Questions on proving, manipulating and applying inequalities.

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2
votes
1answer
730 views

equality of triangle inequality

$z$ 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 ...
2
votes
5answers
364 views

Prove the triangle inequality [duplicate]

I want to porve the triangle inequality: $x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$ I figured out that probably the cases: $x\geq0$ and $y \geq 0$ $x<0$ and $y < 0$ $x\geq0$ ...
2
votes
3answers
965 views

Prove that $x - \frac{x^3}{3!} < \sin x < x$ for all $x>0$

Prove that $x - \frac{x^3}{3!} < \sin(x) < x$ for all $x>0$ This should be fairly straightforward but the proof seems to be alluding me. I want to show $x - \frac{x^3}{3!} < ...
2
votes
3answers
125 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
477 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 $n>2$.
1
vote
0answers
183 views

Does this hold?

Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $$ \min_{1\le i\ne j\le ...
1
vote
1answer
377 views

Maximum inequality

Can anyone show that ...
0
votes
1answer
81 views

$|x|^p+|y|^p\geq |x+y|^p$ for $0<p\leq 1$ [closed]

How to prove such inequality: $|x|^p+|y|^p\geq |x+y|^p$ for $0<p\leq 1$ and $x,y \in \mathbb{R}$?
49
votes
2answers
2k views

Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
22
votes
3answers
806 views

Proving that the triangle inequality holds for a metric on $\mathbb{C}$

Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as: $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} + \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$ I ...
38
votes
10answers
3k views

Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
42
votes
4answers
1k views

AM-GM-HM Triplets

I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
19
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 ...
18
votes
2answers
1k views

Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that:

Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that: $|a_{n}|+|a_{n-1} | \leqslant 2^{n-1}$. ...
18
votes
6answers
797 views

How to prove the inequality between mathematical expectations?

Let $X$ and $Y$ be independent random variables having the same distribution and the finite mathematical expectation. How to prove the inequality $$ E(|X-Y|) \le E(|X+Y|)?$$
15
votes
4answers
964 views

An Inequality problem relating $\prod\limits^n(1+a_i^2)$ and $\sum\limits^n a_i$

Let $(a_1,\space a_2,\space \cdots, \space a_n) \in \mathbb R^n_+$ such that $\displaystyle \prod^n_{i=1 }a_i = 1$. Prove that $$\displaystyle \prod^n_{i=1} (1+a_i^2) \le \cfrac ...
16
votes
3answers
1k views

Stuck trying to prove an inequality

I have been trying to prove (the left half of) the following inequality: $$ \underbrace{\sum_i \sum_j |x_i| \le \sum_i \sum_j |x_i + x_j|}_\textrm{?} \le 2 \sum_i \sum_j |x_i|$$ (All $x_i$s are ...
13
votes
2answers
379 views

Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.

For two positive integer sequences $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_m$ satisfying $x_i\neq x_j\quad \text{and}\quad y_i\neq y_j\quad \forall i,j, i \ne j$ ...
12
votes
2answers
695 views

Inequality. $\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3$

Let $a,b,c$ be positive numbers . Prove the following inequality: $$\sqrt{\frac{11a}{5a+6b}}+\sqrt{\frac{11b}{5b+6c}}+\sqrt{\frac{11c}{5c+6a}} \leq 3.$$ What I tried: I used ...
11
votes
1answer
624 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 ...
11
votes
8answers
2k views

$a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$

If a,b,c are real numbers satisfying $a+b+c =0; a^2+b^2+c^2=1$. Prove that $a^2 b^2 c^2 \le \frac{1}{54}$.
24
votes
4answers
1k views

Prove that $\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$

show that $$\sqrt{7}^{\sqrt{8}}>\sqrt{8}^{\sqrt{7}}$$ and I found $$LHs-RHS=0.017\cdots$$ I have post this interesting problem Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$ can someone suggest ...
21
votes
2answers
493 views

Trigonometric Inequality. $\sin{1}+\sin{2}+\ldots+\sin{n} <2$ .

How can I prove the following trigonometric inequality : $$\sin1+\sin2 +\ldots+\sin n <2$$ with $n \in \mathbb{N}^{*}$. The problem is that I don't know how to start this problem, I try to ...
14
votes
2answers
1k views

Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$

I'm interested in finding an elementary proof for the following sum inequality: $$\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$$ If this inequality is easy to prove, then one may easily prove that the sum ...
13
votes
2answers
192 views

$\forall x,y>0, x^x+y^y \geq x^y + y^x$

Prove that $\forall x,y>0, x^x+y^y \geq x^y + y^x$ A friend of mine told me none of the teachers in my school have succeeded in proving this seemingly simple inequality (it was asked at an ...
11
votes
6answers
1k views

