Questions on proving, manipulating and applying inequalities.

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0
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1answer
20 views

Comparison between my formula and the number 1

It is known that the real numbers $a>0$, $b>0$, and $0<\theta<1$, then is the following inequality correct? $$ \frac{a^\theta\cdot b^{1-\theta}}{\theta\cdot a+(1-\theta)\cdot b}\leq 1 $$ ...
0
votes
1answer
14 views

Prove that $M(|f|,S)-m(|f|,S)\leq M(f,S)-m(f,S)$

Let $M(f,A)=sup\{f(x):x\in A\subseteq[a,b]\}$ and let $m(f,A)=inf\{f(x):x\in A\subseteq[a,b]\}$. Given that $|f(x_{0})|-|f(y_{0})|\leq |f(x_{0}-f(y_{0})|\leq M(f,S)-m(f,S)$ for $x_0,y_0\in S$, prove ...
4
votes
0answers
148 views
+50

Prove $\sum_{cyc}\left(\frac{a^4}{a^3+b^3}\right)^{\frac34} \geqslant \frac{a^{\frac34}+b^{\frac34}+c^{\frac34}}{2^{\frac34}}$

When $a,b,c > 0$, prove $$\left(\frac{a^4}{a^3+b^3}\right)^{\frac34}+\left(\frac{b^4}{b^3+c^3}\right)^{\frac34}+\left(\frac{c^4}{c^3+a^3}\right)^{\frac34} \geqslant ...
2
votes
2answers
65 views

Prove that Triangle ABC is an equilateral triangle iff $\tan{A}+\tan{B}+\tan{C} = 3^\frac32$.

This question is picked from AM GM HM inequalities, so this is to be proved form that concept only, I think it isn't possible because there is no inequality, but if it is please tell me how.
0
votes
2answers
22 views

Prove that $M_1^2\leq 2M_0M_2$, if $2M_1t≤2M_0+M_2t^2$

Let $0\leq M_1,M_2,M_3\in\mathbb{R}$ and $\forall \ t\in\mathbb{R}:\ 2M_1t≤2M_0+M_2t^2$. Prove that $M_1^2\leq 2M_0M_2$. I tried assigning different values to $t$, but this didn't help.
0
votes
2answers
54 views

If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.

I solved it using AM, GM inequalities and reached to $[(1+a)(1+b)(1+c)]^7 > 2^{21}(abc)^\frac72 $ please help how to get $7^7(abc)^4$ in the inequality.
10
votes
2answers
2k views

Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$

As the title says, am searching for a proof of If $A,B \in \mathbb{R}^{n\times n}$ and $AB=0$ then $\mathrm{rank}(A)+\mathrm{rank}(B) \leq n$ I am doing this as preparation for an upcoming ...
6
votes
3answers
1k views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
1
vote
3answers
52 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 $$ ||fgh ||_1\le||f||_p\ ||g||_q\ ||h||_r $$
28
votes
8answers
736 views

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $x^2y+y^2z+z^2x < \frac12$

$x,y,z \geqslant 0$, $x+y^2+z^3=1$, prove $$x^2y+y^2z+z^2x < \frac12$$ This inequality has been verified to be correct according to Mathematica. $\frac12$ is not the best bound. I try to do AM-GM ...
0
votes
3answers
24 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
34 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
11 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 ...
0
votes
1answer
17 views

Inequality involving the Hessian matrix of a convex function

Let $f \in C^2(\mathbb{R}^d)$ be a convex function with Hessian $H$. Is it true that $$ (x^T H(x) - y^T H(y)) (x-y) \ge 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1) $$ for all $x,y \in \mathbb{R}^d$? ...
1
vote
0answers
25 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
42 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
13 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
44 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 ...
3
votes
2answers
70 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
35 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
1answer
53 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
70 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
88 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
119 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
244 views
+50

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
108 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
16 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 ...
8
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
38 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 ...
2
votes
0answers
95 views
+50

Inequality with analytic functions on the unit ball

Let $g(z) = \sum_{n\geqslant 0} a_nz^n$ be an analytic function where $a_n$ only take values in $\{0,1\}$ (not sure if it is a necessary condition, it is just the case I'm considering). Let ...
1
vote
1answer
34 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
443 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
29 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
189 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: ...