Questions on proving, manipulating and applying inequalities.

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2
votes
1answer
186 views

Rudin: Problem Chp3.11 and need advice.

I am working on the following problems and I have a couple of questions. Suppose $a_n>0, s_n = \sum_{i = 1}^{n}$ and $\Sigma a_n$ diverges. RTP (a) $\Sigma \frac{a_n}{1+a_n}$ ...
1
vote
6answers
449 views

Proving $\pi^3 \gt 31$

$$\large \pi^3 \gt 31$$ Using a calculator, $\pi^3/31 \approx 1.0002$, so I thought this may be challenging to do by hand. It is extremely easy with the use of any calculator, so I was wondering ...
16
votes
5answers
2k views

Prove $|e^{i\theta} -1| \leq |\theta|$

Could you help me to prove $$ |e^{i\theta} -1| \leq |\theta| $$ I am studying the proof of differentiability of Fourier Series, and my book used this lemma. How does it work?
14
votes
2answers
420 views

A conjecture concerning primes and algebra

A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb ...
10
votes
2answers
992 views

Proof of an inequality: $\sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}}$ [duplicate]

Possible Duplicate: Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction How do I prove the following? $$\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + ...
7
votes
4answers
531 views

Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.

Given $a,b,c>0$, prove that $\displaystyle \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$. I expanded the LHS, ...
6
votes
5answers
277 views

Show that $\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity

Well, I'm preparing for an undergrad competition that is held in April and because of that I've been trying to solve the inequalities I find on the internet. I found this problem: $$\displaystyle ...
9
votes
3answers
200 views

Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$

So, a,b,c are non-negative real numbers for which holds that $a+b+c=3$. Prove the following inequality: $$4\ge a^2b+b^2c+c^2a+abc$$ For now I have only tried to write the inequality as ...
9
votes
2answers
4k views

Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$

One of fundamental inequalities on logarithm is: $$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$ which you may prefer write in the form of $$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
9
votes
3answers
1k views

Proofs of the Cauchy-Schwarz Inequality?

How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
4
votes
4answers
177 views

Prove that $a^2+ab+b^2\ge 0$

How to prove that $a^2+ab+b^2\ge 0$? Obviously the squares are positive, but how can I be sure that $ab$ doesn't become too negative with a certain combination of $a$ and $b$?
3
votes
5answers
12k views

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
0
votes
3answers
526 views

Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$

If $a, b$ are positive real numbers and $a+b = 1$, prove that : $$\left(a+\frac{1}{a}\right)^2 + \left(b+\frac{1}{b}\right)^2 \geq \frac{25}{2}$$ Thank you.
5
votes
6answers
9k views

Proving by induction: $2^n > n^3 $ for any natural number $n > 9$ [duplicate]

I need to prove that $$ 2^n > n^3\quad \forall n\in \mathbb N, \;n>9.$$ Now that is actually very easy if we prove it for real numbers using calculus. But I need a proof that uses mathematical ...
5
votes
1answer
2k views

Liapunov's Inequality for $L_p$ spaces

Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
5
votes
1answer
1k views

Understanding the relationship of the $L^1$ norm to the total variation distance of probability measures, and the variance bound on it

I am trying to find a bound for variance of an arbitrary distribution $f_Y$ given a bound of a Kullback-Leiber divergence from a zero-mean Gaussian to $f_Y$, as I've explained in this related ...
4
votes
5answers
272 views

Simple AM-GM inequality

Let $a,b,c$ positive real numbers such that $a+b+c=3$, using only AM-GM inequalities show that $$ a+b+c \geq ab+bc+ca $$ I was able to prove that $$ \begin{align} a^2+b^2+c^2 ...
3
votes
3answers
368 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$? [duplicate]

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
3
votes
2answers
179 views

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
3
votes
1answer
555 views

Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$ [duplicate]

Is my proof correct? Prove: $a_n \leq b_n \implies \limsup a_n \leq \limsup b_n$ Proof: Let $a_n$ and $b_n$ be sequences such that $a_n \leq b_n \forall_n$. Suppose $\limsup a_n \nleq \limsup b_n$. ...
2
votes
2answers
463 views

Mean Value Theorem: $\frac{2}{\pi}<\frac{\sin x}{x}<1$

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
2
votes
0answers
75 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: ...
1
vote
2answers
208 views

Does this inequality hold true, in general?

Let $$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$ be the prime factorization of the positive integer $N$. Does the following inequality hold true in general? ...
1
vote
3answers
68 views

What is the maximum value of $\frac{2x}{x + 1} + \frac{x}{x - 1}$, if $x \in \mathbb{R}$ and $x > 1$?

What is the maximum value of $$f(x) = \frac{2x}{x + 1} + \frac{x}{x - 1},$$ if $x \in \mathbb{R}$ and $x > 1$? A 2-D plot of of $f$ for $x \in (\infty, \infty)$ is here. Lastly, note that ...
13
votes
3answers
1k 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 ...
9
votes
5answers
783 views

A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
8
votes
3answers
173 views

Math Induction Proof: $(1+\frac1n)^n < n$

So I have to prove: For each natural number greater than or equal to 3, $$(1+\frac1n)^n<n$$ My work: Basis step: $n=3$ $$\left(1+\frac13\right)^3<3$$ $$\left(\frac43\right)^3<3$$ ...
8
votes
4answers
530 views

Inequality with central binomial coefficients

For every even positive number $N$ we have: $$ {2N \choose N } < 2^N {N \choose N/2 } < 2 {2N \choose N } $$ (Furthermore, $\frac{2^N {N \choose N/2 }}{{2N \choose N }} \to \sqrt{2} $ for ...
6
votes
2answers
76 views

How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
6
votes
3answers
873 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.
5
votes
4answers
206 views

Prove that $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $

Prove that $$\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} \forall n \geq 6 $$ I'm trying induction, this is what I have so far: Basecase $(n=6): 64<720<729$ is true. Inductive Case: Assume ...
4
votes
1answer
197 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
4
votes
4answers
100 views

How can I show that $n! \leqslant (\frac{n+1}{2})^n$?

