Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

2
votes
0answers
32 views

A combinatorial argument to prove a general inequality

Recently, I have seen the following argument: $$f(x) < Dx + f\left(\frac{x}{2} \right)$$ $$\Rightarrow f(x) < Dx + f \left( \frac{x}{2} \right ) < Dx + \frac{Dx}{2} + f \left(\frac{x}{4} \...
2
votes
0answers
67 views

Top eigenvalues of diagonally shifted PSD matrix

Let $A$ be a (symmetric) positive semidefinite $n \times n$ matrix with diagonal $D$. Let $k \in \{1, \dots, n\}$ and $M = A - k\cdot D$. Prove that the sum of the top $n-k+1$ eigenvalues of $M$ is ...
1
vote
1answer
24 views

$a+OA\lt b+OB\lt c+OC$ when $a\lt b\lt c$ in a triangle

In the triangle $\triangle ABC$ of sides $a,b,c$ let $O$ be the incenter. If $a\lt b\lt c$ then (it is easy to prove that) $OC\lt OB\lt OA$. Prove that $$\max \{a+OA, b+OB, c+OC\}=c+OC$$
0
votes
1answer
38 views

How to prove $(1+\frac{1}{n})^n\geq\sum_{k=0}^{m}{(\frac{n-m}{n})^k\cdot \frac{1}{k!}}$ for all $m,n \in \mathbb{N}: n\geq m$?

How do I prove $\left(1+\frac{1}{n}\right)^n\geq \sum_{k=0}^{m}{\left(\frac{n-m}{n}\right)^k\cdot \frac{1}{k!}}$ for all $m,n \in \mathbb{N}: n\geq m$? What I observed: Let $q:=\frac{n-m}{n}$, then $...
0
votes
2answers
56 views

Let $x$, $y$, $z$ be positive numbers. Prove that $\sqrt{x(3x+y)}\ + \sqrt{y(3y+z)}\ + \sqrt{z(3z+x)}\ \leq\ 2(x+y+z) $

Let $x$, $y$, $z$ be positive numbers. Prove that $\sqrt{x(3x+y)}\ + \sqrt{y(3y+z)}\ + \sqrt{z(3z+x)}\ \leq\ 2(x+y+z) $ Professor says Cauchy-Schwarz theory should be used.
0
votes
0answers
31 views

Inequality with an indeterminate form

Let $x\in\big[-\frac{1}{b},\frac{1}{b}\big]^2$ and $y\in U(r)=\{z\in\mathbb{R}^2;\,\|z\|=r\}$ with $r>0$ and $b\geq \frac{\sqrt{2}}{r}$ fixed. I'm not able to prove the following inequality: $$\...
-1
votes
1answer
44 views

Bessels inequality proof

Studying a proof of Bessels' inequality. something confused me here is the proof: Lemma 1: Let $H$ be an inner product space if $\{ e_{1}, e_{2} ... , e_{n} \}$ is an orthonormal set then for all $h \...
21
votes
4answers
601 views
+150

prove this inequality with $x_{1}+x_{2}+\cdots+x_{n}=\pi$

Let $x_{i}>0$, ($i=1,2,\cdots,n$) and such that $$x_{1}+x_{2}+\cdots+x_{n}=\pi.$$ Show that $$ \dfrac{\sin{x_{1}}\sin{x_{2}}\cdots\sin{x_{n}}}{\sin{(x_{1}+x_{2})}\sin{(x_{2}+x_{3})}\cdots\sin{(x_{n}...
0
votes
1answer
56 views

show this inequality $(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$

let $x,y,z,w>0$,show that $$(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$$ it seem use AM-GM inequality to solve it,But I can't it,Thanks
2
votes
2answers
51 views

Calculate the maximum value of $\sum_{cyc}\frac{1}{\sqrt{a^2 + b^2}}$ where $a, b, c > 0$ and $abc = a + b + c + 2$.

$a$, $b$ and $c$ are positives such that $abc = a + b + c + 2$. Caculate the maximum value of $$\large \frac{1}{\sqrt{a^2 + b^2}} + \frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}}$$ This ...
5
votes
1answer
127 views

Taylor expansion of $\sqrt{n-k}$

I am reading a paper which casually assumes the asymptotic $\sqrt{n-k} \simeq \sqrt{n}-\frac{k}{2\sqrt{n}}$. This expression is what Wolfram calls Taylor expansion at infinity and from what I ...
1
vote
3answers
70 views

$a^ab^bc^c…k^k > \biggl(\frac{a+b+c+…+k}{n}\biggr)^{a+b+c…+k}$

If $a,b,c,...k > 0$, and $a,b,c,...,k$ are all unequal positive quantities, then prove that: $a^ab^bc^c....k^k > \biggl(\frac{a+b+c+....+k}{n}\biggr)^{a+b+c...+k}$ No other conditions are ...
0
votes
3answers
31 views

How to find the minimum value of algebraic expression on specific interval?

