Questions on proving, manipulating and applying inequalities.

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14
votes
3answers
626 views

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 $...
3
votes
2answers
46 views

Prove this inequality with trigonometry $9\cos^2{x}-10\cos{x}\sin{y}-8\cos{y}\sin{x}+17\ge 1$

let$x,y\in R$,show that $$9\cos^2{x}-10\cos{x}\sin{y}-8\cos{y}\sin{x}+17\ge 1$$ Maybe use Cauchy-Schwarz inequality can solve it?and I can't Adit it:I think the right hand can replace constant $9$ ...
2
votes
1answer
41 views

Number theory problem, fractions and gcd, please help!!!

The problem says "if a,b are positive integers such that $\frac{a+1}{b}+\frac{b+1}{a}$ is an integer then show that $\sqrt{a+b}\ge$ gcd(a,b)" Adding $\frac{2ab}{ab}$ to $\frac{a+1}{b}+\frac{b+1}{a}$ ...
0
votes
2answers
34 views

How to solve $(2p_1^2-2p_1+1)^n \le 2^{-10}$ where $p_1 = 1-(1-(1/n))^N$.

Let $$S_{n,N}=(2p_1^2-2p_1+1)^n$$ and $p_1 = 1-(1-(1/n))^N$. I would like to solve $S_{n,N} \leq 2^{-10}$ for $n$. This seems hard to do exactly but is there a good approximation one can find? We ...
-1
votes
1answer
61 views

Prove or disprove $x^{a_1}y^{a_2}+x^{a_2}y^{a_1}\ge x^{b_1}y^{b_2}+x^{b_2}y^{b_1}$

Prove or disprove: If $\max(a_1,a_2)\ge\max(b_1,b_2)$, then $$x^{a_1}y^{a_2}+x^{a_2}y^{a_1}\ge x^{b_1}y^{b_2}+x^{b_2}y^{b_1}$$ I can not understand it in the proof Muirhead's inequality ...
0
votes
1answer
36 views

Elementary proofs involving inequalities

So the task of this exercise is to prove each statement. $\forall a \in$ $\mathbb R$: Prove that $a^2 \ge 0$ Does it suffice to say that $a^2 \gt 0$ or $a^2 = 0$, which means that $a \gt 0$ or $...
1
vote
2answers
48 views

Prove the inequality between integral and summation of multiplicative inverse

I want to prove the following inequality: $$ \ln(n) = \int\limits_1^n{ \frac{1}{x} dx } \geq \sum_{x = 1}^{n}{\frac{1}{x + 1}} = \sum_{x = 1}^{n}{\frac{1}{x}} - 1 $$ I ask this question as I'm ...
1
vote
1answer
22 views

How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B) $$ or ...
2
votes
0answers
38 views

Variant of Barrow's inequality

I proposed the conjecture as following: Let $ABC$ be a triangle, let $D$ be a point inside of $ABC$. From $D$ and $ABC$, define $F$, $E$, and $G$ as the points where the internal angle bisectors of $\...
2
votes
3answers
82 views

Prove $5a^2+b^2+c^2\geq 4ab+2ac$

I saw this question recently: Let $a,b,c$ be real numbers. Prove $5a^2+b^2+c^2\geq 4ab+2ac$. I feel like this is something with AM-GM inequality. Can someone help me with it?
5
votes
3answers
1k views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| \...
1
vote
1answer
48 views

Meaning of $Ax \leq b$

I continue to come across $Ax \leq b$ or $Ax= b$ in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ...
1
vote
3answers
34 views

Find the $\sum_{sym}ab$ maximum of the value

Let $a,b,c,d,e\in (0,1)$ and such $$a+b+c+d+e=1$$ find the maximun of the value $$S=ab+ac+ad+ae+bc+bd+be+cd+ce+de$$ I Conjecture the maximun is $\dfrac{2}{5}?$,such $a=b=c=d=e=\dfrac{1}{5}$,so $$S\...
2
votes
2answers
57 views

Prove that for any $n \ge 2$,$1\times3\times5\times \dots \times(2n-1)<n^n$ without induction

Prove that for any $n \ge 2$,$1\times3\times5\times \dots \times(2n-1)<n^n$ without induction I asked for a non induction prove but I am stuck in induction prove too. In induction we should prove ...
36
votes
6answers
2k 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 ...
5
votes
2answers
63 views

An inequality in positive real continuous function

I proposed my conjecture as follows: Let $f(x)$ is a positive real continuous function that is convex on $[m, M]$, let $m \le x_i \le M$, for $i=1,2,...,n$ then show that $$\frac{f(x_1)+f(x_2)+.....+...
-2
votes
2answers
25 views

Find values of $m$ and $n$ such that $m \leqslant 6 \sin x+ \cos (2x) -1\leqslant n$ [on hold]

$$m \leqslant 6 \sin x+ \cos (2x) -1 \leqslant n$$ I have no clue how to do it. please help.
2
votes
3answers
44 views

