Questions on proving, manipulating and applying inequalities.

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2
votes
3answers
55 views

Find two fractions such that their sum added to their product equals $1$

This is a very interesting word problem that I came across in an old textbook of mine. So I managed to make a formula redefining the question, but other than that, the textbook gave no hints really ...
0
votes
1answer
37 views

$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ then Holder's inequality [duplicate]

If $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ and $ f\in L_p $ $g\in L_q $ and $h\in L_r $ so how can I prove $$ ||fgh ||_1\le||f||_p\ ||g||_q\ ||h||_r $$
0
votes
3answers
26 views

Inequality for quadratic function and exponent

I might be a but rusty but while doing probability tasks i got stuck on some inequalities from analysis. The task is to prove that there exists $K>0$, so that: $$ \left|\frac{1}{1+e^{3x}} ...
2
votes
1answer
40 views

Deriving a bounding $\delta$ of an interior point

This question is based on the Baby Rudin's 2.16: Regard $Q$, the set of all rational numbers, as a metric space, with $d(p,q)=\lvert p -q \rvert$. Let $E$ be the set of all $p \in Q$ such that $2 ...
1
vote
0answers
22 views

Doob's maximal inequality with stopping time

I have been searching for a version of Doob's maximal inequality with stopping time insides the time index, i.e. given $\Lambda_n$ is a positive sub-martingale and N is a stopping time is there any ...
1
vote
0answers
28 views

Probability problem related to Markov inequality

Problem Let $p$ be the probability of a person chosen at random to support Bernie Sanders. A sample is taken of $50$ persons chosen at random, each of them is asked if he or she would vote for ...
1
vote
2answers
46 views

Showing there is a constant for which an inequality holds true

I'm supposed to show that for $x>0$ and $p>0$ there is a constant $C$ such that $e^x\ge Cx^p$. The constant $C$ depends on $p$ but not on $x$. After analysing the behaviour of the graphs of ...
0
votes
1answer
16 views

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$ My attempt: It's equivalent to proof the following inequality: $$ ...
0
votes
2answers
46 views

Questions about proof of inequality $(x^p-1)/p <> (x^q-1/q) $

I am working through a proof of the following inequality, where x,p,q are positive, and p and q are integers. $$ \frac{(x^p-1)}{p} \neq \frac{(x^q-1)}{q} $$ Which gives $$ \tag1 ...
0
votes
0answers
34 views

Alignment for equality sign [migrated]

I solved a question and I wrote it in Latex, but the equality sign is not aligned, I searched google, but none of the ways there worked with me. How do I get the equality signs aligned in the below? ...
3
votes
3answers
104 views

A question about $ax = b$

I am studying inequality and come across the following statement. I don't understand it and want to believe the book must have made mistakes. I am going to copy what the book says here exactly. A ...
4
votes
2answers
83 views
+50

Jensen-like averaging inequality on integers

Let $\mathbb{Z}^*=\mathbb{Z}^+\cup\{0\}$. Let $f:\mathbb{Z}^*\rightarrow\mathbb{R}$ be a nondecreasing function such that $f(a+b)\leq f(a)+f(b)$ for all $a,b\in\mathbb{Z}^*$. Is it true that for all ...
2
votes
1answer
36 views

How to prove a set of inequalties in not satisfiable?

For the set of inequalities $$\begin{cases} 10 a - b - c \ge d\\ 5 b - a - c \ge d\\ 2 c - a - b \ge d\\ d \ge a + b + c\end{cases}$$ how can I show these cannot all be satisfied for $a, b, c, d$ ...
0
votes
1answer
14 views

Inequality in recurrence relation

I'm having a mental block understanding what is probably a simple inequality in a guess and check example for a recurrence relation. Would someone please explain to me how they obtain the inequalities ...
1
vote
0answers
60 views