How I can solve the following problem: $$4x-x^2\in{\mathbb Z},$$$$ -2<x\leq 4 \Rightarrow \min(4x-x^2)=?$$
1
vote
1answer
77 views

least value of $\lfloor \frac{a+b}{c}\rfloor+\lfloor \frac{c+b}{a}\rfloor+\lfloor \frac{a+c}{b}\rfloor$ [duplicate]

If $a,b ,c>0$ . Then least value of $$\bigg\lfloor \frac{a+b}{c}\bigg\rfloor+\bigg\lfloor \frac{c+b}{a}\bigg\rfloor+\bigg\lfloor \frac{a+c}{b}\bigg\rfloor$$ Where $\lfloor x\rfloor$ is floor ...
2
votes
2answers
70 views

Solve the inequality $ 2x^{2} + x - 4 \sqrt{2x^{2} + x + 4} < 1 $

Solve the inequality $$ 2x^{2} + x - 4 \sqrt{2x^{2} + x + 4} < 1 $$ Attempt: I get that $2x^{2} + x + 4 > 0$ for all $x$. Let $y = 2x^{2} + x$, then the inequality becomes $$ y - 4 \sqrt{y + ...
0
votes
0answers
26 views

An interesting inequality involving three sets of reals

I found the following inequality on a different site and there wasn't a posted solution. I have run out of ideas and haven't gotten anywhere, so I hope someone can provide the "gotcha" to this problem:...
2
votes
2answers
99 views

Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?

Using the Cauchy-Schwarz Inequality, we have $$ \begin{align} 1 &=\left(\int_n^{n+1}1\,\mathrm{d}x\right)^2\\ &\le\left(\int_n^{n+1}x\,\mathrm{d}x\right)\left(\int_n^{n+1}\frac1x\,\mathrm{d}x\...
0
votes
2answers
54 views

Show that $(x+1)^2 (y+1)^3 (z+1)^4 \geq 4^4$ if $xyz = 1$

Show that $(x+1)^2 (y+1)^3 (z+1)^4 \geq 4^4$ if $xyz = 1$ and $x,y,z \in \mathbb{R}^+$. I have no idea whatsoever about how to even begin with it. Extraction would be too big and useless. AM-GM ...
3
votes
1answer
172 views

Showing that two curves do not intersect

I want to show that $$x+1 \neq (x^3(x+2))^{1/4} + \sqrt{x+1-\sqrt{x^2+2x}}$$ for any real $x>0$. There are two approaches I've taken: showing they are equal and arriving at a contradiction (but ...
4
votes
2answers
93 views

$ \sum_{1 \le i < j \le n} a_{i} a_{j} \ge n(n-1)/2 $ , prove that $a_{1} + … + a_{n} \ge n$ for $n \ge 2$ using AM-QM

Let $a_{1}, a_{2}, ...$ be a sequence of positive real numbers. Let the following relation holds: $$ a_{k+1} \ge \frac{k a_{k}}{a_{k}^{2} + (k-1)}, \:\: k \ge 1$$ Prove that $ S_{n} = a_{1} + a_{2} + ...
0
votes
0answers
58 views

Inequality in a triangle with tan

Again, a construction Given a triangle $ABC$ with all its angles smaller than $90^\circ$, using standard notations ($\tau$ is the semiperimetr, $r$ the radius of the incircle.), prove that : $\...
3
votes
2answers
228 views

Area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$

Find the area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$ (using definite integration). I cannot understand how to find this area. I have graphed the lines and found ...
3
votes
2answers
239 views

Inequality in a Polynomial

This is a problem I made. Let $x^3-ax^2+bx-c$ be a polynomial with real coefficients and three real roots, all greater than $1$. Prove, that $b+c \geq 3a-5$. Due to the discussion made (see the ...
1
vote
2answers
87 views

Calculate the maximum value of $\sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}$ where $x + 2y + 3z = 2$

$x$, $y$ and $z$ are positives such that $x + 2y + 3z = 2$. Calculate the maximum value of $$ \sqrt{\frac{3yz}{3yz + x}} + \sqrt{\frac{3zx}{3zx + 4y}} + \sqrt{\frac{xy}{xy + 3z}}$$ This problem is ...
0
votes
1answer
57 views

