Questions on proving and manipulating inequalities.

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0
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1answer
12 views

Sum of elements in a sequence

Let $a_n$ be a sequence in $\mathbb{R}$ and $a\in\mathbb{R}$. Suppose that $N \in \mathbb{N}$, $\epsilon >0$ and for every $n > N$ $|a_n -a|<\epsilon$. Show that for every $n>N$ the ...
5
votes
1answer
68 views

Prove: $\sqrt[3]{\frac{a^{2}+bc}{b^2+c^2}}+\sqrt[3]{\frac{b^{2}+ac}{a^2+c^2}}+\sqrt[3]{\frac{c^{2}+ab}{a^2+b^2}}\geq 9\frac{\sqrt[3]{abc}}{a+b+c}$

Let $a,b,c\in \mathbb{R^+}$. Prove: $\sqrt[3]{\frac{a^{2}+bc}{b^2+c^2}}+\sqrt[3]{\frac{b^{2}+ac}{a^2+c^2}}+\sqrt[3]{\frac{c^{2}+ab}{a^2+b^2}}\geq 9\frac{\sqrt[3]{abc}}{a+b+c}$ PS: I don't have ...
3
votes
5answers
96 views

Prove that $\sin (\theta) + \cos(\theta) \ge 1$

Let $\theta$ be an arbitrary acute angle. Prove that $\sin (\theta) + \cos(\theta) \ge 1$. $$\big(\sin (\theta) + \cos (\theta)\big)^2 = 1 + 2 \sin(\theta)\cos(\theta)\ge 0$$ so, ...
0
votes
0answers
13 views

Expectation Value and Generalized Holder Inequality

In the context of probability, I need help in interpreting a generalized Holder inequality (wiki): $\| \prod_{k=1}^n f_k \|_r \leq \prod_{k=1}^n \| f_k \|_{p_k}$, where $\sum_{k=1}^n \dfrac{1}{p_k} ...
5
votes
4answers
78 views

What is an exact characterization for the functions $f$ such that $xf'(x) \leq 2f(x)$?

What is an exact characterization for the functions $f$ such that $xf'(x) \leq 2f(x)$? I know, for instance, that the inequality holds for all functions $f(x) = c_0 + c_1x + c_2x^2$, with $c_0, c_1, ...
0
votes
4answers
46 views

Prove that $n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}$

I am trying to see why it holds that, for $n \in \mathbb{N},$ $$n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}.$$ I would appreciate help to see this.
2
votes
2answers
31 views

Differentiating both sides of an inequality with monotonic functions

If $f(x)\le g(x)$ for all real $x$ for monotonic functions $f$ and $g$ (say, both increasing), does it follow that $f'(x)\le g'(x)$? (Note: I've seen several questions asking the same thing without ...
2
votes
3answers
50 views

Maximum of given expression?

Suppose $a,b,c>0$ and further that $a^{2} + b^{2} + c^{2}=2abc + 1 $. The problem is to find $\max \big(a-2bc\big) \big(b-2ca\big) \big(c-2ab\big) $. Give me some help. I've tried $X=a-2bc$, ...
3
votes
1answer
58 views

$A,B$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?

Suppose $A,B \in {M_n}$ be Hermitian.Is this true that $tr[(AB)^2]\le tr(A^2B^2)$?
4
votes
1answer
35 views

An inequality on the series of powers of reciprocals of the primes

Let $p_n$ denote the $n$-th prime $(p_1=2)$ Let $s>1$ Prove that $\displaystyle-1+\ln(\frac{s}{s-1})\leq\sum_{k=1}^\infty\frac{1}{p_k^s}\leq\ln(\frac{s}{s-1})$ Using the classical ...
0
votes
0answers
33 views

integral inequalities and continuous functions [on hold]

Let $f$ be a positive, continuous function on $\mathbb{R}$. Let $c\in (0,1/2)$ be a constant and $\lambda>1$. I want to prove that: (1). for any $a\in\mathbb{R}$, there exists $\delta(a)>0$ ...
3
votes
3answers
49 views

How can I show this inequality: $-2 \le \cos \theta (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$

Show that $$-2 \le \cos \theta ~ (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$$ for all value of $\theta$. Trial: I know that $0\le \sin^2 \theta \le1 $. So, I have $\sqrt3 \le \sqrt{\sin ^2 ...
1
vote
2answers
42 views

Solve Inequality for $ |x| $

Given $$\big|\frac{(x-2)}{(x+3)}\big| < 4,$$ solve for $x.$ \ My solution $$|x - 2| < 4|x + 3|$$ Since, $ |x - 2| \ge |x| - |2| $ and $ |x + 3| \le |x| + |3| $ according to triangle ...
1
vote
2answers
29 views

Inequality involving inner product and an othonormal set of vectors

$ \newcommand{\ip}[2]{\left\langle #1,#2 \right\rangle} $ Here is the statement of the problem: Suppose that $V$ is a real inner product space with an inner product $\langle\cdot,\cdot\rangle$, and ...
1
vote
2answers
33 views

Requesting constructive feedback on my proof of a problem from Apostol Vol.1.

