Questions on proving, manipulating and applying inequalities.

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2
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0answers
28 views

Partitioning a set of integers (with Alice and Bob)

Let $ d_1,\ldots,d_n \in \mathbb{N}_{\ge 2} $ (not necessarily distinct) be given. Define $ D:=\operatorname{lcm}(d_1,\ldots,d_n) $ and $ d:=\sum_{i=1}^n d_i $. (1) Alice claims that whenever $ \...
1
vote
2answers
51 views

What's the mistake on my answer for this inequality $ \frac{\left(x+1\right)}{\sqrt{x^2+1}}>\frac{\left(x+2\right)}{\sqrt{x^2+4}} $

Good evening to everyone! I have the following inequality: $$ \frac{\left(x+1\right)}{\sqrt{x^2+1}}>\frac{\left(x+2\right)}{\sqrt{x^2+4}} $$. I don't know what's wrong with my answer: $$ \frac{\...
2
votes
1answer
34 views

The first step in the proof of the Pólya-Vinogradov Inequality.

The well-known Pólya-Vinogradov Inequality states: $\forall m, n \in \mathbb{N}: \displaystyle \left|{\sum_{k \mathop = m}^{m+n} \left({\frac k p}\right)}\right| < \sqrt p \ \ln p$,. where $(k/p)$...
-4
votes
0answers
30 views

proving: Let x, y, p, q be positive numbers with 1/p + 1/q = 1. Prove that xy ≤ x^p/ p + y^q/q [on hold]

4) Let $x, y, p, q$ be positive numbers with $ 1/p + 1/q = 1$. Prove that $$xy ≤ x^p/ p + y^q/q $$
2
votes
3answers
52 views

Concept of roots in Quadratic Equation

$a$ , $b$, $c$ are real numbers where a is not equal to zero and the quadratic equation \begin{align} ax^2 + bx +c =0 \end{align} has no real roots then prove that $c(a+ b+ c)>0$ and $a(a+ ...
4
votes
1answer
61 views

How prove this inequality

Let $a,b,c>0$ and such $a+b+c=1$ show that $$\dfrac{1}{1-a}+\dfrac{1}{1-b}+\dfrac{1}{1-c}\ge \dfrac{1}{ab+bc+ac}+\dfrac{1}{2(a^2+b^2+c^2)}$$ Let $p=a+b+c=1,ab+bc+ac=q,abc=r$ $$\Longleftrightarrow -...
1
vote
3answers
56 views

Inequality with square root $x+\sqrt{x^2-10x+9}\ge \sqrt{x+\sqrt{x^2-10x+9}}$

Good morning to everyone! The inequality is the following:$$ x+\sqrt{x^2-10x+9}\ge \sqrt{x+\sqrt{x^2-10x+9}} $$. I don't know how to solve it. Here's what I tried: $$x+\sqrt{x^2-10x+9}\ge \sqrt{x+\...
0
votes
2answers
47 views

Where am I going wrong with this inequality?

Good evening to everyone! I got this inequality: $$\frac{x-1}{x-2}<\frac{x-1}{x}.$$ If I try to solve this, it gives me $$ \frac{x-1}{x-2}<\frac{x-1}{x} \Rightarrow \frac{x-1}{x-2}-\frac{x-1}{x}...
3
votes
2answers
73 views

The number of positive integer solutions to the equation $x_1+x_2+…+x_n=n^2.$

I'm working on this problem. To solve it I need this lemma: Let $n\ge2, n\in \mathbb N$. Let $X$ be the number of solutions in positive integers to the equation $x_1+x_2+...+x_n=n^2$. Let $Y$ be ...
-1
votes
2answers
22 views

Solve an easy inequality

How can I find the solution of the following inequality analytically in terms of $x$? $$i_1x^3+i_2x^2+i_3x \ge 0$$ where $i_k$ is a constant value.
2
votes
1answer
47 views

Prove inequality $\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+d^3}{2}}+\sqrt[3]{\frac{d^3+a^3}{2}} \le 2(a+b+c+d)-4$

Let $a,b,c,d$ positive real numbers, such that $$\frac1a+\frac1b+\frac1c+\frac1d=4.$$ Prove inequality $$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+d^3}{2}}+\sqrt[3]...
3
votes
1answer
37 views

Is there an $\alpha\in\mathbb{R}^m$, such that $\alpha_i > 0$ and $A\alpha\in S$?

$A$ is a real $n\times m$ matrix and set $S\subseteq \mathbb{R}^n$ is defined as $$S = \{(x_1,\dots, x_n)\in \mathbb{R}^n\mid \forall(i,j)\in I.\; x_i< x_j\}\text{,}$$ where $I$ is a possibly empty ...
3
votes
2answers
79 views

Find $\lim\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}$ [on hold]

The question arise in connection with this problem Prove that $$\lim_{n\rightarrow \infty}\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\...
2
votes
2answers
39 views

Inequality with $\arctan$

I try to show that $x \cdot \arctan\left( \frac{1}{x^2} \right)$ is monotonically decreasing, but I can't solve this inequality with $\arctan$. Can somebody show me how to do this? $$ x \in [1, \...
2
votes
1answer
37 views

For which values of $a$ does $d\ge ac\ln c\implies d\ge c\ln d$?

