Questions on proving and manipulating inequalities.

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2
votes
2answers
24 views

Which is the greatest (Inequalities)

Here are three different inequations, $$A+B < C+D$$ $$B+D < A+C$$ $$A+D < B+C$$ Which is the greatest among $A,B,C,D$? I do have a hunch on how to solve this, but don't know how the ...
5
votes
3answers
319 views

A difficult inequality involving complex numbers

Suppose that $z_1,\ldots,z_n$ are complex numbers with the property that there is some constant $C$ such that $$\big|z_1^r+\cdots+z_n^r\big|\leqslant C$$ for all integers $r\geqslant0$. Show that ...
4
votes
0answers
36 views

How to understand/remember Holder's inequality

If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}=1$ and if $f \in L^p$ and $g \in L^q$, then $f\cdot g \in L^1$ and $$\int |fg| \leqslant ||f||_p \cdot ||g||_q$$ I think ...
1
vote
2answers
42 views

Prove the inequality $\sum_{i=1}^n x_i^{n}-(n-1)\sum_{i=1}^n x_i^{n-1} \geq-n$

I am talking about a generalization of this inequality posed in this question. let $x,y,z>0$ and such $xyz=1$, show that $$x^3+y^3+z^3+3\ge 2(x^2+y^2+z^2)$$ I am trying to prove the ...
3
votes
2answers
110 views

Using Mean value theorem to prove the inequality $1.995<129^{1/7}<2.005$

How can I prove the following inequality using mean value theorem? $$1.997<129^{1/7}<2.003$$ Progress
1
vote
0answers
21 views

Inequality involving incomplete gamma function

When trying to answer this question: Find minimum $n$ such that $1+z+\frac{z^2}{2!}+\cdots+\frac{z^n}{n!}=0$ has no answer inside the circle of radius $100$ centered at the origin I ended up in what ...
1
vote
0answers
33 views

A question related to Integral and supremum

Let $f\in L_{p}([0,1])$ and 1-periodic on $R^{1}.$ Suppose $[a,c]\subset [0,1].$ Are the following quantities equal? $$ \underset{|h|\leq \delta_{1}}{\sup}\int_{a}^{b}|f(x+h)-f(x)|^{p}dx+ ...
1
vote
5answers
88 views

proving that $(n-1)^n>n^{n-1}$

I want to prove that $(n-1)^n>n^{n-1}$, for $n>4$, $n$ is an integer. So i divided by $n^n$ and got: $(1-\frac{1}{n})^{n}>\frac{1}{n}$ I know that ...
2
votes
5answers
83 views

How to prove $\sin(\frac{1}x) < \frac{1}x$ for all $x$ greater than $1$.

How to prove $\sin(\frac{1}x) < \frac{1}x$ for all $x$ greater than $1$? I was thinking using slopes but I get a contradiction (i.e. $\cos(\frac{1}x) > 1$) when I do some of the algebra.
1
vote
1answer
29 views

$L_1\subset L_p$?

I am trying to check whether the implication $\forall p>1\quad f\in L_p(X,\mu)\Rightarrow f\in L_1(X,\mu)$ is true when $\mu(X)<\infty$. By $L_p(X,\mu)$ I mean the space of Lebesgue integrable ...
3
votes
1answer
66 views

Why does $(n+1)^n\lt n^{n+1} \implies \left(1+\frac{1}{n}\right)^n\lt n$?

During an example done in lecture, I encountered an inequality by the form of $$(n+1)^n\lt n^{n+1}$$ My professor immediately simplified it to $$\left(1+\frac{1}{n}\right)^n\lt n$$ I have attempted to ...
6
votes
3answers
72 views

Proving bounds on an infinite product

Let $p$ be an infinite product, such that $p = 2^{1/4}3^{1/9}4^{1/16}5^{1/25} ...$ Prove that $2.488472296 ≤ p ≤ 2.633367180$. I start this problem by representing p in the infinite product ...
1
vote
2answers
23 views

$f(x)=cx+\dfrac{1}{x^2+3}$. Find the values of c for which f(x) is increasing for all $x$

$f(x)=cx+\dfrac{1}{x^2+3}$. Find the values of c for which f(x) is increasing for all $x$ I found $$f'(x) =c-\dfrac{2x}{\left( x^2+3\right)^2}$$ But I am having a tough time moving from here. I ...
1
vote
2answers
39 views

Simplifying an inequality: $4x(x-2) \lt 2(2x-1)(x-3)$

I have: $$4x(x-2) \lt 2(2x-1)(x-3)$$ For the last part, do I multiply both things in $()$ by two then solve them like I normally would? If I solve them and then multiply will it work the same? Is that ...
0
votes
1answer
60 views

Nth root of n is greater than 1?

A proof I did recently called upon a "fact" which my prof called without giving explanation or proof, which is the "fact" that $\sqrt[n]{n}>1$, how can this be shown?
0
votes
2answers
35 views

If $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{b-a})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$

The problem is as stated in the title. There is the obvious condition that no two are equal. I tried to do this without using brute force and do all the multiplications. There must be a simple way to ...
1
vote
3answers
46 views

Inequality problem involving QM-AM-GM-HM or Cauchy Schwarz inequality

Question: Prove that if $x$, $y$, $z$ are positive real numbers then the following inequality holds: ...
-2
votes
1answer
68 views

Plotting these complex inequalities on a complex plane.

