Questions on proving, manipulating and applying inequalities.

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2
votes
1answer
49 views

Prove that $\forall z \in \Bbb C : \lvert \Re(z) \rvert \le \lvert z \rvert \le \lvert \Re(z) \rvert + \lvert \Im(z)\rvert$

I'm having difficulties trying to prove these two complex inequalities : $\forall z \in \Bbb C :$ $$\lvert \Re(z) \rvert \le \lvert z \rvert \le \lvert \Re(z) \rvert + \lvert \Im(z)\rvert$$ ...
0
votes
0answers
32 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 = ...
2
votes
0answers
21 views

$a^2b+b^2c+c^2a \leqslant 3$

$a,b,c >0$, 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. I have no clue if ...
0
votes
0answers
20 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
25 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( ...
6
votes
0answers
46 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 ...
2
votes
0answers
48 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 ...
3
votes
2answers
72 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!
0
votes
0answers
12 views

An inequality combining Hölder and Euler's aproach in his proof for infinitude primes

By Hölder inequality $$ \left( \sum_{n=1}^N\mu(k)\log k \right)^{q_n}\leq \left(\sum_{k=1 }^N \left| \mu(k) \right|^{p_n} \right)^{\frac{q_n}{p_n}}\cdot \left(\sum_{k=1} ^N (\log k)^{q_n} \right) ...
2
votes
2answers
43 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 $p(1)=a+b+c+d$, ...
3
votes
4answers
39 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?
8
votes
4answers
444 views

How to prove inequality $\frac{a}{a+bc}+\frac{b}{b+cd}+\frac{c}{c+da}+\frac{d}{d+ab}\ge 2$

Question: Let $$a,b,c,d>0,a+b+c+d=4$$ show that $$\dfrac{a}{a+bc}+\dfrac{b}{b+cd}+\dfrac{c}{c+da}+\dfrac{d}{d+ab}\ge 2$$ when I solved this problem, I have see following three ...
7
votes
0answers
110 views
+100

When might some a variable leave the basis?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...
1
vote
2answers
26 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
40 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 ...
1
vote
4answers
59 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} ...
3
votes
2answers
33 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
35 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 $$ax^4±bx^3±cx^2±dx±e \leq 0$$ ?
0
votes
0answers
25 views

How to generalize Tupper's self-referential formula?

How do I generalilze Tupper's self-referential formula so that it can graph arbitrarily big images, and not just $17 \times 106$ pixels ones? $${1\over 2} < \left\lfloor ...
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
56 views

Geometric inequality $180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}$

Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}\le 180^{\circ}$$ My work so far: ...
2
votes
4answers
81 views

An inequality involving $\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$

$$\frac{x^3+y^3+z^3}{(x+y+z)(x^2+y^2+z^2)}$$ Let $(x, y, z)$ be non-negative real numbers such that $x^2+y^2+z^2=2(xy+yz+zx)$. Question: Find the maximum value of the expression above. ...
9
votes
1answer
178 views
+50

Proof of an inequality in $\mathbb{C}$

Let $z\in \mathbb{C}, n \geq 2$. Show this complex inequality $$|z^n-1|^2\le |z-1|^2\left(1+|z|^2+\dfrac{2}{n-1}\Re{(z)}\right)^{n-1}$$ For $n=2$ the inequality is easy to prove: $$|z^2-1|^2\le ...
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 ...
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
0answers
105 views

When is the inequality $\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, a_2)\beta(b_1, b_2)$ true?

Let $\beta(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$. Does there exist some general condition on $a_1, a_2, b_1, b_2\in \mathbb{N}^+$ such that $$\beta(a_1+b_1, a_2+b_2)\ge \beta(a_1, ...
2
votes
1answer
34 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 ...
2
votes
1answer
72 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 ...
1
vote
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
31 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 ...
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 ...
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 ...
4
votes
2answers
51 views

What is the inequality which is used to prove this inequality?

Let $x,y,z,t$ be real numbers such that $x,y,z,t\geq 1$ and $xyzt=16$. How to prove $$x-\frac{1}{x}+y-\frac{1}{y}+z-\frac{1}{z}+t-\frac{1}{t}\geq6$$ I want some hint. thank you very much
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 ...
4
votes
5answers
87 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, ...
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
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 ...
1
vote
2answers
58 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 ...
-4
votes
1answer
40 views

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

Find $x \in \mathbb R$ that satisfy this inequality $4<|x +2| +|x-1|<5$
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 ...
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
votes
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 ...
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 ...
2
votes
0answers
56 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
778 views

Equality in Young's inequality for convolution

I am trying to understand how sharp Young's inequality for convolution is. The inequality says $||f \ast g||_r \leq ||f||_p ||g||_q$ where as $1/p+1/q = 1+1/r$. Actually, there are a couple of papers ...
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
5answers
54 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 ...
1
vote
1answer
199 views

Three related inequalities (the first being $2(|a|^p + |b|^p) \leq |a + b|^p + |a - b|^p \leq 2^{p-1}(|a|^p + |b|^p)$)

A friend told me this interesting problem. It should be easy enough, but I cannot figure it out completely. If $a, b \in \mathbb{R}, p \geq 2, \frac{1}{p} + \frac{1}{q} = 1$, then $2(|a|^p + |b|^p) ...