Questions on proving, manipulating and applying inequalities.

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20
votes
4answers
1k views

Prove $\sqrt{x^2+yz+2}+\sqrt{y^2+zx+2}+\sqrt{z^2+xy+2}\ge 6$, given $x+y+z=3$ and $x,y,z\ge0$

Let $x+y+z=3,x,y,z\ge 0$,show that $$\sqrt{x^2+yz+2}+\sqrt{y^2+zx+2}+\sqrt{z^2+xy+2}\ge 6$$ Additional information I have seen the following problem: $x,y,z>0,x+y+z=3$, prove that $$\sqrt{x^...
6
votes
3answers
1k views

Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$

Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that: $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} \ge{\frac{{...
16
votes
6answers
3k views

If $a > b$, is $a^2 > b^2$?

Given $a > b$, where $a,b ∈ ℝ$, is it always true that $a^2 > b^2$?
11
votes
1answer
669 views

How to prove $\frac{1}{4}(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a})\ge \sqrt[4]{\frac{a^4+b^4+c^4+d^4}{4}}$

Let $a,b,c,d>0$, show that $$\dfrac{1}{4}\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)\ge \sqrt[4]{\dfrac{a^4+b^4+c^4+d^4}{4}}$$ I know this is interesting inequality,...
13
votes
8answers
910 views

prove $\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$ by mathematical induction

How to prove $$\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$$ by Mathematical induction,n$\ge $1
25
votes
2answers
858 views

How to prove this inequality(7)?

let $x,y,z\in\mathbb{R}$, prove that $$4(x^6+y^6+z^6)+5(x^5y+y^5z+z^5x)\ge\dfrac{(x+y+z)^6}{27}$$ I do this sometimes, and I think this problem,is very hard,I hope someone can solve.Thank you By ...
9
votes
1answer
188 views

Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$. [duplicate]

I proved this inequality in the following way: Lemma: $r \in \Bbb N, r \geq 3$. We have $r^r \gt (r+1)^{r-1}$. Proof: We apply the AM-GM inequality to the $r$ positive integers where there are $r-1$...
4
votes
6answers
201 views

Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$ [duplicate]

Having the following inequality $$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$$ To prove it for all natural numbers is it enough to show that: $\frac{1}{(n+1)^2}-\frac{1}{n^2}...
12
votes
2answers
6k 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 ...
11
votes
6answers
10k views

How to prove $\log n < n$?

Sorry if this is a silly question but most books claim $\log n < n$ for $n \geq 1$ without providing any proof, saying it's too obvious. Could someone give me a rigorous proof? Is there some trick ...
10
votes
2answers
1k 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}} + \...
9
votes
5answers
588 views

Improving bound on $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}}$

An old challenge problem I saw asked to prove that $\sqrt{2 \sqrt{3 \sqrt{4 \ldots}}} < 3$. A simple calculation shows the actual value seems to be around $2.8$, which is pretty close to $3$ but ...
8
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?
5
votes
1answer
2k views

Minkowski's inequality

Minkowski's inequality says the following: For every sequence of scalars $a = (a_i)$ and $b = (b_i)$, and for $1 \leq p \leq \infty$ we have: $||a+b||_{p} \leq ||a||_{p}+ ||b||_{p}$. Note that $||x||_{...
2
votes
1answer
189 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) $\Sigma \frac{a_n}{1+a_n}$ ...
12
votes
6answers
464 views

How do I prove that $3<\pi<4$?

Let's not invoke the polynomial expansion of $\arctan$ function. I remember I saw somewhere here a very simple proof showing that $3<\pi<4$ but I don't remember where I saw it.. (I remember ...
7
votes
4answers
563 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, ...
7
votes
5answers
289 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 \...
5
votes
1answer
1k views

Equality of triangle inequality in complex numbers

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
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 $$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$ ...
4
votes
4answers
261 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
14k 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 ...
13
votes
2answers
5k views

Various proofs of Hardy's inequality

For any $p > 1$ and for any sequence $\{a_j\}_{j=1}^\infty$ of nonnegative numbers, a classical inequality of Hardy states that $$ \sum\limits_{k=1}^n\left(\frac{\sum_{i=1}^ka_i}{k}\right)^p\le \...
12
votes
3answers
275 views

When does a variable leave a basis (in linear programming)?