If x is an arbitrary real number, prove that there is exactly one integer n which satisfies the inequalities $n \le x < n+1$. Let S be the set of all $t \in \mathbb{Z}$ such that $t \le x$ for an ...
0
votes
2answers
17 views

Solving inequality(limit)

Can someone explain how we get from $(x - 3) < \varepsilon/8$ and $x < 4$ to: $(x-3)(x+3) < (\varepsilon/8)(4+3) = (7\varepsilon)/8$
13
votes
2answers
101 views

$\forall x,y>0, x^x+y^y \geq x^y + y^x$

Prove that $\forall x,y>0, x^x+y^y \geq x^y + y^x$ A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an ...
2
votes
3answers
33 views

Method for proving polynomial inequalities

Let $x\in\mathbb{R}$. Prove that $\text{(a) }x^{10}-x^7+x^4-x^2+1>0\\ \text{(b) }x^4-x^2-3x+5>0$ Possibly it can be proved in a few different ways, but I have first tried to prove it ...
2
votes
0answers
31 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ and $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
3
votes
1answer
37 views

Inequality and Trigonometric Substitution [duplicate]

Prove that for all positive real $a,b,c$, we have $$(a^2+2)(b^2+2)(c^2+2) \geq 9(ab+bc+ca).$$ Because of the term $a^2+2$, this motiveates me to substitute $a=\sqrt{2}\tan A, b=\sqrt{2}\tan B, ...
11
votes
2answers
98 views

Why is $\int\int f(x)f(y) |x-y|dxdy$ negative?

The Setup Let $f:\mathbb{R} \to\mathbb{R}$ be a smooth function with support in the interval $[-R,R]$ and satisfying $\int f = 0$. By manipulating some integrals, I found the surprising inequality ...
1
vote
1answer
41 views

Is this a correct way to use triangle inequality

If I have: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq f(x^*)$$ Can I proceed to say: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq |g_1(x) - g_2(x)| - |(g_1(a) - g_2(a))|$$ $$ \implies |g_1(x) - ...
0
votes
0answers
24 views

If positive definite matrices $A>B>0$ and $C>0$, then is $AC>BC$ true? [on hold]

Suppose I have positive definite matrices $A$, $B$ and $C$. If $A>B$, can we conclude $AC>BC$?
0
votes
1answer
31 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
4
votes
1answer
27 views

Multiplying two inequalities

Suppose we have two inequalities $$a\leq x\leq b\tag{1}$$ $$c\leq y\leq d\tag{2},$$ where $a,b,c,d>0$. Then can I conclude that $$ac\leq xy\leq bd\quad ?$$ My attempt: Since $a,b,c,d>0$ and ...
1
vote
0answers
32 views

Local estimates for $|(x+\epsilon)^{-1} - x^{-1}|$

I am interested in a local pertubation bound for the reciprocal function. How can you estimate the difference $|(x+\epsilon)^{-1} - x^{-1}|$ where $x > 0$ and $\epsilon > 0$ is small? Even ...
5
votes
2answers
54 views

Can every (convex) polygon be described by a single inequality (involving absolute values)?

For example, $$ |x| + |2x + y| + |x + 2y| + |y| + |x+y| < 4 $$ describes an octagon. I'm wondering whether an equation of this form always exist for any convex polygon, and if so, whether there ...
5
votes
2answers
100 views

$\frac{x}{\sqrt{yz}+\sqrt{3}}+\frac{y}{\sqrt{xz}+\sqrt{3}}+\frac{z}{\sqrt{yx}+\sqrt{3}}\leq \frac{1}{4\sqrt{3}xyz}$

Let $x;y;z>0$ such that: $xy+yz+zx=1$. Prove that: $\frac{x}{\sqrt{yz}+\sqrt{3}}+\frac{y}{\sqrt{xz}+\sqrt{3}}+\frac{z}{\sqrt{yx}+\sqrt{3}}\leq \frac{1}{4\sqrt{3}xyz}$ I think: ...
1
vote
2answers
31 views

How to apply the AM-GM inequality?

What is the minimum value of $8x^3+36x+54/x+27/x^3 $ for positive real numbers x? Express your answer in simplest radical form. I attempted to make an equation between the product of the terms and ...
0
votes
0answers
46 views

How to compute P(|X - E_Y[h(y)]| < c)?

