Questions on proving, manipulating and applying inequalities.

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Prove this inequality $xy+yz+xz\ge xyz$ [on hold]

Let $x,y,z>0$ such $$2xy+2yz+2xz=1+x^2+y^2+z^2$$ show that $$xy+yz+xz\ge xyz$$
6
votes
0answers
99 views

How prove this ineqlity [on hold]

Let $x,y,z,w>0$ and such that $xyzw=1$. Show that $$ \dfrac{1+x}{1+x^2}+\dfrac{1+y}{1+y^2}+\dfrac{1+z}{1+z^2}+\dfrac{1+w}{1+w^2}\le 4. $$
33
votes
4answers
1k 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 ...
3
votes
1answer
37 views

How can I prove this inequality involving logarithm?

$n^2 \geq n \log_{2}n$ I tried like this: $n^2 \geq n \log_{2}n$ $n^2-n \log_{2}n \geq 0$ $n(n-\log_2 n) \geq 0$ I don't know what to do after this?
0
votes
2answers
92 views

Minimum value of $4a+b$

Let $ax^2+bx+8=0$ be an equation which has no distinct real roots then what is the least value of $4a+b$ where $a,b\in \Bbb R$. My Try: I differentiated the given function to get $f'(x)=2ax+b$ now ...
0
votes
2answers
60 views

Solve $|x^2-5|\geq 4$

$$|x^2-5|\geq 4$$ $|x^2-5|\geq 4\Rightarrow$ $x^2-5\geq 4 $ or $x^2-5\leq -4$ Case: $x^2-5\geq 4\Rightarrow x^2-9\geq0\Rightarrow (x-3)(x+3)\geq 0$ so the answer is $x\geq 3$ or $x\leq -3$ case: $...
10
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2answers
381 views
+50

Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$

$x,y,z >0$ and $x+y+z=3$, prove $$\tag{1}\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$$ My first attempt is to use Jensen's inequality. Hence I consider the function $...
1
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5answers
116 views

Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$

$$|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$$ I have tried: $$|a\sin x+b \cos x|\leq |a+b|\leq \sqrt{a^2+b^2}$$ enough to prove: $$|a+b|\leq \sqrt{a^2+b^2}$$ But I can find how to continue from here
1
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2answers
81 views

Prove that: $\forall x\in (0,\frac{ \pi}{4(n-1)})$ $\tan(nx)> n \tan(x)$

Let $n\in \mathbb{N} , n> 1$ Prove that : $\forall x\in (0,\frac{ \pi}{4(n-1)})$ $\tan(nx)> n \tan(x)$ I know: $f(x) = \tan x$ is convex function $f(a x + b y) < a f(x) + b f(y), a+b=...
3
votes
4answers
82 views

Prove that $\frac{\tan{x}}{\tan{y}}>\frac{x}{y} : \forall (0<y<x<\frac{\pi}{2})$

Prove that $\frac{\tan{x}}{\tan{y}}>\frac{x}{y} : \forall (0<y<x<\frac{\pi}{2})$. My try, considering $f(t)=\frac{\tan{x}}{\tan{y}}-\frac{x}{y}$ and derivating it to see whether the ...
1
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0answers
13 views

A false identity involving $2^{\frac{1}{\zeta(s)}}$ for $\Re s>1$, from these particular values of the Riemann Zeta function and its alternating

Yesterday when I was exploring symbolic calculations $\dagger$ about specializations in $z=\frac{1}{n}$ with $n>1$ an integer, of $$\zeta(z)=(1-2^{1-z})^{-1}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^z}:=...
5
votes
1answer
177 views

Inequality with a rational polynomial

Let $$P(x)=x^{n-1}+a_{n-2}\,x^{n-2}+a_{n-3}\,x^{n-3}+\cdots+a_0\in\mathbb{Q}[x]$$ be a monic rational polynomial of degree $n-1$. I want to show that, for every set of $n$ distinct integers $\{x_1,...
10
votes
5answers
285 views

Which number is greater, $11^{11}$ or $9^{12}$?

Which number is greater than $11^{11}$ or $9^{12}$? My work so far: $11^{11}=285311670611>9^{12}=282429536481$. But to verify the validity of equality should be in the range of easily ...
1
vote
0answers
40 views

pound symbol in inequalities

a friend of mine has to solve some equations, but in some of them appears something like a pound symbol. Do you know what it means and how to solve it? or the teacher really lose his wit? Thank You!...
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4answers
96 views
-1
votes
1answer
25 views

Existence of a set function

Consider the set S of all subsets of {1, 2, 3, 4}. Show that there exists a function h : S → [0, 1] that satisfies all following conditions: Condition 1: h(∅) = 0 Condition 2: h({1, 2, 3, 4}) = 1 ...
7
votes
1answer
1k views

