Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

4
votes
3answers
330 views

Proof that $\ln(n^2)(\ln(n) - 1) < n$ for all $n\in\mathbb{N}$

I would like to know which proof strategy to use when proving the next inequality: $\ln(n^2)(\ln(n) - 1) < n,\quad\forall n \in \mathbb{N}$. I have been trying to use this two proved inequalities $\...
21
votes
8answers
36k views

Is a prime factor of a number always less than its square root?

I was going through the fundamental theorem in Number Theory where any non zero integer n can be represented as a product of distinct primes. A related problem with this theorem is to prove that for ...
5
votes
1answer
124 views

Solving for Inequality $\frac{12}{2x-3}<1+2x$

I am trying to solve for the following inequality: $$\frac{12}{2x-3}<1+2x$$ In the given answer, $$\frac{12}{2x-3}-(1+2x)<0$$ $$\frac{-(2x+3)(2x-5)}{2x-3}<0 \rightarrow \textrm{ How do I ...
3
votes
2answers
182 views

Does $a_k,b_k>0$ imply $\left(\sum_{k=1}^n \frac{a_k}{n}\right)^2+\left(\sum_{k=1}^n \frac{b_k}{n}\right)^2\ge \prod_{k=1}^n(a_k^2+b_k^2)^{1/n}$?

Does $$a_k,b_k>0$$ imply that $$\left(\sum_{k=1}^n \frac{a_k}{n}\right)^2+\left(\sum_{k=1}^n \frac{b_k}{n}\right)^2\ge \prod_{k=1}^n(a_k^2+b_k^2)^{1/n}$$?
3
votes
3answers
213 views

How to prove this inequality?

If $\frac1{x}+\frac1{y}=1$, for all $ x>1 $ Then $mn \le \dfrac{ym^x+xn^y}{xy}$ with equality $m^x =n^y$, where $m,n \ge 0$. EDIT: Internet search results say it Hölder's Inequality or a ...
1
vote
1answer
725 views

Lattices - How to prove a simple inequality?

Lattices are kind of new to me and I'm not yet familiar with all of their properties so excuse me if what I'm asking here is extremely basic or easy. How can I prove the following inequality for a ...
9
votes
4answers
322 views

Showing $2(n+5)^2 < n^3$

I stumbled upon this in my homework and can't seem to get it, any help would be great. Find the smallest $n$ within $\mathbb N$ such that $2(n+5)^2 < n^3$ and call it $n_0$. Show that $2(n+5)^...
1
vote
1answer
83 views

Inequality problem

How might one show that $1\over 2m$$[\sigma_p^2+\mathbb E(p)^2]$ +$m\omega^2\over2$$[\sigma_x^2+\mathbb E(x)^2]$ $\geq \omega \sigma_x\sigma_p$? The immediate thing I think would help is $(\sigma_p-\...
5
votes
3answers
590 views

Does the Schur complement preserve the partial order?

Let $$\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$$ be symmetric positive definite and ...
14
votes
5answers
3k views

Olympiad Inequality Problem

Consider three positive reals $x,y,z$ such that $xyz=1$. How would one go about proving: $$\frac{x^5y^5}{x^2+y^2}+\frac{y^5z^5}{y^2+z^2}+\frac{x^5z^5}{x^2+z^2}\ge \frac{3}{2}$$ I really dont know ...
0
votes
2answers
99 views

Need help proving this linear inequality

I'm reading a paper that made a certain assumption as being trivial (I believe), but which I set out to prove. And now I'm kind of stuck. Let $I=[a,b]$ and $\bf{x}$ $= \{ x_1, x_2, \ldots, x_N \} \in ...
3
votes
1answer
443 views

Is $(x+1)^2 = (x+1)^3$ for any $x$?

This is something I've been struggling to understand since the past few days. Let's take an example: Prove/Disprove: $(x+1)^2 = (x+1)^3$ for all real values of $x$. Proof: Let us assume the ...
5
votes
2answers
403 views

$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$

May I know the standard proof technique to prove such kind of inequalities. $2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$ Thanks!
1
vote
3answers
1k views

Solving an exponential inequality

$$(0{,}25)^{3-0{,}5x^2}\leq8$$ Answers given are: $[-3;3]$ Below is where I got with this, I'm pretty sure I took a wrong approach here. Any help at all is appreciated. $$\begin{aligned} (0{,}25)^{...
-2
votes
1answer
1k views

Upper bound to lower bound

Let $x(a)\geq1$ and $y(b)\geq1$. I have a relation $x(a) \leq k(a,b)y(b)$ for all $k(a\geq b) \geq 1$ and $x(a)=y(b)$ when $k(a=b)=1$. Can we conclude that $x(a)\geq y(b)$ ?
1
vote
2answers
346 views

How to find the minimum value of $\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x}$?