Show that this inequality doesn't hold

Given $(a,b,c) \in \mathbb R^3_+$ show that atleast one of the real numbers $a(1-b)$, $ b(1-c)$ and $c(1-a)$ is less than or equal to 1\4. I tried to show it by contradiction i.e Suppose that $$a(1-...
0
votes
1answer
32 views

I'm having trouble understanding what this problem is asking me

This is the problem So my problem is that I dont know how to solve it... I have learned about system of inequalities and that kinda stuff, but I never got anything like this. I do not want anyone to ...
1
vote
10answers
117 views

A clean proof of $x^2 \geq x$, for any integer $x$

I am trying to prove that $x^2 \geq x$ for any integer $x$. Since we know that for any number $n$, $n^2 \geq 0$ we conclude that if $x \leq 0$ the proposition will hold. Next we must prove that the ...
0
votes
2answers
59 views

Solve the following using AM-GM inequality

The least value of $a \in R$ for which $4ax^2 + \frac{1}{x} \ge 1 $for all $x \gt 0 $, is Using AM-GM inequality $$\frac{4ax^2 + \frac{1}{2x} + \frac{1}{2x}}{3} \ge \sqrt[3]{a}$$ $$4ax^2 + \frac{1}{...
-2
votes
0answers
33 views

Hölder inequality application to show that f=1

I want to proof that if $f \in L^{1}_{\mu}(\mathbb{R}), f > 0$ continuous, satisfies $(\int_\mathbb{R} f(x)d\mu)^{3} \le \int_\mathbb{R} f(x)^{3sin^{2}(x)}d\mu * (\int_\mathbb{R}f(x)^{\frac 32cos^{...
4
votes
7answers
93 views

Solution of Inequality $\displaystyle \frac{1}{x-6}\le 3$

Solve the inequality: $\displaystyle \frac{1}{x-6}\le 3$ solution: \begin{align*}\frac{1}{x-6}& \le 3 \\ x-6& \le \frac{1}{3} \\x& \le 6+\frac{1}{3}\\ x&\le19/3\end{align*} but, ...
5
votes
1answer
139 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}...
1
vote
1answer
28 views

Implication of exponential growth: how is it deduced?

Let $L$ be a differentiable function defined on $\mathbb{R}\times\Omega$ with $\Omega\subseteq\mathbb{R}^n$. I will say it has exponential growth if for all $O\subset\subset\Omega$ open there exists a ...
3
votes
1answer
101 views

Minimize $P(x,y,z)=(2x+3y)(x+3z)(y+2z)$, when $xyz=1$

Find the minimum value of the product $P(x,y,z)=(2x+3y)(x+3z)(y+2z)$, when $xyz=1$ and $x,y,z$ are positive real numbers. I don't know how to go about this. AM-GM got really messy, and I don't know ...
0
votes
2answers
38 views

How to prove that ${l \choose a_1,…,a_n}\le n^{l-1} $ , when $a_1+…+a_n=l$.

In the proof of (Corollary 8 chap. 3 ) in the book "Sobolev Spaces on Domains" by Burenkov the following inequality is used : given $a_1,...,a_n \in \mathbb{N}$ such that $a_1+...+a_n=l$, then $${l \...
2
votes
3answers
35 views

Does Gaussian convolution respects order?

Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$ Now let us define the ...
0
votes
2answers
20 views

Solving the inequality involving modulus

Can I change $\frac{1}{|x-2|} \le \frac{1}{|2x-3|}$ to $|x-2| ≤ |2x-3| ? $ If I remembered correctly, I cant change $a \lt \frac{ 1}{|b|}$ to $a|b| \lt 1$ instead, I have to change it to $a-\frac{1}...
0
votes
0answers
31 views

Help with Simplying and equation

I would like some help simplifying and equation. Contraints $C_1 $ is a positive integer Constant $C$ is also a positive integer Constant $x$ and $ y $ are both real real numbers. $x\leq 0$, $y\...
1
vote
2answers
74 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb F^...
0
votes
1answer
34 views

Express $c$ and $d$ in terms of $m$ where $c$ and $d$ are zeroes of $f$ where $m > -2$

Let $$f(x) = x^2 - mx -(6m^2+25m+25)$$ where $m > - 2$ It can be shown that $f(x)$ has two zeroes. Suppose we have $c,d \in \mathbb R$ s.t. $c < d$ and $f(c) = f(d) = 0$, express $c$ and $d$ ...
1
vote
3answers
60 views

Mathematical Induction Inequality problem [on hold]

I am trying to solve the following problem with mathematical induction: $$ \forall n>1,\qquad \frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<\frac{n-1}{n} $$ but since I am new to the concept ...
0
votes
1answer
37 views

how to proceed next in this logarithmic inequality?

