Questions on proving, manipulating and applying inequalities.

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2
votes
3answers
60 views

Proving $n \lt 2^n$ for $n\geq 1$ using induction

Very close to understanding this, hopefully. Via induction, I'm following a proof but can't understand one of the last steps. Claim: $n < 2^n$ for natural numbers $n = 1, 2, 3,\ldots$ For step ...
0
votes
0answers
65 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle ...
1
vote
1answer
29 views

The AM-GM inequality with rational weights

This is Problem 2.2 of Steele's "The Cauchy-Schwarz Master Class": Suppose that $p_1, \ldots, p_n$ are nonnegative rationals whose sum is $1$. Show that for any real numbers $a_1, \ldots, a_n$, one ...
0
votes
3answers
64 views

Induction Inequality with Summation [closed]

I can't seem to figure out this problem. Do you have any ideas? For an integer $n > 1$, show that $$ \sum_{k=1}^n {1\over \sqrt{{n^2}+k}} > {{\sqrt{1+{1\over n}}}\over 2} $$
2
votes
1answer
58 views

Elementary inequality? [duplicate]

Let $x,y>0$, it seems (with numerical simulations) that $x^x+y^y \geq x^y +y^x$. If this is true, it has to be well known by some people. Does this inequality have a name? several proofs?
1
vote
3answers
51 views

Show $|\exp(-x/2) - \exp(-y/2)| \leq |x-y|/4$ for $x,y\geq 0$. [closed]

I am trying to show this inequality: $$ \left|e^{-x/2} - e^{-y/2}\right| \leq \frac{|x-y|}{4} $$ for $x,y\geq 0$. I've gotten stuck and could use some kind assistance. Many thanks in advanced!
-1
votes
3answers
63 views

Prove the inequality by induction [duplicate]

Prove the inequality by induction: $3^n > n^3\ $ for $\ n \geq 4$ Edit: 1) Base case: $n=4$, $3^4>4^3, 81>64$ 2) Assume true for n=k: so $3^k>k^3$ 3) Consider $(k+1)^3$, $(k+1)^3 = ...
0
votes
1answer
31 views

Solution of linear inequality

I have the following system of linear inequality on $x_1, x_2, \dots, x_n$, $x_i \in \mathbb{R} \; \forall i$ $x_i - 2x_j < b \; \forall i, j$ The right hand side of the inequality ($b \in ...
2
votes
0answers
24 views

Schwarz Inequality of function from upper half plane to disc

So I've been working on this problem and I have everything nailed down (I think) except for the very end. In particular I get a bound, but I can't seem to reduce it down to the one the question is ...
9
votes
2answers
141 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$

Edit: This problem 1 is a 2014 Sydney mathematics competition problem (8th grade). It seems difficult to solve. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such ...
1
vote
4answers
64 views

Prove that $\sum_{i=1}^{i=n} \frac{1}{i(n+1-i)} \le1$

$$f(n)=\sum_{i=1}^{i=n} \dfrac{1}{i(n+1-i)} \le 1$$ For example, we have $f(3)=\dfrac{1}{1\cdot3}+\dfrac{1}{2\cdot2}+\dfrac{1}{3\cdot1}=\dfrac{11}{12}\lt 1$ If true, it can be used to prove: ...
0
votes
1answer
28 views

Inequality about Frobenius norms for matrices [closed]

For any square matrix A, but not necessarily symmetric, what are some ways to prove the inequality $$ \|A^2\|_F^2\leq\|A^TA\|_F^2, $$ where $\|B\|_F^2=tr(B^TB)$ is the Frobenius norm of matrix $B$ ?
1
vote
4answers
155 views

Solve: $-\frac{1}{\sqrt{2}} < \sin \theta + \cos \theta < \frac{1}{\sqrt{2}}$

The question is: Solve $$-\frac{1}{\sqrt{2}} \lt \sin\theta + \cos\theta < \frac{1}{\sqrt{2}}$$ for values of $\theta$ between $0^\circ$ and $180^\circ$. I realized that: $$\begin{align} ...
7
votes
0answers
80 views