Proof of $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$

Let $a_n>0$ and $b_n\geq 0$, then $\lim\sup(a_nb_n)\leq \lim\sup(a_n)\limsup(b_n)$ My attempt at a proof is as follows. Let $A_n=\sup\{a_n, a_{n+1},...\}$, $B_n=\sup\{b_n, b_{n+1},...\}$, and $...
0
votes
1answer
23 views

Energy inequality heat equation

Consider $u \in C_1^2(\Omega \times [0,T]), \Omega\subset\mathbb{R}^n$ as a solution of the problem $ u_t - \Delta u = f, \text{ in } \Omega \times (0, T]$, $u = 0, \text{ on } \partial\Omega \...
0
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0answers
42 views

Jensen's inequality for two random variable

Prove: Let $X$ and $Y$ be two random variables in probability space $\left ( \Omega ,\mathcal{F},\mathbb{P} \right )$ , and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a convex function, then $$f\left ( ...
0
votes
1answer
45 views

Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
1
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0answers
33 views

Hardys Inequality on Integrals

My PDE script lists a form of Hardy´s inequality which is not the common one: Let $u\in C^\infty([0,\infty))$ and $\delta < -\frac{1}{2}$. Then: $(\int_0^\infty |r^\delta u(r)|^2 dr)^{\frac{1}{...
2
votes
0answers
101 views

Prove $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$

$x,y,z >0$, prove $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \frac32+ \frac{27}{16}\frac{(y-z)^2}{(x+y+z)^2}$$ This inequality is easier than the other one. Previously, I learned ...
-1
votes
1answer
39 views

Find $a$ and $b$ if $0 < x < 5$ then $a < x + 2 < b$.

I am trying to solve this simple equation but I am not able to figure out how to do it. Can somebody give me a hand? If $0 < x < 5$ then $a < x + 2 < b$.
16
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0answers
542 views

Stronger than Nesbitt inequality

For $x,y,z >0$, prove that $$\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geqslant \sqrt{\frac94+\frac32 \cdot \frac{(y-z)^2}{xy+yz+zx}}$$ Observation: This inequality is stronger than the ...
1
vote
3answers
99 views

Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$

If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$. I don't know how to solve it. And can you please mention the used formulas first. What I can see is that if one ...
0
votes
5answers
77 views

Solve the following $\frac{3x}{x+6} \ge 0 $

Solve $$\frac{3x}{x+6} \geq 0 $$ My work $$(x+6) / 3x <0 $$ $$1/3 + 6/x <0 $$ $$ 6/x <-1/3 $$ $$ x >-18 $$ is that correct
0
votes
1answer
32 views

Prove or disprove that $e^{-x} \, \dfrac{\cos\left( \frac{u-1}{2}\, x\right)}{\cos\left( \frac{u+1}{2}\, x\right)}\geq 1$.

Let $u\in[0,1]$. Prove or disprove that there exists an interval $I\subseteq [0,\pi/2]$ for which the following inequality $$e^{-x} \, \dfrac{\cos\left( \frac{u-1}{2}\, x\right)}{\cos\left( \frac{u+1}{...
1
vote
2answers
44 views

Existence of positive integer solution of a equation

I'm trying to find if the following equation has positive integer solutions $$x + (x+y) + (x+2y) + (x+3y) + \cdots + (x+(n-1)y) = z$$ where $z$ and $n$ are given. I can't progress further. -> $xn +...
1
vote
1answer
57 views

Simple inequality

$$2<\frac{x}{x-1}\leq3$$ What I did is: $$2<\frac{x}{x-1}\leq3\Rightarrow 2<\frac{x-1+1}{x-1}\leq3 \Rightarrow 2<1+\frac{1}{x-1}\leq3\Rightarrow 1<\frac{1}{x-1}\leq 2\Rightarrow$$ $$...
6
votes
1answer
591 views

Minimal set of inequalities

I have a set of $m$ linear inequalities in $R^n$, of the form $$ A x \leq b $$ These are automatically generated from the specification of my problem. Many of them could be removed because they are ...
3
votes
2answers
62 views

Prove inequality: $\frac{P+2004a}{P-2a}\cdot\frac{P+2004b}{P-2b}\cdot\frac{P+2004c}{P-2c}\ge2007^3.$

Let $a,b,c$ the sides of a triangle, $P$ its perimeter. Prove inequality: $$\frac{P+2004a}{P-2a}\cdot\frac{P+2004b}{P-2b}\cdot\frac{P+2004c}{P-2c}\ge2007^3.$$ My attempt: 1) $P=a+b+c$. Then $\...
1
vote
1answer
36 views

Upper and lower bounds log determinant

I found an inequality in Wikipedia that i want to know how to prove it. For a positive definite matrix A, the trace operator gives the following tight lower and upper bounds on the log determinant. $...
4
votes
1answer
46 views