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2})\ge 4$

Prove $\sum_{\mathrm{cyc}} (\frac{a^2}{3bc}+\frac{a(b+c)}{b^2+c^2} )\ge 4$, preferably with SOS. My approach: $\sum_{\mathrm{cyc}} \frac{a^2(b^2+c^2)+3abc(b+c)}{3bc(b^2+c^2)} \ge 4$. Let $f(a,b,c)=\...
3
votes
1answer
90 views

elementary but irritating algebra problem

The question itself is very short and sweet, and requires no background. Find a solution to the following system: $$ \left\{\begin{array}{l} a,b,c,d,e,f>0\\\\ a+b+c+d+e+f=1 \\\\\displaystyle\frac{...
1
vote
0answers
79 views

Prove that $\frac{d}{2^m + 1} + \frac{m}{2}\frac{2^m}{2^m+1} \geq \frac{\log_2 d}{4}$

I am reading Tau and Vu's book Additive Combinatorics, and I came across a step in a proof that I am not able to verify. On page 252, in the last line of the proof of Theorem 6.4, it is stated that ...
2
votes
2answers
59 views

How to prove that for all non-negative $\forall x\in\mathbb R: x\ge \frac{\ln^2(1+x+\sqrt{2x})}{2}$?

I'm trying to prove that for all non-negative $\forall x\in\mathbb R:$ $$x\ge \frac{\ln^2(1+x+\sqrt{2x})}{2}.$$ You can think of it as a tighter inequality than the useful $x\ge \ln(1+x)$ or $e^x\ge ...
0
votes
1answer
27 views

What are Contra Harmonic Mean and Inverse Contra Harmonic Mean? [closed]

Are they related to Inequality? Like the $$ AM \times HM = GM^2 $$
1
vote
1answer
38 views

Tough substitution inequality

Prove that if $x, y, z >0$ and $xyz=x+y+z+2$, then $$ \sqrt{x}+\sqrt{y}+\sqrt{z} \leq \frac{3}{2}\sqrt{xyz}. $$ By the way, the first equation implies the existence of positive $a, b, c$ such ...
2
votes
2answers
63 views

Bound on $x$ for $x-\ln x\ge 1+\epsilon$

I'm looking for a function $x(\cdot)$ where the domain is $\mathbb{R}^+$, that satisfies the following: For all $\epsilon > 0$, the inequality $x-\ln x\ge 1+\epsilon$ is satisfied for all $x \geq ...
0
votes
1answer
24 views

Show that $\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|} $

We must show that $$\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|}. $$ Here is my attempt, however I wondered if there is a one-trick wonder ...
1
vote
3answers
35 views

Simplification of proof for $\exists c>0$ such that $|x+y|\leq c(1+\sqrt{x^2+y^2})$

Question: Consider function $f(x,y)=|x+y|$ for $x,y\in \mathbb{R}$. Show that $\exists c>0$ such that $|x+y|\leq c(1+\sqrt{x^2+y^2})$. My proof: i) if $|x|,|y|\leq 1$, then $|x+y|\leq 2\leq 2(1+\...
1
vote
1answer
30 views

Subset of $[n]$ without chain of leangth $5$ is of size $\leq \mathcal 2\Biggr(\binom{n}{(n-1)/2}+\binom{n}{(n-3)/2}\Biggl)$

Suppose $n$ is odd. Let $\mathcal P([n])$ denote the power set of $[n]$, that is, the $2^n$ subsets of $\{1,...,n\}$. We say that a family of sets $\mathcal F\subseteq \mathcal P([n])$ is nice if $\...
0
votes
3answers
50 views

Calculate the minimum value of $\frac{x^2 + 4}{y^2 + 1}$ where $1 \le y \le 2$ and $2y \le xy + 2$.

$x$ and $y$ are reals such that $1 \le y \le 2$ and $2y \le xy + 2$. Calculate the minimum value of $$\large \frac{x^2 + 4}{y^2 + 1}$$ This problem is adapted from a recent competition... It really ...
2
votes
1answer
50 views

Prove this generalization of Cauchy-Schwarz: $|(a.b)c+(b.c)a-(c.a)b|\le|a||b||c|$.

If you take $c$ unit perpendicular to $a,b$, Cauchy-Schwarz follows. I can prove the inequality in a roundabout way. First normalize to unit vectors. Then show that equality holds for 2-dimensional ...
0
votes
0answers
12 views

Question regarding Inequation with multivariable functions

In order to simplify the notation consider: $$ x=[x_1,x_2,...x_n] $$ Consider the following inequation: $$ s(x)(u(x)+A(x))<0 $$ My goal is to choose the function u(x) such that the inequation ...
8
votes
1answer
220 views

Prove that inequality Hardy inequality

Suppose $n$ is a positive integer, $2n$ reals $x_i, y_i (1\le i \le n) $ satisfy $$\sum_{i=1}^n x_i^2 = \sum_{i=1}^n y_i^2 = 1.$$ positive reals $0 < \lambda_1 \le \lambda_2 \le ... \le \...
0
votes
3answers
64 views

