Questions on proving, manipulating and applying inequalities.

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1
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2answers
24 views

Range of function Involving Modulus Quantity.

If $x,y,z\in \mathbb{R}\;,$ Then Range of $$\frac{|x+y|}{|x|+|y|}+\frac{|y+z|}{|y|+|z|}+\frac{|z+x|}{|z|+|x|}\,$$ $\bf{My\; Try::}$ Here $x,y,z$ Not all Zero Simultaneously. Now Using ...
0
votes
0answers
21 views

Upper-bounding $\sum_{i=1}^n \frac{1}{\sqrt{i+a_i}}$?

Suppose $a_1, ..., a_n \in \mathbb{N}$ are arbitrary integers. Is it possible to bound $$ A = \sum_{i=1}^n \frac{1}{\sqrt{i+a_i}} $$ with either of the following: $$ B = c\sqrt{\sum_{i=1}^n a_i} $$ ...
3
votes
2answers
32 views

Find range of $\alpha$ of $ \frac{4x^2+1}{64x^2 - 96x \sin \alpha +5} \leq \frac{1}{32}$ for all real x.

I simplified it to get $ \frac{64x^2 + 96 x \sin \alpha +27}{64x^2 - 96 x \sin \alpha +5} \leq 0$. I dont have any idea how to proceed further.
0
votes
1answer
30 views

How to solve quartic inequalities?

Could someone please explain to me how to solve quartic inequalities of the form $$ax^4±bx^3±cx^2±dx±e \geq 0$$ or $$ax^4±bx^3±cx^2±dx±e \leq 0$$ ?
1
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2answers
38 views

Show $\sum_{n\le x}\frac1{\sqrt n}=2\sqrt x-c+O(x^{-1/2})$

I am trying to show the asymptotic expansion for $$\sum_{n\le x}\frac1{\sqrt n}=2\sqrt x-\zeta(1/2)+O(x^{-1/2}).$$ (The exact identity of the zeta term is not important, it need only be some $c$.) To ...
2
votes
2answers
38 views

Maximum value of sum of absolute value of $a,b,c,d$

If $p(x) = ax^3+bx^2+cx+d$ and $|p(x)|\leq 1\forall |x|\leq 1\;,$Then Max value of $|a|+|b|+|c|+|d|$ $\bf{My\; Try::}$ Put $x=0\;,$ We get $p(0)=d$ Similarly Put $x=1\;,$ We get $p(1)=a+b+c+d$ ...
1
vote
4answers
58 views

Solving an inequality $\frac{3}{x-1}\lt -\frac 2x$

I've picked up my old book on Calculus, and going through the introductory examples of the preliminaries, I fail to see my mistake for the following exercise: Solve the inequality $\frac 3{x-1} ...
1
vote
2answers
26 views

Show a set is open using open balls

The set is $ \{ (x_1 , x_2) : x_1 + x_2 > 0 \}$ I wanted to solve this using open balls, so I said let $y = (y_1, y_2)$ be in the stated set. Then create an open ball $ B_r (y)$ around this ...
2
votes
1answer
33 views

prove that $\exists\ \epsilon>0$ such that $\forall x\in [0,1] : f(x)>x+\epsilon$

the question itself: Let $f$ be a continuous function in the close interval $[0,1]$ which upholds the rule: $\forall x\in [0,1] : f(x)>x$. prove that $\exists\ \epsilon>0$ such that $\forall ...
1
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2answers
25 views

If three vector such $|a|^2=a\cdot b=b\cdot c=1,a\cdot c=2$,show that $|a+b+c|\ge 4$

Let three vector $\vec{a},\vec{b},\vec{c}$ such $$|\vec{a}|^2=\vec{a}\cdot\vec{b}=\vec{b}\cdot \vec{c}=1,\vec{a}\cdot \vec{c}=2$$ show that $$|\vec{a}+\vec{b}+\vec{c}|\ge 4$$ since ...
4
votes
1answer
30 views

An inequality $\frac1{(n+1)^{1/(n+1)}}-\frac1{n^{1/n}}\le \frac1{n+1}$

I have graphed the functions $f,g:\mathbb{R^+}\to\mathbb{R}$ defined by $$f(x)=\frac1{(x+1)^{1/(x+1)}}-\frac1{x^{1/x}}\mbox{ and } g(x)=\frac1{x+1}$$ and it seems like $f(x)\le g(x)$ for all $x>0$. ...
0
votes
1answer
29 views

