Questions on proving, manipulating and applying inequalities.

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-1
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0answers
18 views

Combinatorical problem

$k$ is a natural constant.Determine $x,y,z$ knowing that $\binom{z+k}{x+y} + \binom{z}{x} \le k$ and $2x+y \le z$.
5
votes
1answer
106 views

Israel tst 2011 geometrical inequality

Inside an equilateral triangle of area $S$ lies a point, whose distances to the vertices are $x,y, z$. Prove that $xy + yz + zx \geq \frac{4}{\sqrt{3}} S$ I haven't got any idea yet. But I guess ...
-1
votes
0answers
14 views

Inequality with $log$ and $d^x$

Please help me to solve this inequality $Log[d] <\frac{(-1 + d^a) (d^a - d^b) b} {d^{a}(-1 + d^b) a (a - b)}$ with $0<\delta<1$, $a\geqq1$, $b>a$ and thus $b>1$. $a$ and $b$ are ...
0
votes
0answers
11 views

Can these bounds, for the deficiency $D(x)=2x-\sigma(x)$ of a deficient number $x>1$, be improved?

Let $\sigma=\sigma_{1}$ denote the classical sum-of-divisors function. Denote the deficiency of the deficient number $x>1$ by $D(x)=2x-\sigma(x)$. Since $x>1$ is deficient, we have $D(x) \geq ...
3
votes
2answers
55 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
48 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
1answer
45 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 $...
0
votes
2answers
39 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
vote
1answer
31 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 ...
1
vote
2answers
51 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
50 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
36 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\...
-1
votes
1answer
63 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 ...
2
votes
2answers
60 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 ...
2
votes
3answers
86 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
2answers
65 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
26 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-...
3
votes
0answers
43 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 $\...
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 ...
0
votes
2answers
60 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}{...
4
votes
7answers
94 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, ...
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 ...
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
3answers
61 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 ...
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
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 $x\in(-3,-2)\cup (-1,\infty)$ ...
0
votes
0answers
28 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
2
votes
2answers
80 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 $z=r\cdot e^{i\theta}$. ...
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 ( ...
0
votes
1answer
35 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 $d$ ...
5
votes
1answer
69 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-...
1
vote
0answers
14 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 ...
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
2answers
39 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 $n$ ...
1
vote
1answer
21 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. ...
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 ...
3
votes
1answer
70 views

Upper bound of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$

Let $n\geq 3$ be a positive integer and let $x_i$'s be non-negative real numbers with $x_1+x_2+\dots+x_n=1$. What is the maximum of $x_1x_2x_3+x_2x_3x_4+\dots+x_{n}x_1x_2$? If the sum were symmetric ...
-10
votes
2answers
50 views

Which quantity is greater? [on hold]

A. -0.1 or B. -0.10101010101 This is actually an evaluation of an expression when plugging certain values in the GRE. I plugged in the value -0.1 and arrived at my doubt
0
votes
1answer
42 views

a inequality similar to geometric means

Let $a$, $b$ be two positive constants. We sure have $$ a^2+b^2\geq 2ab $$ My question: would it be possible to have an inequality like $$ a^2+b^2\geq Ca^{2+\epsilon}b^{1-\eta} $$ where $C$, $\epsilon$...
1
vote
1answer
30 views

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
1
vote
2answers
71 views

Dominance between two functions

Let two functions $f(z)$ and $g(z)$ with $z\in[0,c]$ with $c$ a constant such that $c<b$. I'd like to check whether $f(z)-g(z)>0$. I've tried to set $f(z)$ to its minimal value and $g(z)$ to its ...
5
votes
3answers
139 views

Prove the Inequality $\frac{1}{1-x}-\frac{x(3-x)(2-x)(13x^4-50x^3+89x^2-84x+36)}{4(1-x)(2x(1-x))^2}<1$

Can anyone suggest any hints to prove the following inequality: $$\frac{1}{1-x} - \frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} < 1,$$ for all $x \in (0,1)$?
1
vote
3answers
41 views

Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
-1
votes
0answers
32 views

Solve Equation with max integer [closed]

Solve please $\dfrac{\left[\sqrt{x-[x ]}\right]}{(x+3)(x+4)}\ \geq0$ edit
0
votes
1answer
32 views

Solving inequality of two independent exponentially distributed RVs

I have huge problems solving following excersice: There are two molecules. The decay of the molecules is exponentially distributed with $\alpha_1 = 1$ (for molecule 1) and $\alpha_2 = 2$ (for ...
0
votes
1answer
38 views

Using CS inequality to find maximum of a function

I am trying to us Cauchy-Schwarz inequality to find the maximum of: $$|(a^2)(b^2)(a-b)+(b^2)(c^2)(b-c)+(c^2)(a^2)(c-a)|$$ Where $a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. ...
0
votes
1answer
69 views

Find maximum value of a function [closed]

$a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. I need help finding the maximum value of: $$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big|$$ To be honest, I don't know where ...
0
votes
1answer
44 views

Comparing the roots of two increasing functions

For any $0 \leq x \leq y \leq 1$, define $f(y;x):=\frac{y^2}{2}-\frac{2 y^3}{3}+\frac{y^4}{4} - \frac{x^2}{2} + \frac{x^3}{3}$ and $g(y;x):=\frac{y^2}{3}-\frac{2 y^3}{4}+\frac{y^4}{5} - \frac{x^2}{3} +...