Questions on proving, manipulating and applying inequalities.

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0
votes
3answers
37 views

How do you graph an inequality on Real/Imaginary plane?

Suppose we have $z$ as a complex number, $z \in C$, how do you graph an inequality which has $z$ in it? This kinds of inequalities arise when we need to graph the shape of stability region of a given ...
0
votes
0answers
10 views

bound $\delta_{s+1}$ from $\delta_s - \frac{1}{2\beta \| x_1 - x^\star \|^2} \delta_{s+1}^2$

The origin of the problem is on page 271, Convex optimization: Algorithm and complexity Given a function $f$ convex and $\beta$-smooth. Define $\delta_s = f(x_s) - f(x^\star)$, where $x_s$ is the ...
0
votes
1answer
56 views

For $a,b,c$ positive real numbers can it be true that: $(ab+bc+ca)^3 \ge (a^2+2b^2)(b^2+2c^2)(c^2+2a^2)$

For $a,b,c$ positive real numbers can it be true that: $$(ab+bc+ca)^3 \ge (a^2+2b^2)(b^2+2c^2)(c^2+2a^2)$$ It really seems unlikely because it reminds me the rearrangement inequality when the ...
0
votes
2answers
58 views

Prove that $ \sqrt{\sum\limits^n_{k=1}(x_k-y_k)^2} \leq \sqrt {\sum\limits^n_{k=1}x_k^2} + \sqrt{\sum\limits^n_{k=1}y_k^2} $

Because both sides are $\geq 0$, we square them, and we get $$ \sum\limits^n_{k=1}(x_k-y_k)^2 \leq \sum\limits^n_{k=1}x_k^2 + ...
2
votes
2answers
179 views

Complicated squeeze theorem

This was a question I had on one of my exams and I just could not figure out how to do it, any explanation would be greatly appreciated! Let $f$ be a function such that $-3 \le f(x) \le 5.$ Find ...
1
vote
1answer
22 views

Real analysis - proving inequality using concavity of function

I am struggling with deriving proof for the following lemma: By the use of the concavity of the appropriate function that for $x,y\in\mathbb{R}$ such that $ x+y =1 $ , prove that the following ...
27
votes
3answers
897 views

Olympiad Inequality $\sum_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the question. ...
30
votes
7answers
811 views

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $x^2y+y^2z+z^2x < \frac12$

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $$x^2y+y^2z+z^2x < \frac12$$ This inequality has been verified to be correct according to Mathematica. $\frac12$ is not the best bound. I try to do AM-GM ...
1
vote
0answers
70 views

$a,b,c >0$, and $ab+bc+ca=3$, prove $(a^ab^bc^c)^{\frac{3}{a+b+c}} \geqslant \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$

$a,b,c >0$, and $ab+bc+ca=3$, prove $$(a^ab^bc^c)^{\frac{3}{a+b+c}} \geqslant \sqrt[3]{\frac{a^3+b^3+c^3}{3}}$$ I think the equality is only achieve when $a=b=c=1$. The condition $ab+bc+ca=3$ is ...
7
votes
3answers
107 views

Find all real numbers $a,b$ such that $|a|+|b|\geq\frac{2}{\sqrt{3}}$ and $|a\sin x+b\sin{2x}|\leq 1$ for all real $x$.

Find all real numbers $a,b$ such that $|a|+|b|\geqslant\frac{2}{\sqrt{3}}$ and $|a\sin(x)+b\sin(2x)|\leqslant 1$ for all real $x$. We could write the inequality as $$ ...
4
votes
0answers
110 views

How to prove that $a^2b+b^2c+c^2a \leqslant 3$, where $a,b,c >0$, and $a^ab^bc^c=1$

$a,b,c >0$, and $a^ab^bc^c=1$, prove $$a^2b+b^2c+c^2a \leqslant 3$$ I don't even know what to do with the condition $a^ab^bc^c=1$. At first I think $x^x>1$, but I was wrong. This inequality is ...
1
vote
0answers
59 views

Prove $a^{ab}b+b^{bc}c+c^{ca}a \geqslant \sqrt[6]{5}$

$a,b,c >0$, and $a+b+c=3$, prove $$ a^{ab}b+b^{bc}c+c^{ca}a \geqslant \sqrt[6]{5}$$ I try to substitute $c=3-a-b$ to reduce the number of variables, but cannot further proceed to solve the ...
1
vote
2answers
32 views

Is it true that $ \sum_{i=1}^m \frac{1}{\sqrt{i}} = O \left( \sqrt{ m-1 } \right) $?

Is it true that?: $$ \sum_{i=n}^m \frac{1}{\sqrt{i}} = O \left( \sqrt{ \frac{m-n}{n}} \right) $$ In special case if we have $n = 1$, is it true that?: $$ \sum_{i=1}^m \frac{1}{\sqrt{i}} = O \left( ...
7
votes
0answers
81 views

Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$

If $a,b,c,d >0$, and $a+b+c+d=4$, prove that $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi.$$ I don't think Jensen's inequality will help here, but I think first determining where equality holds ...
1
vote
2answers
29 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 ...
1
vote
2answers
38 views

Prove that if G is a simple graph, $\chi \geq \frac{|V|^2}{|V|^2-2|E|}$

For a simple graph $G=(V,E)$, I have to prove the following bound on the chromatic number of $G$: $$\chi \geq \frac{|V|^2}{|V|^2-2|E|}$$
1
vote
0answers
38 views

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

Suppose $a_1, ..., a_n \in \mathbb{N}$ are arbitrary integers. Is it possible to bound $$ A =\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}} $$ with either of the following: $$ B = ...
3
votes
2answers
39 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.
-1
votes
1answer
39 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
vote
2answers
92 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
61 views

Maximum value of the sum of absolute values of cubic polynomial coefficients $a,b,c,d$

If $p(x) = ax^3+bx^2+cx+d$ and $|p(x)|\leq 1\forall |x|\leq 1$, what is the $\max$ value of $|a|+|b|+|c|+|d|$? 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
77 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
3answers
35 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
37 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 ...
4
votes
2answers
82 views

Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$

I want to prove $$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$ if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$ I have no idea where to start, any advice would be greatly appreciated!
1
vote
2answers
28 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
39 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
31 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
89 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
79 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
8 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
41 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?
5
votes
5answers
94 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, ...
5
votes
1answer
96 views

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
votes
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
votes
0answers
29 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
170 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 ...
0
votes
2answers
30 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
votes
0answers
37 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 ...
1
vote
2answers
60 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
6answers
65 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
58 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 ...
4
votes
1answer
64 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
51 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
41 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
46 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. [closed]

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
45 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} ...