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
0
votes
0answers
80 views

Prove or disprove $2abc(a+b+c)\ge 3(a^2b^2c^2+1)$

Let $a,b,c>0,ab+bc+ca=3$, prove or disprove $$2abc(a+b+c)\ge 3(a^2b^2c^2+1)$$ Now I can't find any counterexample
2
votes
3answers
89 views

How to prove the inequality $ \frac{1-e^{-x^2}}{x^2}e^{-(x-n)^2}<\frac{2}{n^2}$ for $0<x<n$

Can somebody prove that this inequality is true for $0<x<n$? $$ \frac{1-e^{-x^2}}{x^2}e^{-(x-n)^2}<\frac{2}{n^2}$$ I'm pretty much stuck.
0
votes
0answers
16 views

Inequality for all integers greater than 2 .

Fr all integers $n\geq 2$ , define $f_{n} (x) $ = $(x+1)^{1/n} - x^{1/n}$, where $x>0$ . Then as a function of x a.) $f_{n} (x) $ is increasing for all n b.)$f_{n} (x) $ is a decreasing function ...
0
votes
0answers
14 views

Inequality on exponents of positive numbers

Let x and y be positive numbers . Which of the following always implies $x^y \geq y^x $ a.) $x\leq e\leq y$ b.)$y\leq e \leq x$ c.)$x\leq y \leq e$ or $e\leq y \leq x$ d.)$ y\leq x \leq e$ or ...
1
vote
2answers
124 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
4
votes
2answers
273 views

Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
3
votes
4answers
109 views

Minimum of $\frac{x}{1+y^2}+\frac{y}{1+x^2}$ on $x,y\ge 0$, $x+y=2$

let $x,y\ge 0$, and such $x+y=2$ find the minimum $$\dfrac{x}{1+y^2}+\dfrac{y}{1+x^2}$$ I think $x=y=1$ is minimum of the value $1$,How can I prove?
0
votes
0answers
29 views

Solving an inequality systematically

A question states: "Find all $n >3$ such that $$ \frac{1}{n^{1.1}}<\frac{1}{n \ln n}" $$ Here's my step: $$ n^{1.1}>n \ln n $$ $$ n^.1>\ln n $$ $$ n >(\ln n)^{10} $$ Setting $(\ln ...
0
votes
0answers
17 views

A inequality concerns with the Legendre polynomial of n-th degree of $\cos\theta$

I am reading a paper, where the author concluded that $${\left| {{P_n}(\cos \theta )} \right|^2} \leqslant \frac{2} {{n\pi \sin \theta }},\,\,\forall \theta \in \left( {0,\pi } \right).$$ Here ...
9
votes
8answers
2k views

Which of the numbers is larger: $7^{94}$ or $9^{91} $?

In this problem, I guess b is larger, but not know how to prove it without going to lengthy calculations. It is highly appreciated if anyone can give me a help. Which number is larger ...
1
vote
1answer
44 views

Calculus of variations with inequality and non-integral constraints

I have a question on solving an optimization problem with calculus of variations. I am attempting to maximize the functional $$ J[y] = \displaystyle\int_a^b F(x,y,y') \, \mathrm{d}x, \tag{1}$$ ...
1
vote
1answer
23 views

Why $|\alpha|\lt 1$ and $|\beta| \gt 1$?

I'm reading Conway's complex analysis book and on page 117 he writes: I didn't understand why $|\alpha|\lt 1$ and $|\beta| \gt 1$. I could only prove $\beta\lt -1$.
2
votes
1answer
28 views

Maximum of $(ab+cd)(ac+bd)(ad+bc)$

Let $a,b,c,d\ge 0$ satisfy $a+b+c+d=4$. Find the maximum value of $(ab+cd)(ac+bd)(ad+bc)$. When all of the variables are $1$, the value is $8$. Using the AM-GM inequality gives ...
4
votes
3answers
111 views

