Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

0
votes
0answers
20 views

Proving Hardy's inequality for $L^p(\mathbb R^d)$ from the one-dimensional one

I want to prove the following inequality: Let $d\in \mathbb N$ and $1\leq p < \infty, p\neq d$. Then for all $f\in C_c^\infty(\mathbb R^d \setminus \{0 \})$, $$\int_{\mathbb R^d} \frac{\lvert ...
0
votes
1answer
40 views

Using analysis in solving exponential inequalities

Solve the inequality $$ 35^x+20^x+15^x\le28^x+21^x+25^x, $$ for $x\in\mathbb{R}$. I tried to find the solution by hand and to prove that they are the only ones. I saw that $x=0$ verifies the ...
0
votes
1answer
23 views

Help in deducing an inequality theorem

My book says the following: If for every $\eta<0$, there is a $\Delta x$ such that: $\left| \dfrac{f(x+\Delta x)-f(x)}{\Delta x} - g(x) \right| < \eta$ and $\lim\limits_{\...
0
votes
1answer
15 views

Inequality for standard normal distribution with composite function of pdf and inverse cdf

I am reading one paper https://arxiv.org/abs/1207.7209 In proposition 4.1 the author mentioned a fact $$p \sqrt{k_1 \log (1/p)} \leq \phi \circ \Phi^{-1}(p)$$ where $k_1 = 1/2, p \in (0, 1/2], \phi, \...
0
votes
1answer
48 views

On the Proof of the Uniqueness/Existence Theorem

I am trying to show that $$|\underline{x}(t)-\underline{y}(t)|\leq \left|\underline{x}(t_0)-\underline{y}(t_0)\right|+\int_{t_0}^{t}\left|\underline{f}(\underline{x},s)-\underline{f}(\underline{y},s)\...
3
votes
1answer
60 views

Parametric exponential inequality

Find the values of $m$ s.t. $$ \left(\frac{9}{25}\right)^x-m\left(\frac{3}{5} \right)^x+1>0, $$ for all $x<0$. My attempt is the following: let $y=(3/5)^x>1$ and the inequality transforms as ...
1
vote
1answer
37 views

Hermite - Hadamard inequality [on hold]

What is the geometric meaning of Hermite - Hadamard inequality? I.e. $$f(\frac{a+b}{2})\leq \frac{1}{b-a}\int_{a}^{b} f(x)dx \leq \frac{f(a)+f(b)}{2}.$$ Thank you for your help.
1
vote
4answers
58 views

How to prove this inequality for $a,b,c>0$?

How to prove the inequality for $a,b,c>0$ : $$\frac{2a-b-c}{2(b+c)^2}+\frac{2b-a-c}{2(a+c)^2}+\frac{2c-b-a}{2(b+a)^2}\geq 0$$ ?
0
votes
2answers
37 views

Inequality with absolute value function inside absolute value - $||x-2|-3|<4.$ [on hold]

Help me solving this. $||x-2|-3|<4.$ Find all values of x satisfying this.
0
votes
0answers
69 views

Prove that $\sum_{cyc} \sqrt{\frac{a}{8b+c}}\ge 1$

Prove that $$\sqrt{\frac{a}{8b+c}}+\sqrt{\frac{b}{8c+a}}+\sqrt{\frac{c}{8a+b}}\ge 1$$ where $a,b,c>0$ Here is my idea. Please check it for me. Thanks a lot. Let $$x=\sqrt{\frac{9a}{8b+c}};y=\sqrt{...
-8
votes
0answers
71 views

A short problem in geometry 😊 [on hold]

Given an acute triangle $\triangle ABC$. The altitude $AH$ is equal to the median $BM$. Prove: $∠B<60^∘$
0
votes
0answers
25 views

Is the $2$-norm a lower bound for the dual norm?