The question is $$\frac{1}{\log_4{\left(\frac{x+1}{x+2}\right)}}<\frac{1}{\log_4{(x+3)}}$$ I did the first step for defining the arguments of both sides and got $x\in(-3,-2)\cup (-1,\infty)$ ...
2
votes
2answers
73 views

On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
0
votes
0answers
27 views

Find the maximum of the $k$ such $0\le x^2(3-2x)(2x^k+(3-2x)^k)\le 3$

Find $k_{\max}$,such $$0\le x^2(3-2x)(2x^k+(3-2x)^k)\le 3,0\le x\le 1$$ since $$x^2(3-2x)>0\Longrightarrow 2x^k+(3-2x)^k\ge 0$$ it is clear for $k\in R$ and other case it's not easy to solve
21
votes
7answers
2k views

proving $\mathrm e <3$

Well I am just asking myself if there's a more elegant way of proving $$2<\exp(1)=\mathrm e<3$$ than doing it by induction and using the fact of $\lim\limits_{n\rightarrow\infty}(1+\frac1n)^n=\...
5
votes
1answer
68 views

Circles in complex plane.

Find the real value of a for which there is at least one complex number satisfying $|z+4i|=\sqrt{a^2-12a+28}$ and $|z-4\sqrt{3}|\lt a$. My solutions:- Graphical solution:- $|z+4i|=\sqrt{a^2-...
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 $...
2
votes
5answers
89 views

Monotonicity of the sequence $(a_n)$, where $a_n=\left ( 1+\frac{1}{n} \right )^n$

Define $a_n=\left ( 1+\frac{1}{n} \right )^n$ for $n\geq 1$. I want to show that it is increasing. First, we have $$\frac{a_{n+1}}{a_n}=\left ( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right )^n\left ( ...
1
vote
0answers
13 views

Proof using positive (semi)definite matrices and a sharp matrix inequality

Take symmetric and real matrices F, f and f' such that $F \geq f$ and $F>f'$. Here $F \geq f$ means that $F-f$ is positive semi-definite, and $F>f'$ means that $F-f'$ is positive definite. I ...
32
votes
7answers
1k 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 by Mathematica. $\frac12$ is not the best bound. I try to do AM-GM for this one but not ...
0
votes
1answer
15 views

Determining Bounds to calculate mass

Let $E$ be the solid region defined by the inequalities $x \ge 0$, $0\le z \le \sqrt(x^2 + y^2)$, $x^2 + y^2 + z^2 \le 4$ Suppose that $E$ has mass density $\mu(x,y,z) = xz$. Calculate the ...
1
vote
1answer
23 views

Query about an algebraic inequality involving $(a-b)^p$

I want to know if there exists any inequality of the type $(a-b)^p \geq C(a^p -b^p)$ or $(a-b)^p \leq C(a^p -b^p),$ where $a>0,\, b>0,$ $C>0$ is a constant and $0<p<1.$ I am aware of ...
-1
votes
4answers
92 views

Prove $n^{n/2} < n!$ if $n \gt 2$ [duplicate]

Ive been stuck on this question for so long.How do i do it? $n^{n/2} < n!$ if $n \gt 2, n \in \mathbb{N}$. Please help guys.
1
vote
1answer
20 views

Upper and lower bound of the ratio of summation

Consider $x_1,x_2,x_3,....,x_n\in \mathbb{N}^+$ What is the upperbound and lowerbound of the following expression $R=\frac{\sum_{i=1}^{n-1}(x_i + x_{i+1})}{\sum_{i=1}^{n}x_i}$ Here is my trail. ...
1
vote
2answers
38 views

How to prove $\prod_{i=1}^{n}(x-4i+2)(x-4i+1)>\prod_{i=1}^{n}(x-4i+3)(x-4i)$ for all $x\in\mathbb{R}$?

I would like to prove that for $n\in\mathbb{N}$ we have $f_n(x):=\prod_{r=1}^{n}(x-4r+2)(x-4r+1)>\prod_{r=1}^{n}(x-4r+3)(x-4r)=:g_n(x)$ for all $x\in\mathbb{R}$ (actually it would suffice for $n$ ...
3
votes
2answers
134 views

Prove that $(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$

For the non-negative real numbers $a, b, c$ prove that $$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$$ What I did is applying Holder's inequality in LHS:$$(a^2+(\sqrt{2})^2)(b^2+(\sqrt{2})^2)(c^2+(\sqrt{2})...
9
votes
0answers
201 views

Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$

Let $a$, $b$ and $c$ be non-negative numbers such that $(a+b+c)^2(a^2+b^2+c^2)=27$. Prove that: $$\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$$ A big problem here around $(a,b,c)=(1.6185...,...
0
votes
1answer
21 views

Having trouble proving Inequality [duplicate]

I am having trouble proving this inequality: $2ab\leq a^2+b^2$ I can transpose the equation and change around signs. But I am not sure If I need to use k+1 here or just prove the inequality. In ...