Prove inequality $\frac{a_1a_2…a_n}{(a_1+a_2+…+a_n)^n}\le \frac{(1-a_1)(1-a_2)…(1-a_n)}{(n-a_1-a_2-…-a_n)^n}$

Let $n\in \mathbb N, a_1,a_2, ...,a_n\in \left(0,\frac 12 \right]$. Prove inequality: $$\frac{a_1a_2...a_n}{(a_1+a_2+...+a_n)^n}\le \frac{(1-a_1)(1-a_2)...(1-a_n)}{(n-a_1-a_2-...-a_n)^n}$$ My ...
0
votes
0answers
13 views

when $|c|^2=|a|\cdot|b|-|a-c||b-c|$ then find the maximum of the value

In $\Delta ABC$,Let $\overrightarrow{AB}=a,\overrightarrow {CB}=b$. such$S_{ABC}=1$, and the vector $\vec{c}$ such $$\begin{cases}|a|=x|b|\\ |c|=y|b|\\ |c|^2=|a|\cdot|b|-|a-c||b-c|\end{cases}$$ when ...
2
votes
1answer
68 views

Prove this inequality with $a+b+c=3$

Let $a,b,c>0$,and $a+b+c=3$,show that $$\dfrac{a}{2b^3+c}+\dfrac{b}{2c^3+a}+\dfrac{c}{2a^3+b}\ge 1$$ such Use Cauchy-Schwarz inequality we have ...
0
votes
0answers
7 views

Prove the Poincare's inequality on $B^{0}(0,1)$. [duplicate]

Fix $\alpha >0$. Let $U=B^{0}(0,1)$. Show that there exists a constant $C$, depending only on $n$ and $\alpha$ such that $\int_{U} u^{2}\mathrm{d}x\leq C\int_{U} |Du|^{2}\mathrm{d}x,$ provided ...
2
votes
0answers
33 views

Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$. [duplicate]

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it? I see that if we can show $P[1 \le X - Y \le 2] \le P[|X ...
3
votes
4answers
36 views

Show that the $C_n \geq 4^{n-1}/2^{n}$ where $C_n$ is the Catalan number

I write $C_n=\frac{1}{n+1} {2n\choose n}$ and try to prove this claim by induction. But it didn't quite work out. Any idea how to do this without much computation?
4
votes
5answers
86 views

For integer $n>2$, $(n!)^2 > n^n$

Problem: For integer $n>2$, show that $(n!)^2 > n^n$ My attempt: I tried using induction. For $n=3$, the given condition is satisfied. Let us suppose $k!^2>k^k$ for some $k\geq3$. Then, ...
1
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0answers
33 views
+50

Transformation that preserves an increasing ratio between vectors

Consider two vectors $x=(x_1,x_2,\ldots,x_n)$, $y= (y_1,y_2,\ldots,y_n)$ such that all $x_i,y_i>0$ and \begin{align} \frac{y_1}{x_1}\le \frac{y_2}{x_2}\le\cdots\le \frac{y_n}{x_n} \end{align} Now ...
0
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1answer
23 views

$\left | \sum_{n\in \mathbb N} a_n b_n z^{n} \right | \leq C \left | \sum_{n\in \mathbb Z} b_n z^n \right | (z\in \mathbb C)$?

Let $ a_n , b_n \in \mathbb C$ for all $n\in \mathbb N.$ And there is $M>0$ such that $|a_n| \leq M$ for all $n\in \mathbb N.$ Can we expect $\left | \sum_{n\in \mathbb N} a_n b_n z^{n} \right | ...
0
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0answers
25 views

An inequality of first order partial derivatives.

Suppose $f:\mathbb R^2\to \mathbb C$ is $C^2$ with compact support. Show that $$\left\|\frac{\partial f}{\partial x_1}\right\|_p+\left\|\frac{\partial f}{\partial x_2}\right\|_p\le ...
15
votes
2answers
163 views

if $x^y=y^x$ show that $x+y>2e$

Let $0<x<y$, such that $$x^y=y^x$$ show that $$x+y>2e$$ Since $$y\ln{x}=x\ln{y}\Longrightarrow \dfrac{\ln{y}}{y}=\dfrac{\ln{x}}{x}$$ Let $$f(x)=\dfrac{\ln{x}}{x}\Longrightarrow ...
-4
votes
1answer
40 views

Solve $4<|x +2| +|x-1|<5$ [on hold]

Find $x \in \mathbb R$ that satisfy this inequality $4<|x +2| +|x-1|<5$
0
votes
2answers
28 views