For which values of $a>0$ is it true that for all $c,d>0$, $\hspace{.2 in}d\ge ac\ln c\implies d\ge c\ln d$? I believe that this is true for $a\ge2$, (see Showing if $n \ge 2c\log(c)$ then $n\...
2
votes
1answer
36 views

Greatest integer function inequality solution

If $\lfloor x+\lfloor x\rfloor\rfloor \le 2$ then what are all the possible values of $x$? Please tell how to proceed and tell the solution by graph method of possible. Explain how to sketch the graph ...
5
votes
1answer
36 views

Is it true that $\left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c$?

Let $a,b,c\in\mathbb{R}_{>0}$. Is it true that: $$ \left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c $$ I remarked that the inequality is (a bit weirdly) homogeneous, but couldn't use it. ...
1
vote
3answers
91 views

Proof of AM-GM inequality for $n=3$: $\frac{a+b+c}{3}\geq\sqrt[3]{abc}$ [duplicate]

Sorry for bad formatting, I couldn't mark the 3rd root on the right hand side... I've figured this out into the point where (and yeah, the problem is to prove that this applies to all non-negative ...
3
votes
0answers
35 views

On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers

(Note: This question has been cross-posted from MO.) Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$. A number $M$ is called almost perfect if $\sigma(M) = 2M ...
3
votes
0answers
28 views

Can you please comment on and check these couple of induction proofs?

So the following statements need to be proved: 1) $(1+a_1)(1+a_2)\cdots(1+a_n)>1+a_1+a_2+\cdots+a_n$ for $a_i>0,(i=1,2,\ldots,n)$ and $n\ge2$ 2) $(1-a_1)(1-a_2)\cdots(1-a_n)<1-(a_1+a_2+\...
0
votes
0answers
14 views

How to find the maximal length of a system?

Let P be the set of $(a,b,c)^t \in \mathbb{R}$ which satisfies the following inequalities: $-2a+b+c \leq 4$ $a-2b + c \leq 1$ $2a + 2b-c \leq 5$ where $a \geq 1 $, $b \geq 2$, and $c \geq 3 $. ...
3
votes
3answers
36 views

inequality proof

I have come across a problem: Let $a,b$ and $c$ be real numbers where $a > b$. Prove that if $ac \leq bc$, then $c \leq 0$. I tried using the Indirect proof If $a > b$, and $c > 0$, ...
-1
votes
1answer
34 views

Solution of the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$

I just want to find a way to prove that the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$ is true because I need it for a prove. Thanks for your help!
2
votes
1answer
31 views

Upper bound for $\sum_{i=1}^{N}{x_i^{\beta_i}}$

$\{x_i\}_{i=1}^{N}$ a sequence of positive real numbers and $\{\beta_i\}_{i=1}^{N} $ are real numbers such that $\underset{1 \leq i \leq N}\min{\beta_i}$ > 1. Is it possible to find an upper bound of ...
0
votes
3answers
56 views

Solve $3^{2x}-3^x\geq2$

How to solve: $$3^{2x}-3^x\geq2$$ I tried with $y=3^x$ and solved as equation: $y^2-y-2 \geq 0$ and I get: $y<2$ $y>-1$ How should I proceed?
2
votes
2answers
129 views

Let $a, b, c>0$, such that $a+b+c=1$, prove that $\frac{a}{(b+c)^2}+\frac{b}{(a+c)^2}+\frac{c}{(a+b)^2}\ge\frac{9}{4}$ [on hold]

Let $a, b, c>0$, such that $a+b+c=1$, prove that: $$\frac{a}{(b+c)^2}+\frac{b}{(a+c)^2}+\frac{c}{(a+b)^2}\ge\frac{9}{4}$$
1
vote
3answers
62 views

How to solve $x<\frac{1}{x+2}$

Need some help with: $$x<\frac{1}{x+2}$$ This is what I have done: $$Domain: x\neq-2$$ $$x(x+2)<1$$ $$x^2+2x-1<0$$ $$x_{1,2} = \frac{-2\pm\sqrt{4+4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{...
-1
votes
0answers
11 views

Convex subset and linear equalities

Let S denote the set of $(a,b,c)$ $\in$ ${\mathbb{R^3}}$ which satisfies the following equalities: $-2a+b+c \leq 4 $ $a-2b+c \leq 1 $ $2a+2b-c \leq 5 $ $ a \geq 1 $ $ b \geq 2 $ $ c \geq 3 $ ...
-3
votes
0answers
41 views

Lets a, b, c>0 such that a+b+c=6, prove that: [on hold]

Let a, b, c>0 such that a+b+c=6, prove that: $$\sum_{cyc} \frac{a^7+b^7}{a^5+b^5}\ge12$$
4
votes
2answers
63 views

If $x,y,z>0\;,$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$

If $x,y,z>0\;,$ and $xyz=1$ Then find the minimum value of $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$ $\bf{My\; Try::}$Using Titu's Lemma $$\frac{x^2}{y+z}+\frac{y^2}{z+x}+\...
3
votes
0answers
39 views

Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$ This is the solution provided by my textbook: Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ...
0
votes
3answers
42 views

Values of x that satisfy this inequality

$||x-2|-3|>1$. I have made some cases but still the complete values don't come,plus I don't have any idea of how to sketch the graph for the lefy hand side of the inequality.
0
votes
1answer
14 views

The validity of normalization in homogeneous inequalities?

