Questions on proving and manipulating inequalities.

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Minimum of $\left(\frac1x+\frac1y+\frac1z\right)\left(\sqrt x+\sqrt y+\sqrt z\right)^2$

Minimum Value of $\left(\frac1x+\frac1y+\frac1z\right)\left(\sqrt x+\sqrt y+\sqrt z\right)^2$ for $x,y,z\in\mathbb R_{>0}$ Original Question is here, I applied Titus lemma to reduce it to the ...
3
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1answer
29 views

The biggest and smallest integer solution of $\sqrt{(5+2\sqrt6)^{2x}}+\sqrt{(5-2\sqrt6)^{2x}}\le98$ are?

$$\sqrt{(5+2\sqrt6)^{2x}}+\sqrt{(5-2\sqrt6)^{2x}}\le98$$ I noticed that $5+2\sqrt6=(\sqrt2+\sqrt3)^2$ but that hardly helps. After cancelling the square root and the power of two, I tried ...
2
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2answers
50 views

Mean Value Theorem and Inequality.

Using the mean value theorem prove the below inequality. $$\frac{1}{2\sqrt{x}} (x-1)<\sqrt{x}-1<\frac{1}{2}(x-1)$$ for $x > 1$. I don't understand how these inequalities are related. Am I ...
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0answers
24 views

Application of Van der Corput's lemma

By using Van der Corput's lemma, I will prove $$\int_{0}^{1}e^{ix\cdot(t,t^3)}dt\leq C(1+|x|)^{-1/3}$$ for $x$ in $\mathbb R^2$. But I don't know how to get term $(1+|x|)$.
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2answers
58 views

Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$

Let $z_1,\dots,z_n\in\mathbb{C}$ such that $|z_p|\le1$ for every $p\in\{1,\dots,n\}$. Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$. I have firstly ...
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1answer
11 views

Integer solutions to the inequality $\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$

$$\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$$ If $x$ is of the interval $[-8,10]$ Now I solved this, tried to limit $x$ as much as I could but I consistently get that there should be $10$ values of $x$ ...
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1answer
25 views

Show inequality (max, min)

How do I show this inequality $$d(x,z) \leq \max(d(x,y), d(y,z))$$ when $$\mu (x,y) = \min\{n\in\mathbb{N} \ | \ x_n \not= y_n \}$$ and $$d(x,y) = \frac{1}{\mu(x,y)}$$ What I've done so ...
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1answer
28 views

Inequality in matrix norm

Let $\|\cdot\|$ be matrix norm on $M_n$.Why does $\|A\|_2 \le \|A\|^{\frac{1}{2}} \|A^*\|^{\frac{1}{2}}$? ($\|A\|_2 = \displaystyle\max_{\|x\|_2 = 1} \|Ax\|_2$)
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1answer
28 views

Defining a bounded operator on $l^p$

Let $(c_{jk})_{j,k \in \mathbb{N}} \subset \mathbb{C}$ be such that $a:=\sup_{k \in \mathbb{N}} \sum_{j \in \mathbb{N}}|c_{jk}|<\infty$ and $b:=\sup_{j \in \mathbb{N}} \sum_{k \in ...
2
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3answers
38 views

On an Integral inequality.

I am following a proof and I am having troubles with the last inequality stated Specifically could I have some extra passages on this? $$\int_{\delta}^{\pi} [f(w+u) - f(w)] \frac{\sin^2(nu/2)}{2 ...
4
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0answers
71 views

Is there a math basis for what I observe in simulations? [duplicate]

I am stuck in my research with the following problem which occurs when proving a theorem. If $x,y\in \mathbb{R}^+$ and $x>y$, is it true that $(x-y)^{((x-y)/(2x-y))}\times ...
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61 views

Proving using AM-GM inequality

If $x,y\in \mathbb{R}$, and $x>y$, how to show $(x-y)^{((x-y)/(2x-y))}\times (x+y)^{((x)/(2x-y))}>x$? I know I have to use AM-GM inequality, but it is not clear how.
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1answer
18 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 ...
8
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2answers
114 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
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6answers
122 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, ...
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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} ...
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4answers
80 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, ...
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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.
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3answers
37 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
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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
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1answer
60 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
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1answer
45 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 ...
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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 $\delta(a)>0$ ...
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3answers
50 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 ...
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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 ...
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2answers
30 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 ...
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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 ...
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2answers
17 views

Solving inequality(limit)

Can someone explain how we get from $(x - 3) < \varepsilon/8$ and $x < 4$ to: $(x-3)(x+3) < (\varepsilon/8)(4+3) = (7\varepsilon)/8$
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2answers
109 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
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3answers
34 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
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0answers
33 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 ...
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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, ...
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2answers
99 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 ...
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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) - ...
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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 $AC>BC$?
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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
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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 ...
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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 ...
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2answers
56 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 ...
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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: ...
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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 ...
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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 $h(y)$, ...
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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
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4answers
207 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 ...
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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
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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
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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).
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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
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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 ...