For the first part a) it is very clear a circle with center at -i. With common sense using equation of a circle. For the 2nd part i'm having some trouble. My steps.. b) x+2y<3 $$(\sqrt{x})^2 ...
1
vote
2answers
39 views

Show that $|\sin{a}-\sin{b}| \le |a-b| $ for all $a$ and $b$

I've recently been going over the mean value and intermediate value theorems, however I'm not sure where to start on this.
1
vote
1answer
56 views

Proving geometric inequality with algebra

Geometrically, since a straight line is the shortest path from a point to another: $$\sum\sqrt{x_i^2+y_i^2}\le \sqrt{\left(\sum x_i\right)^2+\left(\sum y_i\right)^2}$$ Where $x_i,y_i$ are positive ...
-2
votes
3answers
52 views

Need help with this proof of inequality and absolute value. [closed]

Help! proof that if $x,y \neq 0$ $\left|\frac{x^5+y^5}{x^4+2y^4}\right|<\left|x\right|+\left|\frac{y}{2}\right|$
1
vote
3answers
60 views

a two-variable cyclic power inequality $x^y+y^x>1$ intractable by standard calculus techniques

If $x$ is in the open interval (0,1) and so is $y$, prove that $$x^y+y^x>1$$ A direct two-variable application of maxima and minima seems difficult.
3
votes
2answers
33 views

Inequality between operator norm and Hilbert-Schmidt norm

I have seen the following inequality here but I don't know where I can find a proof for it. Could somebody give me a hint to understand it or guide me to a reference please? $\|AB\|_{HS} \leq ...
2
votes
1answer
29 views

How to compare $e^{i \alpha n^\beta}$ and $\int_n^{n+1}e^{i \alpha x^\beta}\, dx$

I am solving an exercise in my textbook. After some steps, I need to compare $e^{i \alpha n^\beta}$ and $\int_n^{n+1}e^{i \alpha x^\beta}\, dx$, where $\alpha\neq 0,0<\beta<1$, in order to ...
2
votes
0answers
94 views

Lower bound on a polynomial far from its zeros

Let $p(x) = \sum_{i=0}^{d}c_{i}x^{i} \in \mathbb{R}[x]$ and assume that all its zeros are real and in $[-1,1]$. I am interested in lower bounding the value of $|p(a)|$ in case $a \in [-1,1]$ is far ...
2
votes
1answer
85 views

Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$

$a,b,c$ are positive reals with $abc = 1$. Prove that $$\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$$ I try to use AM $\ge$ HM. ...
13
votes
4answers
142 views

Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$

Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ This problem is copied from Math Olympiad Treasures ...
1
vote
2answers
30 views

Linear systems of inequations

Ok so I have a systems with $6$ inequations and $3$ variables, and a point that may or may not solve this system. To check whether this point solves the inequations is straightforward, my problem is ...
2
votes
5answers
109 views

Prove $ \frac {x_1 + \cdots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \cdots x_n}$

I am studying computer science in the first term. I have to proof the following inequality: $$ \frac {x_1 + \cdots+ x_n}{n} \ge \sqrt[n]{x_1 x_2 \cdots x_n}$$ $x$ can be any positive real ...
1
vote
1answer
20 views

Is it possible to bound the fraction of n-tosses that have exactly m-heads by one over m cube?

For positive integers $n$ and $m$, let us define $$ A_{m,n} := \frac{\binom{n}{m}}{2^n}.$$ Does there exist a constant $C$ independent of n, such that $$ A_{m,n} \leq \frac{C}{m^3} $$ for all $n$ ...
0
votes
1answer
33 views

Prove the inequality of $u$ and $v$

If $u \geq v \geq 1$ then prove that, $$9u^3+4v^2+2\leq 6u^2+9uv^2$$ I can't progress at all regarding this inequality. Any idea will be helpful. Is there any non-calculus solution of the ...
1
vote
1answer
31 views

Why this inequality holds?

I see this in my class note. I know the first one comes form the Euler's formula, but I really don't know the reason why the last inequality holds? ...
0
votes
0answers
17 views

A basic inequality problem involving two variables

How do I solve the following inequalities ? $p^{-\frac{1}{p}}\:\le \:k;\:\log _{10}\left(k\right)\:\le \:k^p:\:p\:\le 1\:and\:\:k\ge 1$
0
votes
0answers
27 views

Prove that harmonic numbers satisfy the equality. [duplicate]

The $k$th harmonic number is defined to be $H_k = 1 + 1/2+ 1/3 + · · · + 1/k .$ Prove that harmonic numbers satisfy the equality $H_1 + H_2 + · · · + H_n = (n + 1)H_n − n$ for all $n \in\Bbb N.$
2
votes
0answers
40 views

Jensen's inequality: proof by using linear functions

Here's an extract from Stochastic Calculus for Finance Volume 1 by Shreve. I don't understand the statement that says a convex function is the maximum of all linear functions that lie below ...
0
votes
1answer
37 views