I'm going through a book on inequalities right now, and the author describes normalization with the following example. Prove that $a^2 + b^2 + c^2 \ge ab + bc + ca$ Of course the fundamental ...
0
votes
0answers
41 views

Probabilistic Modeling Parameters Request

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
1
vote
1answer
39 views

Lagrange multipliers: when is local extremum a global extremum?

Consider the following Olympiad problem from the IMO shortlist: Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that: $36 \leq 4 \left(a^3+b^3+c^3+d^...
3
votes
5answers
234 views

Sum of real powers: $\sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$

Let $\{x_i\}_{i=1}^{N}$ be positive real numbers and $\beta \in \mathbb{R}$. Can we say that: $$ \sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$$ I know that this holds if $...
0
votes
2answers
78 views

Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.

So I have to prove $$ \frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}.$$ I rearranged it $$ a^2bc + ab^2c + abc^2 \leq b^2c^2 + a^2c^2 + a^2b^2 .$$ My idea from there is ...
1
vote
1answer
29 views

Prove that $a^2 pq + b^2 qr + c^2 rp \leq $ given a,b and c are sides of triangle and p+q+r=0

The question is asking to prove that $a^2 pq + b^2 qr + c^2 rp \leq 0 $ given that $a,b$ and $c$ are the sides of a triangle and that $p+q+r=0$. I have tried AM GM as well as countless pages of ...
1
vote
1answer
30 views

Proving an inequality about a set of combinations.

Suppose $A$ is a set of $r$ combinations of an $n$ set, with $\alpha \cap \beta \neq \phi$, whenever $\alpha, \beta \in A$. Show that $$|A| \leq \binom{n-1}{r-1}$$ if $r \leq \frac n2$. What does ...
1
vote
1answer
99 views

Inequality $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ [duplicate]

If $\frac {1}{p}+\frac {1}{q}=1$ and $a,b \ge 0$ , then prove $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ . I can't find a simple and short way to prove this. Any hint would work. Thanks in advance!
0
votes
1answer
21 views

Integral Inequality with Monotonic Function

Problem: For continuous, either both increasing or both decreasing functions $f, g$ on $[a, b]$, suppose that $p(x)$ is continuous and positive. Prove that $$\int_a^bp(x)f(x)dx \int_a^bp(x)g(x)...
0
votes
1answer
20 views

Why is $|\psi_n-f|^p \leq2^p |f|^p$ when $|\psi_n|\leq |f|$?

Why is $|\psi_n-f|^p\leq 2^p |f|^p$ when $|\psi_n|\leq|f|$? Is it true that $|a+b|^p\leq 2^p (|a|^p+|b|^p)$?
1
vote
2answers
33 views

Inequality involving a convex function

I am stuck, showing the following inequality in an easy way (using only inequalities or something): Let $x\in [-a,a]$ for some $a>0$ and $p\in (1,2)$. I want to show that there then exists a ...
1
vote
3answers
112 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
3
votes
1answer
49 views

Does this inequality involving inverse tangent (arctan) hold?

I am wondering if the following statement is true for $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $x,y\in\mathbb{R}$: $$\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)\leq\...
8
votes
2answers
125 views

Prove inequality $\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$

For any $n\ge2, n \in \mathbb N$ prove that $$\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$$ My work so far: 1) $$\...
3
votes
3answers
46 views

Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$

Prove that $\frac{1}{4\cdot 1976^3}-\frac{1}{16\cdot 1976^7}>10^{-19.76}$ without using a calculator. I rearraged to get $4 \cdot 1976^4-1 > 10^{-19.76} \cdot 16 \cdot 1976^7$ and so we have ...
0
votes
0answers
16 views

Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \begin{equation} \vert \vert T \vert \...
1
vote
3answers
49 views

Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$

Prove that $\sqrt{n+1}>\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{8n\sqrt{n}}$ if $n>0$. I didn't see an easy way of proving this without doing a lot of algebra and rearranging. Is there an easier ...
2
votes
2answers
43 views

Inequality involving ArcTan

How to prove that for $x\in[0, +\infty]$ the following inequality is true: $$\arctan x\geq\frac{3 x}{1+2\sqrt{1+x^2}}?$$ I don't have idea from where to start, so any hint is welcome. Thanks in ...