From Convex Optimization by Boyd & Vandengergh: Let $||\cdot\|$ be any norm. Then $\|x\|_* \ge \gamma \|x\|_2$ for some $\gamma \in (0,1] $. I start by assuming that $\gamma \gt 1$. Then $$\...
1
vote
1answer
36 views

Hard inequality :$\sum_{cyc}\frac{a}{\sqrt[3]{a+b}}\leq a+b^{\frac{2}{3}}+c$

I'm interested by the following problem : Let $a,b,c$ be positive real numbers such that $a+b+c=1$ and $a\geq b \geq c$ then we have : $$\sum_{cyc}\frac{a}{\sqrt[3]{a+b}}\leq 1-b+b^{\frac{2}{3}}=...
0
votes
0answers
11 views

Proving an inequality for two stopping times

Let $T$ be a stopping time with respect to the filtration $\{\mathcal F_s\}$ and suppose the following two properties hold: (i) for each $t>0$, $P(0<T\le t) > 0$ (ii) for $A \in \...
0
votes
0answers
27 views

Lower bounding a term of the form $(x^k -1)^s$

I have a term of the form $(x^k -1)^s$ and I would like to find a nice looking lower bound for it without any additive terms inside of the parentheses. The reason to that is because I want something ...
-3
votes
1answer
35 views

find the maximum of the value

In $\Delta ABC$ find the maximum of the value $$\sin{A}+\dfrac{\sqrt{10}}{2}\sin{B}-\sqrt{5}\sin{C}$$
0
votes
1answer
47 views

show this inequality $\sum_\mathrm{cyc}\sqrt{1-ab}\ge 2\sqrt{2}$ [duplicate]

let $a,b,c\ge 0$,and such $a+b+c=1$,show that $$\sqrt{1-ab}+\sqrt{1-bc}+\sqrt{1-ac}\ge2\sqrt{2}$$ maybe can use C-S to solve it. My attempt is $$\sum_\mathrm{cyc}\sqrt{a^2+b^2+c^2+ab+2bc+2ac}\ge 2\...
0
votes
1answer
27 views

Show that if $\sigma = \frac {z(1 - \pi)}{x} < 0$ then $- \pi \sigma = \frac {- \pi z (1 - \pi)}{x} > (1 - \pi)$

I'm trying to understand a published economic paper and can't figure out the following steps: $\pi$ and $1-\pi$ are probabilities, z and x are two long terms that I have summarised for simplification,...
1
vote
4answers
46 views

Proving $a = b = c$ under certain conditions

For all real a, b, c, prove a = b = c if $$\frac{a^2+b^2+c^2}{3} = (\frac{a+b+c}{3})^2 $$ The first idea that came to mind would be to prove this inequality by contradiction. However, I am unsure ...
3
votes
2answers
139 views

If $|z^2 + 2019| < 2019$ prove that $|z + \sqrt{2019}| > 31$

I got this from today's test. Let $z\in \mathbb{C}$. If $|z^2 + 2019| < 2019$ prove that $|z + \sqrt{2019}| > 31$ I tried triangle inequality, but doesn't work. I also tried using ...
0
votes
0answers
25 views

Inequality with exponents

Assume $0 \le x, z \le \frac{1}{2}$ and $t \in (0, \epsilon]$, where $\epsilon > 0$ is a small, fixed constant. Let's consider the following inequality for $1< C < 2$ $$\big(1 - e^{-\frac{x}{...
-4
votes
4answers
67 views

$|a^2-b^2|<(ab)^2$ for $a,b\ge 1$ [on hold]

It seems like $|a^2-b^2|<(ab)^2$ for $a,b\ge 1$. I guess this is common knowledge, but I ask you for a proof.
-1
votes
3answers
56 views

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.

Given that $a$ and $b$ are positives such that $\sqrt{a + 2b} = 2 + \sqrt{\dfrac{b}{3}}$, calculate the minimum value of $$\large \frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$$ I have provided ...
0
votes
1answer
25 views

Show $\lim \left| \left( 1-(1-s)\frac{\lambda_n}{n}\right)^n-\left(1-(1-s)\frac{\lambda}{n}\right)^n\right|\le\lim|1-s ||\lambda_n-\lambda |$

As application of convergence theorem in our probability lecture we want to show the generating function of sequence of binomially distributed random variables converges to the generating function of ...
4
votes
1answer
56 views