In the simplex algorithm in linear programming, what are conditions for a variable to leave a basis (not necessarily basis for the/an optimal solution)? I'm supposed to list as many sufficient and ...
10
votes
5answers
7k views

Greatest prime factor of $n$ is less than square root of $n$, proof

I remember reading this somewhere but I cannot locate the proof.
9
votes
5answers
8k views

Negating A Mathematical Statement

Regard this statement $ x \ge 0$. According to my teacher, by negating this statement, it will become $ x < 0$. Why is this so; why does the $\ge$ morph into $<$, and not into $\le$?
4
votes
5answers
285 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 &=\frac{a^2+b^2}{2}+\...
2
votes
3answers
460 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 $a,b&...
2
votes
2answers
490 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 ...
1
vote
1answer
82 views

How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?

Assume that you have any two real numbers $a$ and $b$, and $1\leq k <\infty$, $k \in \mathbb{R}$. How would you prove the inequality $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
9
votes
5answers
806 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.
9
votes
0answers
192 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 $$n!...
8
votes
3answers
176 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: $n=3$ $$\left(1+\frac13\right)^3<3$$ $$\left(\frac43\right)^3<3$$ $$\...
7
votes
5answers
665 views

Is this direct proof of an inequality wrong?

My professor graded my proof as a zero, and I'm having a hard time seeing why it would be graded as such. Either he made a mistake while grading or I'm lacking in my understanding. Hopefully someone ...
7
votes
2answers
459 views

Prove the general arithmetic-geometric mean inequality

Prove that the general arithmetic-geometric mean inequality \begin{equation*} (a_{1}a_{2}...a_{n})^\frac{1}{n}\leq\frac{a_{1}+a_{2}+...+a_{n}}{n} \end{equation*} holds for all $a_{i}$ positive real ...
6
votes
3answers
2k views

Known proofs of Wirtinger's Inequality?

I am looking for proofs of the (Poincare-) Wirtinger inequality which states that if $f:[0,\pi]\to \mathbb{C}$ is $C^1$ and $f(0)=f(\pi)=0$ then \begin{equation} \int_0^\pi |f(t)|^2 dt \leq \int_0^\pi ...
4
votes
8answers
4k 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 $x$.
4
votes
4answers
103 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.
3
votes
4answers
301 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
1answer
601 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 $\limsup a_n \nleq \limsup b_n$. ...
3
votes
1answer
212 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
3answers
155 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
3answers
143 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
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 $n > 1$, $x \ge 1$: $$\frac{1}{n}\...
2
votes
1answer
95 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 now:...
2
votes
4answers
3k views

For what natural numbers is $n^3 < 2^n$? Prove by induction

Problem For what natural numbers is $n^3 < 2^n$? Attempt @ Solution For $n=1$, $1 < 2$ Suppose $n^3 < 2^n$ for some $n = k \ge 1$ It looks like the inequality is true for $n = 0$, $n = 1$ ...
2
votes
2answers
254 views

If $a,b,c,d,e,f$ are non negative real numbers such that $a+b+c+d+e+f=1$, then find maximum value of $ab+bc+cd+de+ef$

$(a+b+c+d+e+f)^2=$ sum of square of each number (X)+ $2($ sum of product of two numbers (Y) $)$ $ab+bc+cd+de+ef \le Y$ since all are positive. Therefore $1\ge X+(ab+bc+cd+de+ef)$ Edit: From AM GM ...
1
vote
2answers
210 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? $$\prod_{i=1}^{\omega(N)}{\frac{...
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 ...