Let $x,y,z\in [1,4]$ such that $x \geq y$ and $x \geq z$. Find the minimum value of this expression: $$ P=\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x} $$
2
votes
2answers
148 views

Prove that $\vert\sin(x)\vert + \vert\sin(x-1)\vert \ge \sin(1)$

While looking into the convergence of the series $\sum_{n=1}^{\infty}\frac{\sin(n)}{n}$ I stumbled into the inequality $\vert\sin(n)\vert + \vert\sin(n-1)\vert \ge \sin(1)$ for all $n\in\mathbb{R}$. (...
7
votes
2answers
205 views

Largest $k$ for which $a^k +b^k \leq c$

I have an inequality of the form $$a^k +b^k \leq c$$ with $a,b,c,k \in\mathbb{Z^+}$. For known $a,b,c$ I want to find out the largest $k$ for which this inequality holds. I am able to write a program ...
13
votes
4answers
2k views

Show that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$

I know about the fact that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$, where $A$ and $B$ are $m \times n$ matrices. But somehow, I don't find this as intuitive as ...
4
votes
8answers
5k views

Maximizing the sum of two numbers, the sum of whose squares is constant

How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal? This result is also true for more ...
10
votes
2answers
330 views

$\left(\frac{a^2+d^2}{a+d}\right)^3+\left(\frac{b^2+c^2}{b+c}\right)^3\geq\left(\frac{a+b}{2}\right)^3+\left(\frac{c+d}{2}\right)^3$

Prove : $$\left(\frac{a^2+d^2}{a+d}\right)^3+\left(\frac{b^2+c^2}{b+c}\right)^3\geq\left(\frac{a+b}{2}\right)^3+\left(\frac{c+d}{2}\right)^3$$ For $a,b,c,d>0$
3
votes
6answers
480 views

Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$

If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$. How to solve this one? (Here's the source of inspiration for the problem.)
4
votes
1answer
189 views

An inequality in $\mathbf R^d$

First let $z, w \in \mathbf R^d$ with $|z - w| < \min\{1, |z|^{-1}\}$. Further let $0 < t < r < \infty$. I wish to obtain an inequality of the form $$\exp \left (- \frac{|e^{-t^2} z - w|^...
10
votes
1answer
484 views

Proof of Hölder inequality by differentiation

I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure ...
13
votes
5answers
10k views

Solve by induction: $n!>(n/e)^n$

To Prove : $n! > (n/e)^n$ The question seems easy but it ain't; anyone up for it ?
11
votes
2answers
1k views

Beginner - Mathematical induction - help understanding example?

So: $$ (1+x)^n ≥ 1 + nx $$ So he checks for 1, and get: $$ 1+x ≥ 1+x $$ Next for variable k: $$ (1+x)^k ≥ 1 + kx $$ Then the book wanna prove: $$ (1+x)^{k+1} ≥ 1 + (k + 1)x $$ And here ...
27
votes
6answers
6k views

Elementary central binomial coefficient estimates

How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}<\binom{2n}{n}<\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ > 1 ? Does anyone know any better elementary estimates?
2
votes
3answers
145 views

Inequality system problem

Say there is a cat that in three days eat 12 fishes. Say that each day the cat eats more than the day before. Say that the last day the cat has eaten less than the addition of the two previous days. <...
5
votes
1answer
4k views

Minkowski's Inequality

I am wondering how to prove inequality $$\left\| \int f(.,y)dy \right\|_{p}\leq \int \left\| f(.,y) \right\|_{p}dy~~~~?$$ Here, $f$ is an integrable function on $\mathbb{R}^n$ and $\displaystyle \...
1
vote
2answers
203 views

Solve this Inequality

I am not sure how to solve this equation. Any ideas $$(1+n) + 1+(n-1) + 1+(n-2) + 1+(n-3) + 1+(n-4) + \cdots + 1+(n-n) \ge 1000$$ Assuming $1+n = a$ The equation can be made to looks like $$a+(a-1)+...
7
votes
3answers
174 views

Can I replace one inequality with another?

Suppose I have a collection of real numbers $x_b$ where $b \in \{1, ..., n\}$, and a constant $C$ with $1/n \le C \le 1$. Further suppose that for all $b$, $x_b \le C \sum_{a} x_a$ Does it follow ...
10
votes
5answers
7k views

Determining if a quadratic polynomial is always positive

Is there a quick and systematic method to find out if a quadratic polynomial is always positive or may have positive and negative or always negative for all values of its variables? Say, for the ...
3
votes
2answers
847 views

How to calculate volume given by inequalities?

I need to find the volume of the 3d space that is given by the following conditions: \begin{array}{c} 0 < x_1 < 1\\ 0 < x_2 < 1\\ 0 < x_3 < 1\\ x_1 + x_2 + x_3 < a. \end{array}...
5
votes
2answers
1k views

A proof for Landau inequality and similar cases

Landau inequality is about the bounds of derivatives of a real(complex)-valued function defined on some interval of the real line (I heard this from a lecture). I learned the simplest case is: $$ \| ...
2
votes
1answer
91 views

Is this inequality correct?