Inversion of the inequality sign when raising to a negative power

How come since $e>1 \implies e^{-1/2} < 1^{-1/2}$. I know that one reverses the inequality signs when we take reciprocal of both sides or multiplies by a negative number. I have never seen ...
0
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0answers
36 views

Cauchy Schwarz look alike

Let $0<r<1<R$ be two fixed numbers. Suppose that there exist real numbers $x_1$, $y_1$, $z_1$, $x_2$, $y_2$, $z_2$, such that $x_i^2+y_i^2+(z_i-R)^2=r^2$ and $z_i\ge\frac{R^2+r-r^2}{R}$ for ...
-3
votes
0answers
25 views

Show that $1\leq \sqrt{x+1} \ln\left(1+\frac{1}{\sqrt{x}}\right) $ y $-1\leq \sqrt{x+1} \ln\left(1-\frac{1}{\sqrt{x}}\right)$ for $x\geq2$. [on hold]

Show that $$1\leq \sqrt{x+1} \ln\left(1+\frac{1}{\sqrt{x}}\right) \qquad \mbox{ y } \qquad -1\geq \sqrt{x+1} \ln\left(1-\frac{1}{\sqrt{x}}\right)$$ for $x\geq2$. Remark: My interest is to see an ...
1
vote
2answers
57 views

Proving that $\left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3$, given $1\leq a<b$

If $1\leq a<b$, then $$ \left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3.$$ Proceeding by integration by parts; let $u(x)=\sin(x)$ and $dv(x)=1/x$, then $u'=\cos(x)$ & $v(x)=\log(x)$. We ...
0
votes
0answers
35 views

If $f(x)\le 1$ implies $f(x)\le 1/2$ then $f(x+\delta)\le 1$?

Let $f(x)$ be a nonnegative continuous function of $x\in [0,K)$ with $f(0)\le 1/2$, and satisfies "$f(x)\le 1$ implies $f(x)\le 1/2$". Let $x_0\in[0,K)$ (so that $f(x_0)\le 1$ implies $f(x_0)\le ...
1
vote
5answers
53 views

Prove algebraically that, if $x^2 \leq x$ then $0 \leq x \leq 1$

It's easy to just look at the graphs and see that $0 \leq x \leq 1$ satisfies $x^2 \leq x$, but how do I prove it using only the axioms from inequalities? (I mean: trichotomy and given two positive ...
2
votes
0answers
54 views

Inequality involving fourth powers .

I have been into inequalities lately and I am stuck with this. I used a famous inequality at first $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge 3 (\frac{a^4+b^4+c^4}{3})^{\frac{1}{4}}$. From this ...
0
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0answers
36 views

Computng the floor of a sum [on hold]

Compute: $$\left\lfloor \frac {2^{1/{\sqrt 2}}+ 2^{1/{\sqrt 3}}+... +2^{1/{\sqrt {100}}}} {10} \right\rfloor.$$
3
votes
1answer
55 views

Prove that : $\frac{a+b+c+d}{a+b+c+d+f+g}+\frac{c+d+e+f}{c+d+e+f+b+g}>\frac{e+f+a+b}{e+f+a+b+d+g}$

Prove inequality for positive numbers: $$\frac{a+b+c+d}{a+b+c+d+f+g}+\frac{c+d+e+f}{c+d+e+f+b+g}>\frac{e+f+a+b}{e+f+a+b+d+g}$$ My work so far: Lemma: If $x>y>0, t>z>0$, then ...
1
vote
1answer
50 views

Prove $(1+x)^p+(1-x)^p \ge 2(1+x^p)$ for $0\le x\le1$ and real number $p\ge2$.

I don't know how to prove the following questions: If $p\ge2$ is real, then $$ (1+x)^p+(1-x)^p \ge 2(1+x^p) \quad \text{for } 0\le x\le1; $$ if $1\le p<2$, then opposite direction of the inequality ...
1
vote
2answers
37 views

If $a,b,c>0$ and $abc=1\;,$ Then minimum value of Expression.

If $a,b,c>0$ and $abc=1\;,$ Then minimum value of $$\frac{a^2}{a^2+2}+\frac{b^2}{b^2+2}+\frac{c^2}{c^2+2}$$ $\bf{My\; Try::}$ Using $\bf{Cauchy\; Schwarz}$ Inequality ...
-4
votes
1answer
44 views

Inequality that just won't go up!

