Questions on proving and manipulating inequalities.

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10
votes
5answers
463 views

Is -5 bigger than -1?

In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion. I am a mathematics ...
11
votes
2answers
315 views

Is $f(2x)/f(x)$ nonincreasing for concave functions with $f(0)=0$?

I have a question about concave functions. Let $f:R_+\rightarrow R_+$ be any nonidentically zero, nondecreasing, continuous, concave function with $f(0)=0$. Do we have that the ratio function ...
4
votes
1answer
148 views

how to solve the following inequality?

I have the following inequality: $$n \ge \frac{K_n^2}{\epsilon^2} \frac{\log K_n}{\epsilon},\text{ where }K_n = (\log n)^3.$$ I would like to solve it, even numerically. I thought that numerically ...
22
votes
2answers
1k views

An information theory inequality which relates to Shannon Entropy

For $a_1,...,a_n,b_1,...,b_n>0,\quad$ define $a:=\sum a_i,\ b:=\sum b_i,\ s:=\sum \sqrt{a_ib_i}$. Is the following inequality true?: $${\frac{\Bigl(\prod a_i^{a_i}\Bigr)^\frac1a}a \cdot ...
3
votes
1answer
500 views

Finding tight constraints on a linear inequality

I have $a^\intercal M b > 0$, where $\forall a_i > 0$, $\forall b_j > 0$, and M is known. I'd like to find a tight linear constraint on $b$ which is independent of $a$ (other than the ...
3
votes
2answers
226 views

How to prove the inequality $\Theta(x,y)\le \Theta(x,z)+\Theta(z,y)$?

Let $x, y$ be two complex vectors, $$\cos\Theta(x,y):=\operatorname{Re} \frac{y^*x}{\|x\|\|y\|} .$$ Then I want to prove that $$\Theta(x,y)\le \Theta(x,z)+\Theta(z,y) .$$
2
votes
1answer
121 views

Bounding a Complex Polynomial

Given the complex polynomial $P(z) = z^2 + a_1z + a_0$ and the constraint that $|z| > 1$, I'm trying to show that $|P(z)| \geq |z|^2 - |a_1||z| - |a_0|$. The obvious thing to do here of course is ...
2
votes
0answers
245 views

Two vague steps in the proof of Harnack inequality

I am reading the book Elliptic and Parabolic Equations and the proof is excerpted from page 133-136. In Theorem 5.1.3: it claims that ...
2
votes
2answers
150 views

Raising the floor function to a power

I've made a few plots and noticed that $\lfloor Is it true that for positive $x > 1$ and $n \in \mathbb N,\quad n>=2$ the following holds: $$(\lfloor x \rfloor + 1)^n >= \lfloor x^n ...
7
votes
1answer
317 views

Hölder Inequality

I am wondering how I get $$ ...
3
votes
2answers
140 views

Calculate max/min of $x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$

What is a good way to calculate max/min of $$x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$$ where $x_1+y_1+z_1+w_1=a$ and $x_2+y_2+z_2+w_2=b$ and $x, y, z, w, a, b \in \mathbb{N} \cup \{0 \}$, and please explain ...
5
votes
1answer
252 views

Proof of inequality

I have problems with proving inequality : $${a^{2}}+b^2+c^2+\frac{2}{5}abc<50$$ where $a,b,c$ are the lengths of triangle's sides, and the circumference of the triangle is $10$. Thanks.
2
votes
3answers
91 views

Does $c^n(n!+c^n)\lt (n+c^2)^n$ hold for all positive integers $n$ and $c\gt 0$?

I am not sure whether the following inequality is true? Some small $n$ indicates it is true. Let $n$ be a positive integer and $c\gt0$, then $$c^n(n!+c^n)\lt(n+c^2)^n.$$
2
votes
1answer
776 views

Sufficient Conditions for a Bounded Feasible Region in the Linear Programming Problem

I am working on a problem where it would be nice to prove that the feasible region of a LP problem is bounded, but where it is not necessary to solve any particular problem. In particular, given an ...
13
votes
3answers
796 views

Motivation for triangle inequality

Triangle inequality is used in one context or the other in analysis. To list a few $$ \|x+y\| \leq \|x\| + \|y\| $$ $$ d(x,y) \leq d(x,z) + d(z,y) $$ $$ \mu(A \cup B) \leq \mu(A) + \mu(B) $$ What ...
2
votes
2answers
135 views

Help on Inequality Proof

I'm trying to solve a proof for this inequality. I already have the solution, but have a question about the solution. Here's what I have so far: Prove $2^n > n^2$ for $n > 4$ Base Case $n=5$. $32 ...
0
votes
2answers
121 views

Algebraic Inequality

$(a + x)^{1/2} + (a - x)^{1/2} \gt a$ for any real $a\gt 0$.
6
votes
3answers
363 views

How do I prove this inequality?: $a+b+c+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq 3+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ where $a,b,c>0$ and $abc=1.$

I have been thinking over this problem for a couple of days, but I have no idea how to solve it in a simple way. I am interested if there is a way only using elementary methods to prove it. Using the ...
0
votes
2answers
198 views

Quadratic inequalities - why does the solution seem different for $(x-a)(x-b) <0$ and $(x-a)(x-b) >0$?