A matrix with a dense submatrix - application of Chernoff’s Inequality

I am trying to solve an exercise from this book, which I will post here for convenience. I have a bit of a problem understanding how the hint of using Chernoff's bound implies the claim. Specifically ...
5
votes
2answers
70 views

Proving $x \ln^2 x - (x-1)^2<0$ for all $x\in(0,1)$

I want to prove that for all $x\in(0,1)$,$$f(x):=x \ln^2 x - (x-1)^2<0$$ Using the derivative ($f'(x)=-2x+\ln^2 x+2\ln x +2$), I tried to prove that $f$ is monotonically increasing in $(0,1)$, and ...
2
votes
1answer
38 views

Integral inequality similar to Hardy's

I am trying to solve following puzzle: We are given functions $f$, where $f(x) > 0$ and $F := \int_0^x f(t) dt$ and some real $p>1$. Does $\int_0^\infty f(x)^p e^{-x}dx < \infty$ imply ...
0
votes
1answer
57 views

Area of the figure within the circle and outside a polygon

For which values of the parameter $c \in \mathbb{R}$, the area $S$ of the figure $F$, consisting of the points $(x,y)$ such that $$\begin{gathered} \max \{ \left| x \right|,y\} \geqslant 2c \hfill ...
16
votes
1answer
349 views

On the inequality $\frac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
1
vote
1answer
23 views

Prove that $ m_{a} \geq m_{g} \geq m_{h} $ using strict inequalities unless $ a = b $.

$ m_{a} = \frac{1}{2} (a + b) $ $ m_{g} = \sqrt{ab} $ $ \frac{1}{m_{h}} = \frac{1}{2}(\frac{1}{a} + \frac{1}{b})$ Attempted Solution: I believe I have shown the first step, which is $ m_{a} ...
1
vote
4answers
54 views

Prove that If $0<x<\ln 2$ then $x^2\geq e^x-x-1$

If $0<x<$ln $2$ then $x^2\geq e^x-x-1$ I got this problem while reading a proof. Tried to prove it but failed. $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$. So $e^x\geq 1+x$ for all $x$ but ...
1
vote
6answers
72 views

Given $a\leq b \leq c \leq d $ .How to prove $ c - b \leq d-a$

Given $a\leq b \leq c \leq d $ .How to prove $ c - b \leq d-a$. I am facing this inequality again and again in Riemann integration ,but i couldnot figure it out Thanks
0
votes
1answer
63 views

An inequality: $(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)$

Question: Find the smallest possible integer value of $n$, for which the following inequality holds true:- $$(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)$$ Where $x,y,z \in R $. I used the Cauchy inequality ...
0
votes
4answers
50 views

Find the minimum value

If $a,b,c,d$ are positive real numbers and $abcd=1$ then Find the minimum value of $(4+a)(4+b)(4+c)(4+d)$. Find the condition when minimium value holds. I've used AM-GM Inequality $4+a \ge 2 ...
1
vote
1answer
89 views

What is the non-inductive proof of this inequality? [duplicate]

$$\dfrac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} < \dfrac{1}{\sqrt{3n+1}}.$$ However I've non-inductive proof of $\dfrac{1 \cdot 3 \cdot 5 \cdots(2n-1)}{2 \cdot 4 \cdot 6 ...
3
votes
1answer
78 views

Inclusion - exclusion-like inequality

Let $(\Omega,\mathcal{F},\mathbb{P})$ be some probabilistic space and $A_1,\ldots,A_n\in \mathcal{F}$. Is it true that: $$\sum\limits_{i=1}^{n} \mathbb{P}(A_i)^2 - \sum\limits_{1\le i<j\le ...
0
votes
0answers
41 views

