+1}}=0$, then $|\sum x_kx_{k+1}|\le (\max|x_k|- \min|x_k|)\sum x_k$

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
315 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 ...
2
votes
2answers
130 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
53 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 $x, y\geq 1$?
10
votes
11answers
448 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
0
votes
1answer
33 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
34 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
74 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.
6
votes
1answer
158 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
68 views

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

How is it possible to prove the following inequality? ...
3
votes
2answers
97 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
167 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
39 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
418 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
358 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 ...
5
votes
8answers
225 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
126 views

$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4$ prove that $ab+bc+ac-abc \leq 2$ [duplicate]

$a,b,c \geq 0$ 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
163 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
114 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 ...
6
votes
3answers
289 views

To control first derivative with the function itself: $f'(x)^2\leq Cf(x)$ near where $f(x_0)=f'(x_0)=f''(x_0)=0$.

Let $f$ be a compactly supported nonnegative $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2\leq C f(x) $ by showing that for every point ...
1
vote
3answers
82 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 ...
5
votes
2answers
146 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 ...
1
vote
2answers
68 views

Trigonometric problem: $2^{\sin{x}} + 2^{\cos{x}} \ge 2^{(1-1/{\sqrt2})}$

Show that: $$\large2^{\sin{x}} + 2^{\cos{x}} \ge 2^\left({1-\frac{1}{\sqrt{2}}}\right)$$ This looks like an am gm problem to me where we should be using the fact that am is more that or equal to gm ...