Proving that $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$

How am I suppose to prove that: $$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{100}} < 20$$ Do I use the way like how we count $1+2+ \cdots+100$ to estimate? So $1/5050 \lt ...
9
votes
1answer
216 views

To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$

I found this question while browsing through "The Probabilistic Method", by Noga Elon. Let X and Y be 2 independent and identically distributed real valued random variables. Prove that: $$P(|X-Y| ...
7
votes
2answers
332 views

How prove this inequality $\sum_{cyc}\frac{a^2}{b(a^2-ab+b^2)}\ge\frac{9}{a+b+c}$

let $a,b,c>0$, show that $$\dfrac{a^2}{b(a^2-ab+b^2)}+\dfrac{b^2}{c(b^2-bc+c^2)}+\dfrac{c^2}{a(c^2-ca+a^2)}\ge\dfrac{9}{a+b+c}$$ My try: since this inequality is homogeneous ,without loss of ...
12
votes
8answers
791 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
10
votes
2answers
393 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
5
votes
1answer
154 views

How prove this number theory inequality $\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\frac{1}{q}$

show that: for any positive numbers $k$ and $N$, have $$\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\dfrac{1}{q}$$ where $\displaystyle\sum_{q\le N}$ is ...
15
votes
3answers
821 views

Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ ...
15
votes
3answers
1k views

Inequality. $\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab$

I want to prove the following inequality : $$\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab, $$ $a,b,c,d \in \mathbb{R}_{+} .$ In my book, at the answers chapter the author uses AM $\geq$ GM, ...
12
votes
7answers
560 views

Prove that $2^n < \binom{2n}{n} < 2^{2n}$

Prove that $2^n < \binom{2n}{n} < 2^{2n}$. This is proven easily enough by splitting it up into two parts and then proving each part by induction. First part: $2^n < \binom{2n}{n}$. The ...
11
votes
2answers
1k views

Relationship between diameter and radius of a point set

Consider a set of $n$ points in $\mathbb{R}^k$. The diameter of this set is the maximum distance between two of its points; its radius is the radius of the smallest (closed) k-ball that contains all ...
4
votes
1answer
329 views

Prove that: $\cos(x) -1 < -\frac{x^2}{2} + \frac{x^4}{24}$

Prove that: $$\cos(x) -1 < -\frac{x^2}{2} + \frac{x^4}{24}$$ for $x \ne 0$ I need to prove this using Cauchy's mean value theorem. What I did: $f(x) = \cos(x) -1$ $$g(x) = ...
3
votes
1answer
129 views

Show that $\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 … (2n-1)}{2\cdot 4\cdot 6 … (2n)} \le \frac1{\sqrt{n \pi}} $

Show that, if $n$ is a positive integer, $$\frac1{\sqrt{(n+\frac12) \pi}} \le\frac{1\cdot 3\cdot 5 ... (2n-1)}{2\cdot 4\cdot 6 ... (2n)} \le \frac1{\sqrt{n \pi}} . $$ This result is in a current ...
3
votes
3answers
193 views

Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$.

Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$. How to prove inequality $$ ab^2+bc^2+ca^2\le 4.\tag{*} $$ In other words, if $a,b,c$ are non-negative real numbers, then how to prove ...
14
votes
2answers
366 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 ...
12
votes
6answers
450 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
12
votes
2answers
322 views

Prove that $\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$

Let $a,b,c>0$ and $a^3+b^3+c^3=3$. Prove that $$\dfrac{a}{b^2+5}+ \dfrac{b}{c^2+5} + \dfrac{c}{a^2+5} \le \dfrac 12$$ I have an ugly solution for this solution.
10
votes
3answers
396 views

Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$?

Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?
6
votes
1answer
126 views

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

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 ...
6
votes
1answer
159 views

How prove $\frac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\frac{1}{t^2}<e$

Let $k,n\in \mathbb{N},n\ge k$, prove that $$\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\dfrac{1}{t^2}<e.$$ I got the impression that this inequality is very sharp. My idea: ...
5
votes
2answers
65 views

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$

Prove the inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$ Where $a,b,c$ are sides of a triangle. It is clear that $c+a-b$ is positive but how to use it?
4
votes
4answers
196 views

Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+…+\frac1{2^n - 1} \le n$

$\dfrac{n}{2} \le \dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n - 1} \le n $ I've Tried for hours but didn't got any striking idea. I don't have any efforts to show rather than induction. ...
4
votes
2answers
170 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
3
votes
4answers
7k 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 ...
1
vote
1answer
75 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...
18
votes
3answers
555 views

An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$

The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $$ B_n^2 \leq B_{n-1}B_{n+1} $$ for $n \geq ...