Behaviour of the sum of divisors function via logarithmic means versus an elementary problem equivalent to the Riemann Hypothesis due to Lagarias

It is known the following (see [1], here is an open access in his homepage www.math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the ...
1
vote
3answers
79 views

Prove that $n^a < a^n$ for $a>1$ and $n$ big enough

How can I solve this? I'm trying to prove using limits but it's not working.. Thanks
5
votes
4answers
126 views

$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{2}{3}$

This is from the book Problems in Mathematical Analysis I by Kaczor and Nowak: Show that, for $n\in \mathbb{N}$, $$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{2}{3}$$ The solution in the ...
1
vote
3answers
37 views

Inequalities Theorem Proof

While studying, I read that the solutions of $f(x)/g(x)<0$ is equal to the solutions of $f(x)*g(x)<0$. I don't get why this is so. Could someone explain the reason the solutions to these two ...
1
vote
4answers
32 views

how to solve inequality using logarithm

I was given the following expression :$$(0.87)^n\leq 0.1$$ And the next step was: $$n\geq \frac{log(0.1)}{log(0.87)}$$ What was the steps betweens?
2
votes
3answers
121 views

How can I find the minimum value for $F(x,y,z,w)=x^2+y^2+z^2+w^2+xy+zw-xz-yw-yz$

Let $x,y,z,w$ be integer numbers,and $xw=yz+1$ Find this minimum of the value $$x^2+y^2+z^2+w^2+xy+zw-xz-yw-yz$$ This is how did it and I would like to know if I made a mistake Let ...
9
votes
2answers
108 views

If $x+y+z=3$, then $\sum_{\text{cyc}}\frac{x^2}{2y^2-y+3}\ge\frac{3}{4}$

Let $x,y,z>0$, be such that $x+y+z=3$. Show that $$\dfrac{x^2}{2y^2-y+3}+\dfrac{y^2}{2z^2-z+3}+\dfrac{z^2}{2x^2-x+3}\ge\dfrac{3}{4}.$$ I've tried many things but all have failed. ...
0
votes
1answer
33 views

Bound for non-integer power of sum

Let $x > 1$, $y \in (0,1)$ and $z \in (0,1)$. I need to bound $$(x+y)^z - x^z \leq B_z(x)$$ where I guess something like $B_z(x) \approx x^{z-1}$. Is there anything known on these non-integer ...
2
votes
1answer
37 views

Parabolic range conditions proof

This problem is getting the better of me, since I have no idea where to start: The equation of a curve is $y=ax^2-2bx+c$, where a, b and c are constants with $a>0$. Given that the vertex of the ...
0
votes
1answer
22 views

How does $|g(z)-k|<1/2|k|\Rightarrow |g(z)|>1/2|k|?$

I'm reading Ian Stewart's Complex Analysis, page 27, where he proves $\lim_{z\rightarrow z_0}(1/g(z))=1/k$, and he has the following step: $$z\in ...
3
votes
3answers
97 views

If $\frac{a}{b}<\frac{c}{d}$ and $\frac{e}{f}<\frac{g}{h}$, then $\frac{a+e}{b+f} < \frac{c+g}{d+h}$.

If $\frac{a}{b}<\frac{c}{d}$ and $\frac{e}{f}<\frac{g}{h}$ where $b+f>d+h$, then $\frac{a+e}{b+f} < \frac{c+g}{d+h}$. I thought it is easy to prove. But I could not. How to prove this? ...
0
votes
0answers
12 views

Inequality generator?

Is there any computer software for generating new Inequalities? maybe something to be called computer-aided-problem-design or something like this? for example suggesting a function in three variables ...
0
votes
2answers
63 views

If $d_1(x,y)$ and $d_2(x,y)$ are metrics, prove that $d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$ is a metric.

$$d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$$ The first three properties are trivially proven. The triangle inequality, not so much. I tried using the triangle inequalities that apply to $d_1$ and $d_2$, ...
0
votes
2answers
36 views

Explanation for 2 inequalities with same solutions.