I've gotten this result in an exam question on economics, and I can't seem to get this to make sense. Here, $Y$ is unknown. So, how do we know that this is true? $$(1-Y)(1-C) < (1+Y)(1-P), \quad P ...
-1
votes
1answer
42 views

I want to show that $x^2 - x + C\epsilon\ge 0$ under some assumption.

Let $x\ge 0$. For sufficiently small $\epsilon>0$, assume that the property $x\le \sqrt\epsilon$ implies $x\le \frac{1}{2}\sqrt{\epsilon}$. Then I want to show that $$x^2 - x + C\epsilon\ge 0 $$ ...
1
vote
1answer
43 views

Inequality with sum of numbers

A have found a very interesting inequality in a Romanian magazine which I use to prepare for the Lithuanian Mathematical Olympiad. Let $a_1,a_2,...,a_n$ be positive real numbers such that $$\frac {1} ...
1
vote
1answer
50 views

A polynomial inequality

Let $f(x)\in\mathbb{R}[x]$ be a polynomial of degree $n$, which has only real zeros. I would like to show that $$(n − 1)(f'(x))^2 \geq nf(x)f''(x),$$ where $f'$ and $f''$ denote the first and second ...
2
votes
3answers
106 views

Prove inequality with $e^x$ and $\ln$ on the same side [duplicate]

The problem is to prove the following inequality: $$ (e^x - 1) \ln(1+x) > x^2 , \quad\text{ for } x >0 $$ Let me introduce notation $f(x) > g(x)$. At $x=0$ both sides are equal to $0$. So, ...
0
votes
0answers
49 views

If $x$, $y$, $z$ are unequal positive quantities, then show that $(1+xyz)^3\geq (1+y^3)(1+z^3)(1+x^3) $ [on hold]

$$(1+xyz)^3(1+x^3) \geq (1+y^3)(1+z^3)$$ I think this problem is based on AM-GM inequality.
0
votes
0answers
18 views

Solving an inequality with ceiling [on hold]

I'm stuck on the following equation: $\lceil q/14 \rceil < m\lceil q/40\rceil$ From other answers here, I think you can do: Let $x = \lceil q/14 \rceil$ and $y = \lceil q/40 \rceil$ then ...
2
votes
1answer
63 views

Inequality about sums and products

Let $x_1,x_2,...,x_n$ be positive real numbers. Show that $$\frac {1} {2^n \times \sqrt {x_1 \times x_2 \times ... \times x_n} } + \sum_ {k=1}^n \frac {x_k} {(x_1+1)(x_2+1)...(x_k+1)} \ge 1.$$ I tried ...
1
vote
0answers
46 views

$1+x^4\leq 2(y-z)^2$ and switching of $x,y,z$

Find all triples of real numbers $x,y,z$ such that $1+x^4\leq 2(y-z)^2$, $1+y^4\leq 2(z-x)^2$, and $1+z^4\leq 2(x-y)^2$. Beside $(1,0,-1)$ and permutations, I can't find any others. We cannot have ...
1
vote
5answers
56 views

Prove that $\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$

Is it possible to prove that $$\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$$ without evaluating the integral?
0
votes
0answers
16 views

Bounding the $q$-th moment of a Gaussian random variable

I have come across an inequality which confuses me: Suppose $X$ has a Normal$(0,\sigma^2)$ distribution. Then $$ (\mathbb{E}|X|^q)^{1/q} \leq \text{const.} \sqrt{q} \sigma $$ for $q\geq 1$. I ...
1
vote
1answer
21 views

How to combine inequalities

If $u,v$ are real numbers and $|u-3|<1/3$ and $|v-3|<2/3$ then show that $|v-u|<1$. I'm unsure about how to combine this inequalities and simplify. Thanks in advance.
11
votes
5answers
177 views

How to show $\frac{19}{7}<e$

How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$? Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$ ...
1
vote
0answers
40 views

Prove that if $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ than $|f(x)| \leq c \left|\frac{1}{x}\right|^N$

The task is: Knowing that $\forall \lambda >0, x \neq 0$ $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ prove that: $|f(x)| \leq c \left|\frac{1}{x}\right|^N$ I would really appreciate ...
1
vote
2answers
23 views

Matrix norm inequality proof - does this use Cauchy-Schwarz?

The matrix norm for $A : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (so $A$ is an $m \times n$ matrix) is given by $$\|A\| = \sup_{X \in \mathbb{R}^n \setminus \{0\}} \frac{|AX|}{|X|}$$ where $| \cdot |$ ...
1
vote
0answers
69 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...