AM-GM Inequality Involving Squares and Proof

Prove: $$(a^2 + b^2 + c^2)/3 \geq ((a + b + c)/3)^2$$ OR $$(a^2 + b^2 + c^2)/3 \leq ((a + b + c)/3)^2$$ for all $a, b, c \geq 0.$ The problem wants me to find which inequality is correct and then ...
2
votes
3answers
44 views

Prove $(2a + b + c)(a + 2b + c)(a + b + 2c) ≥ 64abc$ using the AM-GM method and establishing when inequality holds

Prove: $$(2a + b + c)(a + 2b + c)(a + b + 2c) ≥ 64abc$$ for all $a, b, c ≥ 0$. Also, calculate/prove when equality holds. To prove this, the first thing that came to mind was the Arithmetic Mean -...
1
vote
1answer
64 views

Prove that $|\frac{a+b}{1+ab}| < 1$ given that $|a|<1$ and $|b|<1$

Can I please get a hint/solution for why, if true, $$\bigg{|}\frac{a+b}{1+ab}\bigg{|}<1$$ given that $|a|<1$ and $|b|<1$, where $a,b \in \mathbb{R}$. I've tried the usual things like ...
2
votes
0answers
34 views

Proof of Khintchine's inequality on $\mathbb{C}$?

I'm trying to understand this proof of Khintchine's inequality for the complex case ($a_n\in\mathbb{C}$). The author claims that the complex case follows from the real case by taking absolute values ...
1
vote
1answer
23 views

Taking assigning an unknown variable $\theta = 0.5$ to solve an inequality

In order to solve the following inequality for $n$, where $0 < \theta < 1$, they use a trick that I do not understand $n \geq (\frac{1.96}{0.03})^2 \theta(1 - \theta)$ Since $\theta$ is ...
2
votes
1answer
28 views

The volume of a cover of an interval is greater than the volume of the interval

Let $A \subset \mathbb{R}^n$ be a $n-$dimensional interval, i.e. $A = [a_1,b_1] \times [a_2,b_2] \times \cdots \times [a_n, b_n]$, where $[a_i, b_i]$ is an interval in $\mathbb{R}$, for every $i \in \{...
1
vote
1answer
56 views

Solve the following Algebraic Logarithmic inequalities [closed]

Solve the following logarithmic inequality with all log base $10$. $$\left(\frac12\right)^{(\log x^2)} + 2 > 3\times2^{(-\log(-x))}$$ I have done many logarithmic inequalities but ...
1
vote
1answer
55 views

$\sum_{k=1}^n \Bigl(\frac{a_1+\dots+a_k}k\Bigr)^p\le\frac{p}{p-1}\sum_{k=1}^n\Bigl(\frac{a_1+\dots+a_k}k\Bigr)^{p-1}a_k$ for nonnegative $(a_k)$

I am struggling with the following task: Let$$1<p<\infty , n\in \mathbb{N}, a_1\geqslant 0,\dots,a_n\geqslant 0.$$ Prove that $$\sum_{k=1}^n \Bigl(\frac{a_1+\dots+a_k}{k}\Bigr)^p \le \...
-1
votes
1answer
37 views

proof of inequality with absolute-value and function

Prove that if $|f'(x)| \leq M$, for all $x \in [a,b]$, then $$f(a)-M(b-a)\leq f(b) \leq f(a)+M(b-a)$$ What I tried so far: Proof. Let $x \in [a,b]$ Assume $|f'(x)|\leq M$ Show $f(a)-(Mb-Ma)\leq f(...
2
votes
1answer
52 views

Given $ a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$, prove $ S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2 $

Suppose a sequence of positive real numbers with $$ a_{k+1} \ge \frac{k a_{k}}{(a_{k}^{2} + k-1)}, \:\: k > 0$$ prove that $$ S_{n} = a_{1} + .. + a_{n} \ge n, \:\: n \ge 2 $$ Solution: I will ...
4
votes
1answer
106 views

Prob. 5, Sec. 6.2, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: How to show this function is strictly decreasing using derivative

Here is Prob. 5, Sec. 6.2, in the book Introduction To Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition: Let $a > b > 0$ and let $n \in \mathbb{N}$ satisfy $n \geq 2$. ...
2
votes
1answer
103 views

Nice olympiad inequality

Let's go for an olympiad inequality : let $a,b,c>0$ then we have : $$\sum_{cyc}\frac{ab}{a+b}\geq \frac{3\sqrt{3}}{2}\sqrt{\frac{abc}{a+b+c}}$$ My proof : $$\sum_{cyc}\frac{ab}{a+b}=\frac{(a^...