I want to ask about way of solving exponential inequalities, I am going to show you two similar examples, but their solving is kinda different. First example: $$3^{2x}-10\cdot3^x+9>0$$ ...
1
vote
2answers
321 views

A question on inequality of arithmetic and geometric means

Let $x_i>0, i=1,...,n$ and $x_1+..+x_n=K$. From the inequality of arithmetic and geometric means, we have $$x_1x_2...x_n\le \left( \frac{x_1+x_2..+x_n}{n} \right)^n$$ The equality holds if and only ...
0
votes
2answers
160 views

Bounding probability where Markov/Chernoff bounds seem to fail

This is related to the question I have asked yesterday: Expected value of max/min of random variables. Assume you have $n$ urns and $k$ balls. Each ball is placed uniformly at random in one of the ...
13
votes
7answers
787 views

What does $\ll$ mean?

I saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically. http://en.wikipedia.org/wiki/German_tank_problem EDIT: It looks like this can use TeX commands. ...
1
vote
1answer
129 views

Solving inequalities comparing $f(x)$ to $0$ where $f$ is an elementary function

Any inequality comparing elementary functions can be rearranged to compare some elementary function $f$ to $0$. What is the best way to approach, in general, solving such inequalities at the ...
5
votes
0answers
149 views

Bounding function involving Beta functions

Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + ...
1
vote
1answer
70 views

Can one prove $\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min(1,\frac{c}{t})$?

Let $c>1/2$ be an arbitrary big fixed constant. Can one prove that for all $t\geq 1$: $$\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min\left(1,\frac{c}{t}\right)$$ for some small constant ...
3
votes
4answers
2k views

Range of values of f(x) using quadratic inequalities (need intuition)

I'm working on an exercise from a book in the chapter on quadratic inequalities: "Find the set of possible values of the given function $\frac{x - 2}{(x + 2)(x - 3)}$". The answer in the book is "all ...
3
votes
2answers
516 views

Non-negative integral solutions of $X_1+X_2+X_3+X_4<n$

The number of non-negative integral solutions of $X_1+X_2+X_3+X_4<n$ (where $n$ is a positive integer) is?
0
votes
3answers
175 views

Proving inequalities

Im really bad when it comes to proving inequalities. I have prove this: these are all positive $\sum_{k=1}^n a_k \leq \sum_{k=1}^n ka_k \leq n \sum_{k=1}^n a_k$ Where would i start with this? ...
2
votes
2answers
1k views

Finding set of values using inequalities

I'm attempting a question in my math book (self teaching so don't have a personal tutor to ask). I'm getting confused as to what I'm supposed to be doing. Here's the question: What is the set of ...
6
votes
3answers
318 views

How to find the minimum value of this expression?

Given that $b_1+b_2+\dots+b_n = 1$, how do I find the minimum value of $$\frac{x_1+x_2+\dots+x_n}{x_1^{b_1}x_2^{b_2}\dots x_n^{b_n}}?$$ For $n=2$ I used calculus and found the answer to be ...
6
votes
2answers
159 views

$x\le\lceil\sqrt{x}\rceil\left\lceil\frac{x}{\lceil\sqrt{x}\rceil}\right\rceil$?

$x\le\lceil\sqrt{x}\rceil\left\lceil\frac{x}{\lceil\sqrt{x}\rceil}\right\rceil$ How do I show this? I made a plot, and it looks true:
5
votes
2answers
362 views

An inequality on a convex function

An exercise in my textbook asked to prove the following inequality, valid for all $a,b,c,d \in R $ $(a/2 + b/3 + c/12 + d/12)^4 \leq a^4/2 + b^4/3 + c^4/12 + d^4/12$ There is a straightforward proof ...
2
votes
3answers
88 views

Incoherence in dealing with quadratic inequalities

I have made a rather obvious yet peculiar observation while calculating with quadratic inequalities. Take a simple quadratic inequality like the one below $\frac{x^2+1}{x}>1$ by multiplying both ...
5
votes
2answers
239 views

how to prove this inequality?

