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Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

2
votes
1answer
33 views

Techniques for upper bounding square of sum of square roots?

Say I have $x_1,...,x_n$ such that $0 \leq x_i \leq 1$ for all $i \in n$ and $\sum_{i=1}^n x_i = 1$. Let $\epsilon > 0$ be some small number. Let $y_1,...,y_n$ be $\epsilon$-close approximations of ...
0
votes
3answers
28 views

Prove $\forall k\geq 4,\log(1+x_k)-x_k\leq {-1\over6k}$ where $x_k={(-1)^k\over\sqrt k}$

The question: Prove $\forall k\geq 4,\log(1+x_k)-x_k\leq {-1\over6k}$ where $x_k={(-1)^k\over\sqrt k}$. The above inequality holds iff $$\begin{align} &\log(1+x_k)\leq x_k-{1\over 6k}\\ &\...
1
vote
4answers
25 views

Show that $(x-5)^2+\frac{9(4-x)}{4x}>1$ on the interval $(0,4)$

Consider the function $f$ given by $$f(x)=(x-5)^2+\frac{9(4-x)}{4x}, $$ for $x\in (0,4)$. I'm asked to show that $f(x)>1$ on the interval $(0,4)$. I've started by recognising that $(x-5)^2>0$ ...
0
votes
1answer
38 views

If PQRS are four sides of a quadrilateral then find the minimum value of $(p^2+ q^2 + r^2) / s^2$ with logic

If PQRS are four sides of a quadrilateral then find the minimum value of $(p^2+ q^2 + r^2) / s^2$ with logic Please help me with this. I need this immediately.
0
votes
0answers
19 views

Analysis of complex transcendental inequalities

There is an inequality of the following form: $(1/(w^2T^2))$ * $(w^2/(w^2T^2+1))$ * $(k^2/(w^2T^2+1)^2)$ Suppose it is necessary that this inequality be strictly < $m$. Is it possible to determine ...
1
vote
1answer
34 views

Inequalities for standardized central moments of probability distributions

It's known that the standardized central even moments of any probability distribution with a density symmetric around the mean form a non-decreasing series, the lower bound (when all are equal to 1) ...
0
votes
3answers
50 views

Find the maximum value of $x^3 + y^3 + z^3$ where $x, y, z \in [0, 2]$ and $x + y + z = 3$. [duplicate]

Given that $x, y, z \in [0, 2]$ and $x + y + z = 3$. Calculate the maximum value of $$\large x^3 + y^3 + z^3$$ I'm done. Should you have different solutions, you could post them down below. Having a ...
0
votes
2answers
99 views

Prove $ \sum_{cyc}a^3- \sum_{cyc}a^2b \geqq 0$ with $k= 0$

Prove with $a+ b,\,b+ c,\,c+ a\geqq 0$ $$k= constant= 0$$ is the only non-negative $k$ such that $$\left \{ \sum\limits_{cyc}\,a^{\,3}- \sum\limits_{cyc}\,a^{\,2}b \right \}\geqq k(\,a- b\,)(\,a- c\,)(...
2
votes
5answers
115 views

Rearrangement inequality and minimal value of $\frac{\sin^3x}{\cos x} +\frac{\cos^3x}{\sin x}$

For $x \in \left(0, \dfrac{\pi}{2}\right)$, is the minimum value of $\dfrac{\sin^3x}{\cos x} +\dfrac{\cos^3x}{\sin x} = 1$? So considering ($\dfrac{1}{\cos x}$, $\dfrac{1}{\sin x}$) and ($\sin^3x$, $\...
4
votes
5answers
139 views

To prove $(\sin\theta + \csc\theta)^2 + (\cos\theta +\sec\theta)^2 \ge 9$

I used the following way but got wrong answer $$A.M. \ge G.M.$$ $$ \frac{\sin \theta + \csc \theta}{2} \ge \sqrt{\sin \theta \cdot \csc \theta}$$ Squaring both sides, \begin{equation*} (\sin\theta + ...
1
vote
2answers
107 views