In Tao's proof of the Hölder’s inequality

(Hölder’s inequality) Let $f \in L^p$ and $g \in L^q$ for some $0 < p,q \leq \infty$. Then $fg \in L^r$ and $\|fg\|_{L^r} \leq \|f\|_{L^p} \|g\|_{L^q}$, where the exponent $r$ is defined by the ...
4
votes
3answers
204 views

Holder's inequality $ \sum_{i=1}^n |u_i v_i| \leq (\sum_{i=1}^n |u_i|^p )^{\frac{1}{p}}(\sum_{i=1}^n |v_i|^q )^\frac{1}{q} $

Using the fact $xy \leq \frac{1}{p}x^p + \frac{1}{q}y^q$ for all $x,y >0$ and $p,q > 0$ with $\frac{1}{p} + \frac{1}{q} = 1$. How can I proof the Holder's Inequality? $$ \sum_{i=1}^n |u_i v_i| ...
0
votes
8answers
91 views

Least value of $a$ for which $4ax^2 + \frac{1}{x} \geq 1$

Find the least value of $a \in R$ for which $4ax^2 + \frac{1}{x} \geq 1$, for all $x>0$. The equation will transform into (Using $x>0$) $4ax^3-x+1\geq 0$ But I don't know how to deal with ...
0
votes
2answers
1k views

Combining inequalities into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
1
vote
1answer
35 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
0
votes
0answers
12 views

Is there a more relaxed bound for this inequality?

I have the following inequality: $$4p^2-||sa+qb||^2>0$$ where $p,s,q$ are real scalars and $a,b$ are real vectors. I know that $||a||\le F_1$ and $||b||\le F_2$. I want to express the inequality ...
1
vote
0answers
46 views

Is it true that: $\frac {1 } {10 }(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2)^2+(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1)^2) \le (y_1-x_1)^2+(y_2-x_2)^2$?

Is it true that: $$\frac {1 } {10 }\left(\left(\sin(y_1+y_2)-\sin(x_1+x_2)+y_2-x_2\right)^2+\left(\cos(x_1+x_2)-\cos(y_1+y_2)+x_1-y_1\right)^2\right) \le (y_1-x_1)^2+(y_2-x_2)^2$$ I think I should ...
3
votes
1answer
446 views

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle. Also find the center and radius of the circle. Here's my solution, ...
0
votes
2answers
30 views

Prove that $\sqrt{A_1A_2} + \sqrt{B_1B_2} \leq \sqrt{A_1 + B_1}\sqrt{A_2 + B_2}$

I am trying to prove the below inequality using the AM-GM inequality, but I can't see how to get it to come out. $$\sqrt{A_1A_2} + \sqrt{B_1B_2} \leq \sqrt{A_1 + B_1}\sqrt{A_2 + B_2}$$
3
votes
1answer
62 views

Show that $\|f_1+f_2\| \leq \|f_1\| + \|f_2\|$ using Minkowski's inequality

I am trying to show that: $\|f_1+f_2\| \leq \|f_1\| + \|f_2\|$ using the Minkowski inequality. for: $$ \|f\| = \left(\int_0^1 \left[|f|^2 + |f'|^2\right]\ dx\right)^{1/2}.$$ I dont see how I can ...
4
votes
1answer
190 views

Can we use matrix to solve this inequality?

Let $$f(x)=\begin{cases} 1&0\le x\le 1\\ 0&\rm{others} \end{cases}$$ Let $x_{i},a_{i}(i=1,2,\cdots,n)$ be positive real numbers, show that: ...
0
votes
1answer
55 views

How prove this determinant can't zero

Let $x,y,z\neq 0$ be real numbers, show that $$f(x,y,z)=\begin{vmatrix} \sqrt{x^2+y^2}&|x|&|y|\\ |y|&\sqrt{y^2+z^2}&|z|\\ |x|&|z|&\sqrt{x^2+z^2} \end{vmatrix}\neq 0$$ or it ...
3
votes
0answers
79 views

