Questions on proving and manipulating inequalities.

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2
votes
1answer
61 views

Irrational number not ocurring in the period of rational numbers

Write each rational number from $(0,1]$ as a fraction $a/b$ with $\gcd(a,b)=1$, and cover $a/b $ with the interval $$ \left[\frac ab-\frac 1{4b^2}, \frac ab + \frac 1{4b^2}\right]. $$ Prove that the ...
2
votes
1answer
47 views

Hidden Cauchy-Schwartz inequality

If $x_{i}\gt 0$ and $x_{i}y_{i}-{z_{i}}^2\gt 0$ for $i\le n$, then prove that $$\frac{n^3}{(\sum_{i=1}^nx_{i})(\sum_{i=1}^ny_{i})-(\sum_{i=1}^n{z_{i}}^2)} \le \sum_{i=1}^n\frac 1{x_{i}y_{i}-z_{i}^2}$$ ...
1
vote
1answer
30 views

Proving inequality contrary to Cauchy Schwartz using additional term

Let $0 \lt p\le a,b,c,d,e\le q$. Prove that $$(a+b+c+d+e)\left(\frac 1a + \frac 1b +\frac 1c+\frac 1d+\frac 1e\right)\le 25+6\left(\sqrt {\frac pq}-\sqrt {\frac qp}\right)^2 $$ In the solution they ...
1
vote
1answer
22 views

Confusion regarding derivation of triangle inequality from Schwarz' inequality

I was going through the proof of triangle inequality as a consequence of Schwarz inequality here: http://en.wikipedia.org/wiki/Triangle_inequality#Example_norms I find somethinng odd in the third ...
1
vote
1answer
39 views

a simple question about inequality [on hold]

Is n > $\sqrt {n+1} * \sqrt {n+2}$? Well, I am pretty sure it is but how I can prove it. I need it for determining a limit by showing it is a zero sequence.
4
votes
2answers
85 views

Inequality involving $\frac{\sin x}{x}$

Can anybody explain me, why the following inequality is true? $$\sum_{k=0}^{\infty} \int_{k \pi + \frac{\pi}{4}}^{(k+1)\pi-\frac{\pi}{4}} \left| \frac{\sin \xi}{\xi} \right| \, \text{d} \xi \geq ...
-2
votes
3answers
78 views

Comparison of the integral $\int_{0}^{10000} x^{1/2} \,dx$ and the sum $\sqrt{1} + \sqrt{2}+ \cdots+\sqrt{10000}$ [on hold]

I am stuck in this problem Let $S = \sqrt{1} + \sqrt{2}+ \cdots \sqrt{10000}$ and $$I =\int_{0}^{10000} x^{1/2} \,dx$$ Show that $I \leq S \leq I +100$
7
votes
1answer
85 views

Arithmetic mean, geometric mean, and vieta's formulas

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. A while ago I noticed that if you form the polynomial $$ P(x) = (x - a_1)(x-a_2) \cdots (x-a_n) $$ then: The arithmetic mean of $a_1, \ldots, ...
7
votes
1answer
105 views

Inequality $\frac{1}{1-abc} + \frac{1}{1-bcd} + \frac{1}{1-cda} + \frac{1}{1-dab} \le \frac{32}{7}$

If $a,b,c,d$ are positive real numbers such that $a^2+b^2+c^2+d^2 = 1$, Prove that: $$\frac{1}{1-abc} + \frac{1}{1-bcd} + \frac{1}{1-cda} + \frac{1}{1-dab} \le \dfrac{32}{7}$$ I saw this problem is ...
2
votes
3answers
113 views

Show without differentiation that $\frac {\ln{n}}{\sqrt{n+1}}$ is decreasing

Show that the function $\displaystyle \frac {\ln{n}}{\sqrt{n+1}}$ is decreasing from some $n_0$ My try: $\displaystyle a_{n+1}=\frac{\ln{(n+1)}}{\sqrt{n+2}}\le ...
2
votes
2answers
75 views

Show without derivative that function $\frac{\ln{n}}{ n\ln{\ln{n}}}$ is decreasing

I have a problem with showing the function $\displaystyle \frac{\ln{n}}{n \ln{\ln{n}}}$ is decreasing. I came to form $(n+1)^{\ln{\ln{n}}}<(n)^{\ln{\ln{(n+1)}}}$ and I don't know how to show that ...
0
votes
0answers
63 views

If this inequality involving prime numbers holds for $n$ larger than some $N$, what is an upper bound to $N$?

Let $p_n$ be the $n$-th prime and set $k_1=10;k_2=6;k_3,k_4=4;k_5,k_6=3;k_7,\ldots ,k_{n+1}=2$. For large enough $n$, prove or disprove that ...
4
votes
0answers
32 views

An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...
0
votes
2answers
88 views

If a,b,c are the 3 edges of a triangle then prove that $2<\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}-\frac{a^3+b^3+c^3}{abc}≤3$?