Prove $ \prod_{cyc}\left ( 2+ a^{\,2} \right ) + abc\geqq 28$

For $a,\,b,\,c\geqq 0$, prove that $$\left \{ \prod\limits_{cyc}\,\left ( 2+ a^{\,2} \right ) \right \}+ abc\geqq 28$$ with $a+ b+ c= 3$. Let $a+ b+ c= 3\,u= 3,\,ab+ bc+ ca= \dfrac{3\,u}{X},\,abc= \...
0
votes
0answers
25 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 ...
3
votes
1answer
63 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 ...
0
votes
1answer
27 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 ...
5
votes
1answer
131 views
+50

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 \...
3
votes
2answers
976 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
0
votes
1answer
48 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
20 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, \...
1
vote
4answers
64 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
0answers
74 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{...
0
votes
1answer
54 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)\...
4
votes
1answer
60 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 +...
1
vote
1answer
43 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.
0
votes
2answers
38 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
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 $$\...
0
votes
1answer
105 views

How to prove $\frac{a}{1998}+\frac{b}{1997}+\frac{c}{1996} \ge 0$ using Calculus only? [on hold]

Let a,b,c are in $\mathbb{R} $ such that $ax^{2}+bx+c \ge0$ for every $x \ge 0$, show that $$\dfrac{a}{1998}+\dfrac{b}{1997}+\dfrac{c}{1996} \ge 0$$
1
vote
1answer
39 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}}=...
-9
votes
0answers
82 views

A short problem in geometry 😊 [on hold]

Given an acute triangle $\triangle ABC$. The altitude $AH$ is equal to the median $BM$. Prove: $∠B<60^∘$
-3
votes
1answer
37 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
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
3answers
160 views

prove this inequality with $\sum\limits_\text{cyc}\sqrt{1-xy}\ge 2$

Let $x,y,z\ge 0$,and $x+y+z=2$, show that $$\sqrt{1-xy}+\sqrt{1-yz}+\sqrt{1-xz}\ge 2$$ Mt try: $$\Longleftrightarrow 3-(xy+yz+zx)+2\sum_\text{cyc}\sqrt{(1-xy)(1-yz)}\ge 4$$ or $$\sum_\text{cyc}\sqrt{(...
0
votes
1answer
51 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\...
5
votes
3answers
105 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 $m,n \in S$ ...
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
142 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 ...
3
votes
3answers
80 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}+ (\,-\...
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
74 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
59 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 ...
1
vote
1answer
194 views

Theta notation from the inequality $c_1\lg(n) \leq \lg(k) \leq c_2\lg(n)$

Consider the inequality $$ c_1\lg(n) \leq \lg(k) \leq c_2\lg(n),\text{ for } n \geq n_{0} $$ With $c_1,c_2,n_0 > 0$, $\lg(k) = \Theta(\lg(n))$ By deriving the actual relationship of $k$ with $n$,...
0
votes
1answer
602 views

Chebychev inequality generalization, essential supremum

$\DeclareMathOperator{\esssup}{ess\,sup}$ I don't understand fully why would this be true. $P$ is probabilty here, and $\mathcal{E}X$ means expected value of $X$. Let $g: \mathbb{R} \to \mathbb{R}_+$ ...
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}|=...
4
votes
2answers
87 views

Estimation on a summation

Suppose we have two vectors $x$ and $y$ in $\mathbb{R}^n$ that satify $\|x\|=\|y\|=1$ $<x,y>=0$ $\sum_{i=1}^{n}{x_{i}}=\sum_{i=1}^{n}{y_i}=0$ That is $x$ and $y$ are of norm 1, $x\perp y$ and $...
2
votes
1answer
150 views
+50

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 ...
-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.
6
votes
2answers
201 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
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
0answers
57 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 ...