Show that $(1+x)^n>\dfrac{n(n-1)}{2}x^2$ for all $x>0, n \in \mathbb{Z}$

Show that $(1+x)^n>\dfrac{n(n-1)}{2}x^2$ for all $x>0, n \in \mathbb{Z^+}$ I actually have no idea how to start - I've expanded the left side to get: $1+{n\choose 1}x+{n\choose 2}x^2+...+x^n$ ...
4
votes
6answers
147 views

Proving basics of $(a+b)^2$

I need to prove this: Consider the following inequality: $$a^2+ab+b^2 > 0$$ I know that $^2$ makes $a$ and $b$ positive numbers, so it always be $>0$, but i got stuck with the ab thing. ...
2
votes
2answers
26 views

Inequality with modulus

I would be glad if someone will help me to understand how to solve inequalities as the following one: $$\vert 6-3x\vert+x \leq \vert x+2\vert$$ I remember that I need to see where the modulus is ...
1
vote
1answer
47 views

Maximization of a ratio

Edit: Removed solved in title, because I realize I need someone to check my work. Ok, so the problem is a lot more straight forward than I originally approached it (which was a false statement -- so ...
0
votes
1answer
75 views

If $a$ and $b$ are positive real numbers, then $a + b \geq 2 \sqrt{ab}$.

If $a$ and $b$ are positive real numbers, then $a + b \geq 2 \sqrt{ab}$. I know how to do the direct proof, but in this case, I want to try proving it by contradiction. I have tried manipulating the ...
1
vote
2answers
33 views

A few inequality problems I can't seem to get; Cauchy and the Mean Inequality Chain

Prove that $ \sqrt{\frac{2x^2-2x+1}{2}}\geq\frac{1}{x+\frac{1}{x}} $ for $ 0 < x < 1. $ This one seems reminiscent of the quadratic mean on the left, maybe $\sqrt{\frac{(x-1)^2+x^2}{2}}$, but ...
0
votes
3answers
15 views

inequality and logic

$|x-2|-|x+4| > 2 $ So I have to check for a. $x\leq -4$ b.$-4<x\leq2$ c.$x>2$ for a.$x\leq -4$ it is $-(x-2)--(x+4)>2\rightarrow -x+2+x+4>2\rightarrow 0x>-4$ so it is $x\leq -4$ ...
10
votes
2answers
250 views

$\sum x_{k}=1$ then, what is the maximal value of $\sum x_{k}^{2}\sum kx_{k} $

Let $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$, and $\sum\limits_{k=1}^{n}x_{k}=1$. then what is the maximal value of ? $$\sum_{k=1}^{n}x_{k}^{2}\sum_{k=1}^{n}kx_{k} .$$ I think, Maybe we could ...
3
votes
0answers
45 views

A contest inequality

The following inequality appeared in the AMTI contest in India, which was held a couple of days ago. If $x$ and $y$ are positive reals such that $x^{2014}+y^{2014}=1$ prove that ...
2
votes
0answers
84 views

How to derive the following from Azuma's inequality?

This is claimed in Proposition 1 in the paper http://arxiv.org/abs/1409.6110 Let $A$ be a $n \times d$ matrix. $A$ can have only $K$ different types of rows i.e. rows of $A$ are chosen from a set of ...
8
votes
0answers
50 views

An Inequality with complex numbers and $1/\pi$ [duplicate]

Let $\displaystyle \{z_1,z_2, \ldots, z_n\}$ be $n$ complex numbers such that: $\displaystyle \sum\limits_{k=1}^n|z_k| = 1$ Then we have to show that, there is a subset $S$ of $\{1,2,\ldots,n\}$, ...
2
votes
3answers
43 views

How to solve $\left|\frac{x+4}{ax+2}\right| > \frac1x$

How to solve: $$\left|\frac{x+4}{ax+2}\right| > \frac{1}{x}$$ What I have done: I) $x < 0$: Obviously this part of the inequation is $x\in(-\infty, 0), x\neq \frac{-2}{a}$ II) $x > 0$: ...
4
votes
1answer
50 views

An inequality relating to a continuous and twice differentiable function

Suppose that $f$ is a continuous and twice differentiable function in $[0,1]$. Please show that $$ \int_0^1 \vert f'(x) \vert dx \leq 9\int_0^1 \vert f(x) \vert dx + \int_0^1 \vert f''(x) \vert dx $$ ...
0
votes
0answers
25 views

Linear Programming question involving a data set of consumer purchases

I am from Netherlands and preparing for an interview with Two Sigma Capital, which for the position I am applying for is notorious for asking linear programming questions. I was trying to solve this ...
6
votes
1answer
45 views

Finding a bound for $\sum_{n=k}^l \frac{z^n}{n}$

For $z\in\mathbb{C}$ such that $|z|=1$ but $z\neq1$ and $0<k<l$, I'm trying to prove that: $$\left|\sum_{n=k}^l \frac{z^n}{n}\right| \leq \frac{4}{k|1-z|}$$ It's more of a game that slowly ...