Show that $$n! \leqslant (\frac{n+1}{2})^n \quad \hbox{for all } n \in \mathbb{N}$$ I know that it can be done by induction but I always find line where I do not know what to do next.
4
votes
1answer
66 views

Simple upper bound for $\binom{n}{k}$

I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar ...
4
votes
8answers
3k views

Prove that $e^x\ge x+1$ for all real $x$ [duplicate]

Without using differentiation, logarithmic function, rigorously, prove that $$e^x\ge x+1$$ for all real values of $x$.
3
votes
4answers
189 views

Triangle Inequality with Complex Numbers

I was wondering how to prove the triangle inequality with complex numbers: Verify that the function $d(z_1, z_2)$ is a distance funtion on $\mathbb{C}$ and also on any subdomain on $\mathbb{C}$. I ...
3
votes
2answers
330 views

Prove that $x^2<\sin x \tan x$ as $x \to 0$ [duplicate]

$$x^2<\sin x \tan x \quad as \; x \to 0$$ I made the substitution $x \to \arctan x$ . $\arctan^2 x<x\sin (\arctan x)$ $\arctan x < \large \frac{x}{(x^2+1)^{\frac 14}}$ There are two ...
3
votes
3answers
150 views

Prove an inequality with a $\sin$ function: $\sin(x) > \frac2\pi x$ for $0<x<\frac\pi2$

$$\forall{x\in(0,\frac{\pi}{2})}\ \sin(x) > \frac{2}{\pi}x $$ I suppose that solving $ \sin x = \frac{2}{\pi}x $ is the top difficulty of this exercise, but I don't know how to think out such ...
3
votes
1answer
1k views

How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$ [duplicate]

The inequalities are: $$\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$$
2
votes
1answer
97 views

Proof of an inequality including logarithms

$0 <x,y \le 1$ $|y \log y-x\log x|\le|x-y|^{(1-\frac{1}{e})}$ I want to know how to solve this inequality.
2
votes
1answer
85 views

Will $\kappa_1, \kappa_2, m$ cardinals. Given $\kappa_1 \leq \kappa_2$. prove: $\kappa_1 \cdot m \leq \kappa_2 \cdot m$

Will $\kappa_1, \kappa_2, m$ cardinals. Given $\kappa_1 \leq \kappa_2$. prove: $\kappa_1 \cdot m \leq \kappa_2 \cdot m$. Hi, I would be happy if someone could help me with this. What I did until ...
2
votes
3answers
135 views

How would you prove $\sum_{i=1}^{n} (3/4^i) < 1$ by induction?

How would you prove this by induction? $$\sum_{i=1}^{n} \frac{3}{4^i} < 1 \quad \quad \forall n \geq 2$$ I can do the base case but don't know how to to finish it.
2
votes
2answers
591 views

How to prove $n < n!$ if $n > 2$ by induction?

I am stuck with the question below, Prove by mathematical induction that $n<n!$ for $n>2$.
0
votes
2answers
76 views

In $ \triangle ABC$ show that $ 1 \lt \cos A + \cos B + \cos C \le \frac 32$

Here is what I did, tell me whether I did correct or not: \begin{align*} y &= \cos A + \cos B + \cos C\\ y &= \cos A + 2\cos\left(\frac{B+C}2\right)\cos\left(\frac {B-C}2\right)\\ y &= ...
10
votes
5answers
1k views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
9
votes
0answers
168 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
8
votes
4answers
4k views

Proof of Bernoulli's inequality

The question reads $$U_n = (1+x)^n - 1 - nx$$ Show that $U_2 \geq 0$ Hence or otherwise show that $(1+x)^n \geq 1 + nx$ for all $x \gt -1$. Obviously the $U_2 \geq 0$ is very easy, I can do ...
7
votes
3answers
1k views

Prove that in an ordered field $(1+x)^n \ge 1 + nx + \frac{n(n-1)}{2}x^2$ for $x \ge 0$

In an ordered field show that $x \geq 0 \implies (1+x)^{n} \geq 1+nx+ \frac{1}{2}n(n-1)x^2$ for every positive integer $n$. I know that $(1+x)^{n} \geq 1+nx$ (Bernoulli's inequality). To get the ...
6
votes
2answers
488 views

A version of Hardy's inequality involving reciprocals.

How can one prove for any sequence of positive numbers $a_n, n\ge1,$ we have $$\sum_{n=1}^\infty \frac{n}{a_1+a_2+a_3+\cdots+a_n}\le 2\sum_{n=1}^\infty \frac{1}{a_n}$$ Added later: Apparently, ...
5
votes
2answers
568 views

How do I prove $(1 + \frac{1}{n})^n < n$ by mathematical induction?

$\displaystyle(1 + \frac{1}{n})^n < n$ for $n \gneq 3$ yes for $n = 1$ it is true I assume it is true for $n = k$ and get $\displaystyle(1+\frac{1}{k})^k < k$ I then go to $\displaystyle(1 ...