Consider the discrete random variable $Y$, the continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, ...
0
votes
1answer
26 views

For which $x$ the inequality $ax+be^{x/2}>c$, where $a,b,c,x>0$ and $c>2b$ holds [on hold]

For which $x$ the inequality $ax+be^{x/2}>c$, where $a,b,c,x>0$ and $c>2b$ holds. Can someone help me for this. Thank you.
12
votes
4answers
206 views

Prove that $\sinh(\cosh(x)) \geq \cosh(\sinh(x))$

Prove that $$\sinh(\cosh(x)) \geq \cosh(\sinh(x))$$ I tried to tackle this problem by integrating both lhs and rhs, in order to get two functions who show clearly that inequality holds. I've ...
-1
votes
2answers
37 views

Calculate the greatest inequality solution. [on hold]

Can someone help me with this task, please? :) By the way, it's not my homework or etc. I want to learn some new things. :) Calculate the greatest inequality $10^x \leq 16 \cdot 5^x$ solution.
1
vote
1answer
40 views

Varied use of the AM-GM inequality

This question appeared in the IMO some year. I have done it in 2 different ways that seem absolutely correct. Please tell me which one is right and why. I fell both are very interesting. The question ...
2
votes
2answers
51 views

Bound on the integral of a function with multiple zeros

This is a follow-up to this If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that ...
1
vote
1answer
23 views

tight estimate for a log-linear inequality

Given $q>0$ and $p$, how do we get a tight estimate for the smallest $x$ such that $x\log(x)+px \geq q$? (such an $x$ always exists).
-2
votes
0answers
18 views

I have 4 in-equations with 4 variables in each of the in-equations. how to find the minimum value of each variable?? [on hold]

Please tell me the answer with solution. I don't know how to start it.completely blank.
0
votes
3answers
30 views

Are there any integral solutions to this inequality?

Are there any integral solutions to this inequality? $$\frac{n\sqrt{3} + 1}{n\sqrt{3}} + {\left(\frac{2n}{n + 1}\right)}^{1/2} < 1 + \sqrt{3}$$ WolframAlpha appears to give an inconsistent ...
1
vote
3answers
56 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
3
votes
1answer
31 views

If $q^k n^2$ is an odd perfect number with Euler factor $q^k$, can $q = 73$ hold?

Call a number $N$ perfect if $\sigma(N)=2N$ where $\sigma$ is the classical sum-of-divisors function. If $N = q^k n^2$ is an odd perfect number, can $q = 73$ hold? Here is my attempt: Since ...
1
vote
0answers
22 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
2
votes
1answer
70 views

Proof: $|||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| $?

I am looking for a proof of the following: \begin{equation*} |||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| \end{equation*} Where A, B are positive, hermitian matrices, and $|||⋅|||$ ...
1
vote
3answers
80 views

Prove inequality $ab+bc+ca\ge 3,\ abc=1$

How can I prove \begin{equation*} ab+bc+ca\ge 3,~a,b,c \in\mathbb{R},~ a,b,c>0\ \end{equation*} and the product $abc=1$? I obtained only $(a+b+c)^2-(a^2+b^2+c^2)\ge6\ and \ ...
1
vote
0answers
22 views

Poincarè inequality in probability

I'm looking for a proof of the poincarré inequality in a probabilitic setting. That is to say, let $\mu$ be a probability on $\Bbb R^n$, what are the hypothesis in order to have, for any f smooth ...
1
vote
0answers
23 views

First moment inequality and time-average limits

Suppose $\{A(t)\}_{t \geq 0}$ and $\{B(t)\}_{t \geq 0}$ are two non-negative stochastic processes such that $$ \frac{1}{T} \int_{s=0}^T A(s) \, {\rm d} s \stackrel{\text{a.s.}}{\rightarrow} a \in ...
-2
votes
4answers
91 views

Prove the inequality $a^2 + b^2 +c^2 \ge ab +bc +ac$ [on hold]

How do I prove the inequality \begin{equation*} a^2 + b^2 +c^2 \ge ab +bc +ac \end{equation*} where $ a,b,c\in\mathbb{R} $ and $a,b,c>0$? I obtained only $(a+b+c)^2\ge 3(ab+bc+ac)$ and some ...
1
vote
0answers
19 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i), \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
-4
votes
2answers
26 views

help with alternating series test [closed]

How do I show that $$\large{\frac{\ln(n)}{n} \geq \frac{\ln(n+1)}{(n+1)}}$$ for $\large{n \geq 1}$?
0
votes
0answers
36 views

Inequality on complex polynomial

For every $a\geq 0$, let $p_a(z)=1-z+az^3$. What is the maximal value of $a$ such that $$ p_a(|z|)\leq |p_a(z)| $$ for all $z\in \mathbb C$? EDIT: I claim that $a=\frac{4}{27}$ is the maximal value. ...
3
votes
6answers
105 views

Visualize $z+\frac{1}{z} \ge 2$

As we know, always $$z+\frac{1}{z} \ge 2,~~~~~~~~~ z\in \mathbb{R}^+$$ However, is there any geometric way to visualize this equation for some one who is not that expert in math? I know this ...