What I found, that is Since the sum of any 2 sides of a triangle is greater than the third if follows that $a + b > c$ or $\dfrac{a+b}{c} > 1$ and so ...
1
vote
1answer
30 views

How to find out if any point of a 2d line segment satisfies a system of linear inequations of degree 1 with one parameter?

Non formally, I want to know how to find out if a line segment intersects the area bounded by three axis-parallel lines. Line segment is not necessarily axis-parallel. I could always bruteforce a ...
1
vote
1answer
42 views

$0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$

this is related to that one the limits of $a_n $and $b_n$ Let for $n\geq 2\quad a_{n}=\prod\limits_{k=2}^{n}\cos\left(\dfrac{\pi }{2^{k}}\right)$ and $b_{n}=a_{n}\cos\left(\dfrac{\pi ...
0
votes
0answers
44 views

Inequality involving algebra

$S = A1 + A2 + ... + Ak + ... + An$ where $1 > A1> A2> ... > Ak> ... > An> 0, n >=3$ Assume that $k < S < k + 1 $ I want to make sum S between -1 and +1. I can do it by ...
1
vote
2answers
36 views

How I can solve this inequality $(R\gamma^T(1/(1−\gamma))≤\epsilon)$ for $T$?!

Can someone solve this inequality for $T$ $$ R\gamma^T(1/(1−\gamma))≤\epsilon $$ In a paper it solved for $T$ and the inequation below is the result, but I can not prove how the inequation above can ...
2
votes
2answers
56 views

Find a limit using mean value theorem

Using Mean Value Theorem how could I find the limit $$\lim_{x\to\infty}\frac{f(x+2)-f(x)}{x}$$ where $|f^\prime(x)|\leq\sqrt{x}$ for all $x.$
1
vote
1answer
33 views

Basic AM,GM,HM Inequality problem

Prove that $a/(a-b+c) + b/(a-c+b) +c/(b+c-a) \leq 3$ if $a,b,c$ are positive integers using AM-GM-HM or otherwise .
4
votes
1answer
75 views

Strange algebraic inequality

Let $x, y, z$ be real numbers such that $-1< x + y + z < 1$ and $x^2 + y^2 + z^2 < 1$. Prove the inequality or give a counter example: $$(x^2 + 2yz)^2 + (y^2 + 2xz)^2 + (z^2 + 2xy)^2 < ...
3
votes
1answer
89 views

prove that $a^2 b^2 (a^2 + b^2 - 2) \ge (a + b)(ab - 1)$

Good morning help me to show the following inequality for all $a$, $b$ two positive real numbers $$a^2 b^2 (a^2 + b^2 - 2) \ge (a + b)(ab - 1)$$ thanks you
1
vote
0answers
50 views

Proving that $ \prod_{i=1}^{n} (1 + a_{i}) \ge \left(1 + \prod_{i=1}^{n} a_{i}^{1/n} \right)^{n} $ [duplicate]

I'd like some help on proving the following inequality $$ \prod_{i=1}^{n}\left(1+a_{i}\right) \ge \bigg(1 + \prod_{i=1}^{n} a_{i}^{1/n}\bigg)^{n}, $$ given that $ a_{i} > 0\,\, \forall\, ...
11
votes
0answers
110 views

How prove this integral inequality $\int_{0}^{1}(f(x)+g(x))dx\ge\int_{0}^{1}f(g(x))dx$? [duplicate]

let two function $f(x),g(x):[0,1]\to [0,1]$ are Continuous function,and $f(x)$ strictly monotone increasing, show that $$\int_{0}^{1}(f(x)+g(x))dx\ge\int_{0}^{1}f(g(x))dx$$ I am not ...
2
votes
3answers
26 views

Minimum value of a differentiable function at some point

Let $f(x)$ be differentiable for all $x\in \mathbb{R}$ and let $f(0)=2$ and $f^\prime(x)\leq -2$. How could i find the minimum value of $f(-1).$
2
votes
0answers
59 views

Looking for an existing proof for a property of triangles

In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote. Lemma 1: Let the nodes ...
1
vote
1answer
56 views

A problem from KVS 2014

I was doing the the following problem- Prove that $$\frac { \sqrt { a+b+c } +\sqrt { a } }{ b+c } +\frac { \sqrt { a+b+c } +\sqrt { b } }{ c+a } +\frac { \sqrt { a+b+c } +\sqrt { c } }{ a+b } \ge ...
10
votes
2answers
167 views

How prove this function inequality $xf(x)>\frac{1}{x}f\left(\frac{1}{x}\right)$

Let $f(x)$ be monotone decreasing on $(0,+\infty)$, such that $$0<f(x)<\lvert f'(x) \rvert,\qquad\forall x\in (0,+\infty).$$ Show that ...
6
votes
1answer
134 views

IMO 2015 warm up problem

I get this problem from IMO 2015 facebook page. Let $x_i$ be positive integers for $i=1,2,...,11$. If $x_i+x_{i+1}\geq 100$, $|x_i-x_{i+1}|\geq 20$ for $i=1,2,...,10$. And $x_{11}+x_{1}\geq 100$, ...
4
votes
1answer
30 views