Let $a_1,a_2,b_1,b_2\in R$ such that $a_1b_1=-a_2b_2$. Is it correct that $$|\alpha_1a_1b_1+\alpha_2a_2b_2|\le 2\frac{\alpha_1-\alpha_2}{\alpha_1+\alpha_2}\frac{(\alpha_1a_1^2+\alpha_2a_2^2)(\...
3
votes
1answer
108 views

Bound on Derivatives

Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function with the following properties. $\phi(x) = 1$ if $|x| \leq 1$ $\phi(x) = 0$ if $|x| \geq 2$ $0 \leq \phi \leq 1$ $\phi$ is ...
10
votes
1answer
443 views

How to prove Halmos’s Inequality

How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from http://...
1
vote
1answer
247 views

$ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$ and $ b_1 = 1$, show that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}{n^2}$ [closed]

In the recursion $ b_{n + 1} = \frac {b_n^2 + 2b_n}{b_n^2 + 2b_n+2}$, with $ b_1 = 1,$ how can one prove that $ \left|\frac{2}{n}-\frac{2\ln{n}}{n^2}-b_n\right|\leq\frac{1}{n^2}$?
3
votes
2answers
683 views

Proof of cardinality inequality: $m_1\le m_2$, $k_1\le k_2$ implies $k_1m_1\le k_2m_2$

I have this homework question I am struggling with: Let k1,k2,m1,m2 be cardinalities. prove that if $${{m}_{1}}\le {{m}_{2}},{{k}_{1}}\le {{k}_{2}}$$ then $${{k}_{1}}{{m}_{1}}\le {{k}_{2}}{{m}_{2}}$$ ...
5
votes
3answers
636 views

In a acute angled triangle, we have $\tan(A)\cdot\tan(B)\cdot\tan(C) \geq 3\sqrt{3}$

How to show: In a acute angled $\triangle \ ABC$ show that $$\tan(A) \cdot \tan(B)\cdot \tan(C) \geq 3\sqrt{3}$$ Any ideas?
5
votes
3answers
255 views

How to prove the following inequality without expansion

$M^k \le 2^r < M^{k+1}$ where $M>1 , k>0$ for some $r$. It simply tells you that there exists a $2^r$ between $M^k$ and $M^{k+1}$. for example: if $M=3$, $k=1$ then $$M^k = 3, \quad M^{k+...
6
votes
4answers
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{{...
2
votes
2answers
183 views

How do I show $\|(e^{-2\pi ihx_j} - 1)/h \| \leq 2 \pi \|x\|$?

How do I show $\|(e^{-2\pi ihx} - 1)/h \| \leq 2 \pi \|x\|$? for each h and x? I thought about using taylor expansion of Euler's formula, but it did not work out. Thank you in advance.
2
votes
2answers
502 views

How to solve these inequalities?

How to solve these inequalities? If $a,b,c,d \gt 1$, prove that $8(abcd + 1) \gt (a+1)(b+1)(c+1)(d+1)$. Prove that $ \cfrac{(a+b)xy}{ay+bx} \lt \cfrac{ax+by}{a+b}$ Find the greatest value ...
15
votes
2answers
13k views

Prove that $\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a$

Let $a, b, c$ and $d$ be positive real numbers such that $a+b+c+d = 4$. Prove that $$\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a .$$ How can I approach this using only the AM - GM ...
13
votes
6answers
6k views

Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction

I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$ ...
4
votes
3answers
174 views

Different approaches for proving $a^a \times b^b \gt (\frac{a+b}{2})^{(a+b)}$,where $a \gt b \gt 0$

Proving $a^a \times b^b \gt (\frac{a+b}{2})^{(a+b)}$,where $a \gt b \gt 0$ One method that could be used here is using the inequality $$(1+x)^{(1+x)} \times (1-x)^{(1-x)} \gt 1$$ nothing is wrong ...
19
votes
2answers
7k views

Hölder's inequality with three functions

Let $p,q,r \in (1,\infty)$ with $1/p+1/q+1/r=1$. Prove that for every functions $f \in L^p(\mathbb{R})$, $g \in L^q(\mathbb{R})$,and $h \in L^r(\mathbb{R})$ $$\int_{\mathbb{R}} |fgh|\leq \|f\|_p\...
2
votes
1answer
275 views

How to prove $ - |x| \leqslant \sin x \leqslant |x|\quad,\quad\forall x \in \mathbb{R}$?

How to formally prove $$ - |x| \leqslant \sin x \leqslant |x|\quad,\quad\forall x \in \mathbb{R} \; ?$$ without using derivatives and graphs of real functions. Please, could anyone respond? Thank ...
1
vote
3answers
281 views

A “fast” way to ,find the maximum value of $(x^2) \times (y^3)$, if $3x+4y=12$ for $x,y \ge 0$

If $3x+4y=12$ $\forall x,y \ge 0$, the maximum value of $(x^2) \times (y^3)$ is $6 \times (6/5)^5$ $3 \times (6/5)^5$ $ (6/5)^5 $ $7 \times (6/5)^5$ How to approach this problem? I thought of ...