$a^{|b-a|}+b^{|c-b|}+c^{|a-c|} > \frac52$ for $a,b,c >0$ and $a+b+c=3$

$a,b,c >0$ and $a+b+c=3$, prove that $$a^{|b-a|}+b^{|c-b|}+c^{|a-c|} > \frac52$$ What I did: It is cyclic inequality so I assume $c= min\{ a,b,c \}$. I consider the first case where ...
5
votes
1answer
102 views

Prove $\frac{ab}{4-d}+ \frac{bc}{4-a}+\frac{cd}{4-b}+\frac{da}{4-c} \leqslant \frac43$ for positive $a,b,c,d$

$a,b,c,d \geqslant 0$ and $a^2+b^2+c^2+d^2=4$, Prove $$\frac{ab}{4-d}+ \frac{bc}{4-a}+\frac{cd}{4-b}+\frac{da}{4-c} \leqslant \frac43$$ I try some reverse AM-GM techniques but fail. I don't think ...
0
votes
1answer
28 views

$L^p$ Norm of product of two bounded functions

If $f$ and $g$ are bounded functions in $L^p[a,b]$, does the following inequality hold in $L_p$ spaces? $$\|fg\|_p\leq\|f\|_p\|g\|_p$$
4
votes
1answer
66 views

Tricky Multivariable Inequality

Given 100 positive real numbers $x_1, x_2, \cdots, x_n$ that satisfy $x_1^2+x_2^2+\cdots+x_n^2>10000$ and $x_1+x_2+\cdots x_n\le 300$, prove that there exist three numbers from this set such that ...
0
votes
1answer
34 views

An inequality with complex numbers.

Given $n$ complex numbers $z_1,\ldots,z_n$, is it true that $$ |z_j|\sum_{k=1}^n|z_k|\leq\sum_{k=1}^n|z_k|^2 $$ for $j\in\{1,\ldots,n\}$ ? Thank u for any help!
1
vote
2answers
92 views

Prove that $(x+(y+z^{1/4})^{1/3})^{1/2}\geqslant(xyz)^{1/32}$

Please help with the inequality $$\large\sqrt[2]{x+\sqrt[3]{y+\sqrt[4]{z}}}\geqslant\sqrt[32]{xyz}.$$ I've tried with Cauchy's theorem. And don't know what to do later.
4
votes
1answer
88 views

Prove $x+y^2+z^3 \geqslant x^2y+y^2z+z^2x$ for $xy+yz+zx=1$

$x,y,z \geqslant 0$ and $xy+yz+zx=1$, prove $$x+y^2+z^3 \geqslant x^2y+y^2z+z^2x$$ What I try $$x+y^2+z^3 \geqslant x^2y+y^2z+z^2x$$ $$\Leftrightarrow x(xy+yz+zx)+y^2+z^3- x^2y-y^2z-z^2x \geqslant ...
0
votes
1answer
26 views

Maximum area of a triangle when perimeter is fixed.

I can't solve the following problem: Show that amongst all triangles with perimeter $3p,$ the equilateral triangle with side $p$ has the largest area. Further show that $9p^2\ge 12\sqrt{3}\Delta.$ ...
2
votes
1answer
34 views

find the maximum of the function $f(x)=a+b\sqrt{2}\sin{x}+c\sin{2x}$

let $a,b,c\in R$,and such $a^2+b^2+c^2=100$, find the maximum value and minimum value of the function $$f(x)=a+b\sqrt{2}\sin{x}+c\sin{2x},0<x<\dfrac{\pi}{2}$$ Use Cauchy-Schwarz inequality?
2
votes
1answer
49 views

Is this estimate true or not true?

Let $\varepsilon>0$. Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ the standard normal density function. Then $$\lim_{\varepsilon\to 0}\int_0^1 \frac{1}{\sqrt{x}}\left[ ...