Inequality problem

$$|x|^{x^2-x-2}<1$$ I tried to do this problem by taking different cases like first taking $x$ and then $-x$. For $+x$, the exponent should be less than $1$ so that the whole thing becomes a ...
1
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2answers
39 views

Prove that order is antisymmetric. (for natural numbers)

Prove that order is antisymmetric.(for natural numbers)i.e. If $ a \leq b$ and $b\leq a$ then $a=b$. I do not want a proof based on set theory. I am following the book Analysis 1 by Tao. It should be ...
0
votes
2answers
90 views

How to prove $(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$

Q: How to prove $$(\cos (x)+1)^{\sin (x)+1}>(\sin (x)+1)^{\cos (x)+1},(0<x<\frac{\pi}{4})$$ What should I do here? I don't even know where to start from. Please help me by giving me a hint.
2
votes
3answers
57 views

Solving $x^2 - 16 x+55> 0$ for $x$

Solving $x^2 - 16 x+55> 0$ for $x$ my work $$(x-11)(x-5) > 0$$ then x >11 and x > 5 is that correct ???
0
votes
0answers
29 views

$|P(z)|\le r^n C$

I've found that proof, first answer, using that $|P(z)|\le r^n C$ for all $|z|=r>1$ where $P(z)=\sum_{k=0}^{n}a_kz^k$, $C=\max\{|a_0|,...,|a_n|\}$. I do not understand why that is true. I only get ...
0
votes
1answer
26 views

What is the logical way to elaborate this identity expression (special expression) : |a| = -a (when a is negative)?

I am a beginner trying to resolve inequalities. However, I am having difficulty comprehending the stated expression. Isn't the absolute value of any expression positive or negative supposed to be a ...
2
votes
2answers
41 views

Interior of a preimage of a continuous function

Let $ f:\mathbb{R}^n\rightarrow \mathbb{R} $ be convex. Let there exist a point $ x_0 $ with $ f(x_0)<0 $. Prove that $$ \operatorname{int}\left\lbrace f(x)\ge 0 \right\rbrace = \left\...
2
votes
1answer
26 views

Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
1
vote
2answers
113 views

Proof of inequality between sums

Let $\{y_i\}_{i=1}^N, \{z_i\}_{i=1}^N$ be two sets of real numbers s.t. $y_i, z_i \ge 0$, $\sum_{i=1}^N y_i = 1$, $\sum_{i=1}^N z_i \le 1$. I have been asked to show that $$ \sum_{i=1}^N y_i \log \...
2
votes
0answers
30 views

Find the best constant $C_{n}$ such this complex inequality

nd we can consider this problem In general? if $|z_{1}|=|z_{2}|=\cdots=|z_{n}|=1$ if there exist complex $z(|z|=1)$ such $$\sum_{i=1}^{n}\dfrac{1}{||z-z_{i}||^2}\le C_{n}$$ find the best $C_{n}$? $$...
1
vote
3answers
77 views

About a complex number $z\in S^1$ fulfilling an inequality

Let $a,b,c$ be complex numbers, such that $\|a\|=\|b\|=\|c\|=1$. I have to show that there exists a complex number $z$, with $\|z\|=1$, such that: $$\dfrac{1}{\|z-a\|^2}+\dfrac{1}{\|z-b\|^2}+\dfrac{1}{...
0
votes
1answer
79 views

Three inequalities with a common constraint

A question struck me today and to be able to answer it I wrote it down as a mathematical expression and have been able to simplify the question to the following: Given $$x > 1,y > 1,z > 1$$ $...
0
votes
1answer
25 views

What is meant by strictly in this statement?

If $n$ is prime, then $n$ is not divisible by any prime number between 1 and $\sqrt{n}$ strictly. (Assume that $n$ is a fixed integer that is greater than 1.). I searched online and found that "...
0
votes
2answers
79 views

$\left| x \right| \le 3\left[ {\sqrt x } \right]$ [closed]

Let $\left| x \right| \le 3\lfloor {\sqrt x } \rfloor$. What is the answer to this inequality?
-2
votes
0answers
52 views

Need help in solving a few inequalities using $AM\ge GM$ [closed]

Can someone solve these questions purely using $AM\ge GM$? I tried and had no luck with these questions. $(1)$ If $a,b,c,d$ are positive numbers, such that $a+b+c+d=4$, then prove that $$\quad\frac 4{...
-3
votes
2answers
72 views

proving $\frac{1}{n+3}+\frac{1}{n+4}+…+\frac{1}{2n+4}>\frac{1}{2}$

how can one prove that: $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ For all natural $n$, without using induction? thank you.
-2
votes
2answers
45 views

Real Analysis (Proof)

I'm thinking that maybe this is an application of the Mean Value Theorem. But I'm not sure how to do it. Please help. >.< i) Let $a>0$ and $n>2$. If $$\frac{a}{1+2a}< \frac{1}{n}$$ , ...