Inequality with arctangent between shortened taylor expansion and equation similar to the taylor expansion

How do you prove that for $0<x\leq1$, it is true that $$x-\displaystyle\frac{x^3}{3}<\arctan x<x-\displaystyle\frac{x^3}{6}$$
0
votes
0answers
52 views

Inequality regarding sequence approximation square root

I was reading a proof in which the square root function on $[0,1]$ is approximated by a sequence of functions (polynomials) defined on $[0,1]$. The induction hypothesis is that you have functions ...
4
votes
3answers
192 views

Inequality of the Fibonacci sequence and the golden ratio

How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. Any help is really appreciated
2
votes
1answer
78 views

Inequality.such as Nesbitt

Let $a,b,c >0 $ , prove that: $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} \leq \dfrac{a^2}{b^2+c^2}+\dfrac{b^2}{c^2+a^2}+\dfrac{c^2}{a^2+b^2}$$
5
votes
3answers
81 views

Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.

I can show that for $x > 0$ and $r_{i} > 0$ we have $$ \left(\, x + r_{1}\,\right)\ldots\left(\, x + r_{n}\,\right)\ \geq\ \left[\, x + \left(\, r_{1}\ldots r_{n}\,\right)^{1/n}\,\right]^{n}.$$ ...
2
votes
2answers
65 views

show that inequality holds for $n \ge 10$

I want to prove that for $n \ge 10$ holds: $$(n+1)^{\sqrt{n+1}}<n^{\sqrt{n+2}}$$ I know that holds $(n+1)^{{n+1}}<n^{{n+2}}$ which can be proven by induction, but here I don't know how to deal ...
0
votes
2answers
24 views

How to solve this inequality and sketch this graph?

I tried squaring and simplifying and got a solution set different to the one it says in the answer so I'm not really sure what I'm doing wrong. Also not sure how to sketch the graph any help?
1
vote
0answers
29 views

System of linear diophantine modular inequalities

How can we best find a numerical solution to a system of $m\ge2$ linear diophantine modular inequalities $$\big((a^j x+b_j)\bmod n\big)<c\;\text{ for }1\le j\le m$$ where $x$ is the only unknown, ...
1
vote
1answer
45 views

How to simplify this inequality

I have the following inequality where $i$, $N$ and $p$ are constants, $j$ is a variable and $p_j$ is the chance that 'event' $j$ is happening: $$i\geq -pi+((1-p)\cdot \sum ^N _{j=0}(j\cdot p_j))+\sum ...
0
votes
1answer
45 views

Cauchy Schwartz inequality and absolute value

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$ Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?
0
votes
1answer
40 views

Dubiety About An Inequality Proof

In Principles of Mathematical Analysis, the author is attempting to demonstrate that, if $x>0$ and $y<z$, then $xy<xz$, which essentially states that multiplying by a positive number does not ...
1
vote
1answer
32 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
1
vote
1answer
33 views

Prove/disprove number of zeros inequality

Having a continuous differentiable function $f(x)$, and denote $Z(\cdot)$ number of zeros (assume real line), and $(\cdot)^\prime$ first derivative, I would like to know if following inequality ...
1
vote
1answer
40 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
0
votes
2answers
22 views

Region given by these inequalities in XY Plane

Given region as $ 0\leq x \leq y $ , $ x+y \leq 1$ . I did this as Is this correct ?
1
vote
2answers
95 views

Permutation of positive real numbers

Consider a set of positive real numbers $\{P_1,P_2,\dots,P_n\}$ and a permutation of this set $\{Q_1,Q_2,\dots,Q_n\}$. Is it possible to find a permutation such that ...
1
vote
0answers
23 views

Entropy proerty

Let $a,b,c>0$ be distinct postive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
0
votes
1answer
15 views

Young's inequality implies $L^p$ convergence of convolution

I am reading a material which states: If $f_n \to f$ in $L^1(\mathbb{R})$, $g \in L^P(\mathbb{R})$. Then $f_n*g \to f*g$ in $L^p(\mathbb{R})$ by Young's inequality. But I cannot see why Young's ...
1
vote
1answer
36 views

How can I determine the bounds for this inequality?

I have the following inequality: $$ -40 < \bigg(\frac{a}{2^{52}}\bigg) (2^{b}) < 40 ,\ \text{with}\ a \in [-2^{52}, 2^{52}]\ \text{and}\ b \in [-1024, 1024].$$ How can I "thin" the range of ...
-1
votes
1answer
41 views

How can I rearrange $|a-b|<|b|/2$ to get $a^2>(b^2)/4$? [closed]

How can I rearrange $|a-b|<|b|/2$ to get $a^2>(b^2)/4 $?
1
vote
0answers
33 views

Non-trivial inequality

Equation (1) on page 7 of http://arxiv.org/pdf/1312.7308v1.pdf claims that: $$\frac{1}{t}\log \left(\frac{\log ((1+\epsilon)t)}{\omega} \right) \geq c \Rightarrow t \leq \frac{1}{c} \log \left( ...