While studying, I read that $(x-a)^{2k \pm 1}f(x)>0$ has the same solution as $(x-a)f(x)>0$. I do not get why this is true. Could someone please explain why these two inequalities have the same ...
1
vote
2answers
23 views

Inequalities for f(x) is always positive

Given that $f(x)=4x^2-1$ Find the range of values of $x$ so that $f(x)$ is always positive. My attempt, $4x^2-1>0$ $4x^2>1$ $x^2>\frac{1}{4}$ $x>\pm\frac{1}{2}$ So ...
7
votes
3answers
148 views

Show that $p \in \left[\frac{4^m}{2\sqrt{m}},\frac{4^m}{\sqrt{2m+1}}\right]$

If the number of ways in which $m$ identical apples can be put in $2m$ boxes, so that no box contains more than one apple, is $p$, prove that $$p \in ...
9
votes
6answers
521 views

How to prove that $7^{31} > 8^{29}$

How can I prove that $7^{31}$ is bigger than $8^{29}$? I tried to write exponents as multiplication, $2\cdot 15 + 1$, and $2\cdot 14+1$, then to write this inequality as $7^{2\cdot 15}\cdot 7 > ...
0
votes
2answers
74 views

Solving this inequality with integral

We have function $f:\mathbb{R}-\{2 \}\to\mathbb{R}$ $$f(x)=\frac{x^2}{x-2}$$ Show that $8\le\int\limits _3^4f\left(x\right)dx\le9$ I solved the definite integral and got $\int\limits ...
1
vote
1answer
23 views

Bound of the Complex Expression

Here, $x$, $y$ and $\alpha$ are all complex numbers such that $|x|<\epsilon$ and $|y|<\epsilon$. Now what would be upper bound of the following expression: $|\frac{\alpha+y}{x y - \alpha}|$? ...
0
votes
2answers
41 views

Proving Lower bounds on an Approximately Linear Function

We are looking for a lower bound on the function, $\frac{1.31}{e^{\frac{1.31}{x+1}} - 1}$ for $x \geq 2$. This function seems to behave linearly. We believe that the following statement holds: ...
0
votes
4answers
34 views

If $d(x,y)$ is a metric, how does the following inequality apply?

I'm interested if someone can formally type out why this is. I thought it was trivial, but the professor wanted a more detailed explanation: $${d(x,y)\over {1+d(x,y)}}\leq ...
1
vote
2answers
37 views

Squaring both sides of an inequality: attempt to prove a general rule

I have attempted to produce a proof of the intuitive rule for squaring inequalities, according to which, given any two numbers x and y and regardless of their sign, 1) if |x| < |y| then ...
3
votes
3answers
47 views

Let $A \subset \mathbb Z^3$ / $|A| < \infty$. Prove that: $|A| \le \sqrt{|A_x| |A_y| |A_z|}$

Here is the problem statement word by word: $1)$ Prove that if $a_{ij}$, $b_{jk}$ and $c_{ki}$ are non-negative reals with $1 \le i,j,k \le n$, then: $$\sum_{i,j,k = 1}^n \sqrt{a_{ij} \times ...
0
votes
4answers
58 views

Show $ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$

I am tasked with proving the following limit: $$ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$$ using the definition of the limit. I think I have done so correctly. I was hoping to have someone ...
3
votes
2answers
62 views

this inequality $\prod_{cyc} (x^2+x+1)\ge 9\sum_{cyc} xy$

Let $x,y,z\in R$,and $x+y+z=3$ show that: $$(x^2+x+1)(y^2+y+1)(z^2+z+1)\ge 9(xy+yz+xz)$$ Things I have tried so far:$$9(xy+yz+xz)\le 3(x+y+z)^2=27$$ so it suffices to prove that ...