Given $x>0$, $y>0$ and $x + y =1$, how to prove that $\frac{1}{x}\cdot\log_2\left(\frac{1}{y}\right)+\frac{1}{y}\cdot\log_2\left(\frac{1}{x}\right)\ge 4$ ?
3
votes
2answers
146 views

Inequality problem

Prove that if $$|x-x_0| < \min\left(\frac{\epsilon}{2(|y_0| + 1)}, 1\right)$$ and $$|y-y_0| < \frac{\epsilon}{2(|x_0| + 1)},$$ then $|xy - x_0y_0| < \epsilon.$ I am doing some problems in ...
1
vote
0answers
83 views

Single solutions to an inequality

Suppose we had an inequality $ax < by < cx$ where $a,b,c,x,y \in \mathbb Z$. If we fix $a,b,c$ and let $x,y$ vary how can we find the values of $x$ for which only a single $y$ satisfies the ...
0
votes
1answer
434 views

Extension of reverse triangle inequality

Using Reverse Triangle Inequality, one can write for $x,y\in R^1$ $$ ||x|-|y||\leq |x-y| $$ Is there any suitable inequality doing the following $$ ||x|^p-|y|^p|\leq f_p(|x-y|) $$ for $1 \leq p ...
9
votes
2answers
554 views

Proof that $t-1-\log t \geq 0$ for $t > 0$

Using basic calculus, I can prove that $f(t)=t-1-\log t \geq 0$ for $t > 0$ by setting the first derivative to zero \begin{align} \frac{df}{dt} = 1 - 1/t = 0 \end{align} And so I have a critical ...
1
vote
4answers
124 views

Solving equality to find upper limit

I need to find a sensible upper limit for a part of an algorithm in a program I am writing. I have boiled it down to this. Given $a$, $b$ and $c$, find $x$ in $a^{x-1}b < c < a^{x}b$. But I ...
2
votes
1answer
139 views

Inequality involving random variables

I have random variables $X_1,\ldots,X_n \in \{0,1\}$ which are dependent and $$\mathbb{P}[X_i=1 | X_1=x_1, \ldots, X_{i-1}=x_{i-1}, X_{i+1}=x_{i+1}, \ldots X_{n}=x_n] \geq p$$ for any ...
4
votes
3answers
257 views

How to prove inequality?

What is an easy way to show that for positive integers $i,n$, a real $p \in (\frac12,1)$ and $\epsilon \in [0,p]$, $$p^i(1-p)^{n-i} \geq (p-\epsilon)^i(1-(p-\epsilon))^{n-i}.$$ (I have a complicated ...
1
vote
1answer
104 views

How can I find $\sup -\frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$ for $x_1 x_2 > 0$?

I need to find a constant $a$ such that for all $x_1 x_2 > 0$: $$a > - \frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$$ that is to say the supremum of the term on the right hand side. My question is how to ...
2
votes
1answer
81 views

Given $a>0$ and $ac-b^2>0$ show

Given $a>0$ and $ac-b^2>0$ show $cy^2+a[(x+\frac{by}{a})^2-(\frac{by}{a})^2] > 0$ I'm completely confused about this, I've tried a few approaches. I end up getting stuck saying that I know ...
0
votes
1answer
71 views

$r^a \leq r^b+r^c$ given $a \leq b+c$?

Apologies for the (maybe obvious) question. It's come up as part of a proof of the triangle inequality for a metric function I'd like to work with. For real number $r \geq 1$ and positive integers a, ...
1
vote
1answer
125 views

Some preliminary inequality

Consider a ratio $\int_A f(x,a)dx/\int_B f(x,a)dx$ where $A, B \subset [0,1]$ and $a \in R$. Suppose for any $x' \in A$ and $x \in B$, $f(x',a)/f(x,a) > f(x',b)/f(x,b).$ Then can we say ...
11
votes
2answers
463 views

An inequality : is it true if it is then how to prove it?

consider positive numbers $a_1,a_2,a_3,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$. does the following in-equality holds and if it does then how to prove it ...
4
votes
4answers
480 views

logarithm inequality

Why $\frac{1}{n+1}<\log(n+1)-\log(n)<\frac{1}{n}$?
1
vote
1answer
127 views

lower bond for $\log(n!)$

By the inequality here or using integration I can get the following lower bound for $\log(n!)$. How can I get a better lower bound for $\log(n!)$? $$\log(n!)>\log\left(\frac{(n+1)^n}{e^n}\right)$$ ...
5
votes
3answers
144 views

Proving $\frac{1}{2}(n+1)<\frac{n^{n+\frac{1}{2}}}{e^{n-1}}$ without induction

I want to show that $\displaystyle\frac{1}{2}(n+1)<\frac{n^{n+\frac{1}{2}}}{e^{n-1}}$. But except induction, I do not know how I could prove this?
5
votes
4answers
165 views

Inequality understanding

My textbook says that: $$ \frac{(n+1)^n}{n!}=\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)^2\cdots\left(1+\frac{1}{n}\right)^n<e^n $$ But I do not understand this. Can you please enlighten ...