Maximize an unweighted sum given a weighted sum

My problem boils down to the following: Given real numbers $c_i \geq 0$ $$\begin{array}{ll} \text{maximize} & f(x_1)+f(x_2)+\cdots+f(x_n)\\ \text{subject to} & c_1x_1+c_2x_2+\cdots +...
2
votes
3answers
51 views

$|\cos(z)|\leq e^{|z|}$

Problem is proving inequality for all $z \in \mathbb{C}$ $$|\cos(z)|\leq e^{|z|}$$ My attempt: We know $$|\cos(z)|=|\frac{e^{iz}+e^{-iz}}{2}|\leq |\frac{e^{iz}}{2}|+ |\frac{e^{-iz}}{2}|$$ $$|e^{iz}|=...
3
votes
3answers
74 views

Prove $x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$

Prove $$x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$$ I have a solution, and I hope to see more nicer ones, thanks for your interest! We have $$(\,x+ y+ z\,)^{\,2}+ (\,-\...
-2
votes
0answers
44 views

Why is there a Semi-Colon in this problem, and what does it mean? [on hold]

The problem that I am so concerned about, and that I don't understand at all is this- 4(x+3)>20;2 I don't understand what the Semi-Colon is for, and what it does. Please help me.
0
votes
1answer
26 views

Trouble with Holder Inequality Gymnastics: $\left(\int fg^{q}\right)^{r-p}\leq\left(\int fg^p\right)^{r-q}\left(\int fg^r\right)^{q-p}.$

$\textbf{The Problem:}$ Let $f$ and $g$ be nonnegative and measurable and $0<p<q<r<\infty$. Prove that $$\left(\int fg^{q}\right)^{r-p}\leq\left(\int fg^p\right)^{r-q}\left(\int fg^r\...
1
vote
1answer
34 views

Prove that $3 \le \sum_{cyc}a\sqrt{b^3 + 1} \le \sum_{cyc}ab^2 + 3$ where $a, b, c \ge 0$ and $a + b + c = 3$.

$a$, $b$ and $c$ are non-negatives such that $a + b + c = 3$. Prove that $$\large 3 \le a\sqrt{b^3 + 1} + b\sqrt{c^3 + 1} + c\sqrt{a^3 + 1} \le \frac{ab^2 + bc^2 + ca^2}{2} + 3$$ This problem is ...
1
vote
1answer
65 views

simple inequality problem not using differentiation

consider these positive reals x,y such that $\frac{1}{x}+\frac{8}{y}=1$ What is minimum of $$x^2+y^2$$ I can solve this using differentiaion or lagrange multipliers but can someone suggest more ...
0
votes
0answers
24 views

New bounds for convex function of 2 variables

It's related to my post : New bound for Am-Gm of 2 variables In fact I have discovered (maybe it's already knew) a new formula for convex function this is the following : Let $f(x)$ be a twice ...
3
votes
1answer
104 views

if $a^2+b^3+c^4+2019\ge k(a+b+c)$ find the maximum of the $k$

Let $k$ be postive integers,and for any postive real number $a,b,c$ such $$a^2+b^3+c^4+2019\ge k(a+b+c)$$ find the maximum of $k$. It seem use AM-GM inequality to solve it, but I can't find ...
0
votes
0answers
54 views

Show equality of an integral

Let $0\le K_N < N_{\gamma}\sim N$ and $0<n<N_{\gamma}$. We want to show as $N\to \infty$ we have $$\int_{0}^{N_{\gamma}} \frac{1}{y}(1-(1-y/N_{\gamma})^{N_{\gamma}})(1-y/N_{\gamma})^{n-1}dy \...
5
votes
3answers
104 views

Set of positive integers

Let $A$ be a set of positive integers with the following properties: a) If $n \in A$ then $n \leq 2018$ b) If $S$ is a subset of $A$ with $|S|=3$ then there are two elements $m,n \in S$ ...
4
votes
1answer
77 views

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\displaystyle \sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$. My attempt was to use firstly AM-GM in the denominator, like $a^3+5 \geq 3a+3$ and the ...
2
votes
0answers
27 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
59 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
23 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
54 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
30 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
39 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 \...
6
votes
2answers
196 views

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
45 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
48 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
94 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
69 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
70 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 ...