Questions on proving and manipulating inequalities.

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0
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0answers
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positive Maclaurin polynomials

Consider even degree Maclaurin polynomials $M[n;2k]$ for $(1+x)^n$ where degree $= 2k < n$ and $n$ is a positive integer. Examples: (1) The quadratic #$M[3;2] = 1 + 3x + 3x^2$ is clearly ...
30
votes
10answers
3k views

Old oxford scholarship question: $a^ab^b \ge a^bb^a$

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
3
votes
3answers
55 views

Problem similar to Kolmogorov's inequality using martingale.

Suppose that $X_k$ is a sequence of independent random variables with mean zero and variance $1$. Let $S_k=X_1+\cdots+X_k$ and let $$ h(\lambda)=\limsup_{n \rightarrow \infty}P\left(\max_{1\leq k\leq ...
2
votes
2answers
24 views

Why this power inequality for sums of real numbers holds?

$$\left|\sum_{i=1}^nx_i\right|^p \leq \begin{cases} \sum_{i=1}^n|x_i|^p & p\in(0,1]\\ n^{p-1}\sum_{i=1}^n|x_i|^p & p>1 \end{cases}$$ Can it be generalized for arbitrary sequences ...
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2answers
39 views

Prove the inequality $(n+1)^4 < 4n^4$ for $n\geq 3$ by induction

The inequality I'm concerned with is $(n+1)^4 < 4n^4,\ n\geq 3$. I'm not sure how induction is supposed to work here. If I assume $(k+1)^4<4k^4$, I cannot see how this helps show ...
0
votes
1answer
48 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
0
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2answers
38 views

Factorial inequality $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n$ using induction

I want to show $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n.$ Assume that it holds for some positive integer $k\geq 1$ and we will prove, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
0
votes
2answers
994 views

Use mean value theorem to prove the following

Use the mean value theorem to prove that: $$\cos(x)>1-\frac{x^2}{2}$$ for all $$x>0$$
3
votes
1answer
31 views

Inequality $a^2b^2+2(a+b)\geq 4ab+1$

Let $a,b\geq 1/2$. Prove that $$a^2b^2+2(a+b)\geq 4ab+1.$$ We know that $(ab-1)^2\geq 0$ implies $a^2b^2+1\geq 2ab$, so the inequality reduces to $2(a+b)\geq 2ab+2$, or $a+b\geq ab+1$. But this is ...
5
votes
1answer
44 views

prove this martingale inequality

The problem is like this: Let $Y_1,Y_2,\ldots$ be nonnegative i.i.d. random variables with $E(Y_m)=1$. Let $X_n=\prod_{m\leq n} Y_m$, show that $\lim_{n\rightarrow \infty}X_n=0$ if $P(Y_m=1)<1$. ...
0
votes
1answer
71 views

When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn, p.64. For any four complex numbers $a$, $b$, $c$, $d$, the following identity ...
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4answers
73 views

Question on Inequalities [closed]

Prove that if $a$ and $b$ are two positive integers, then $$a^2(b+1)+b^2(a+1)≥4ab.$$
2
votes
6answers
104 views

$2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$

If $2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$. This is a question from a regional math olympiad and thus there must exist solutions without application of calculus. I have no idea how to ...
2
votes
1answer
47 views

Inequality$\Big|\sum_{j=1}^n a_{1j} x_j \Big|^2 \leq \sum_{j=1}^n |a_{1j}|^2 \sum_{j=1}^n |x_j|^2$

Let ${\bf A}$ be a $m \times n$ matrix with entries $a_{ij}$, and ${\bf x}$ be a $n \times 1$ vector with entries $x_{i}$. Then how can I show $$ \left\vert\,\sum_{j\ =\ 1}^{n} ...
6
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2answers
105 views
+50

How prove this inequality $\prod_{1\le i<j\le 5}|z_{i}-z_{j}|^2\le 5^5$

let $z_{1},z_{2},z_{3},z_{4},z_{5}$ are complex numbers,and such $$|z_{1}|^2+|z_{2}|^2+|z_{3}|^2+|z_{4}|^2+|z_{5}|^2=5$$ ...
5
votes
1answer
62 views

If $f(0)=0$ and $f(1)=1$, prove that $\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If $f'$ is continuous, prove that $$\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$$ Progress I let ...
0
votes
1answer
19 views

Proof if $n_k < n_{k+1}$ for all $k \in \mathbb{N}$, then $n_k \geq k$ for all $k \in \mathbb{N}$.

So if we proceed by induction on $k$, the base case $k = 1$ works since $n_1 \geq 1$ is true because $1$ is the smallest integer in $\mathbb{N}$. For the induction hypothesis, we have that $n_k \geq ...
0
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1answer
19 views

$\left|(1+R^2e^{2i\theta})^2\right| \geqslant (R^2-1)^2$ in complex integration

I need to prove: $$\lim_{R\to +\infty} \left|\int_0^\pi \frac{e^{iaR(\cos\theta+i\sin\theta)}}{(1+R^2e^{2i\theta})^2}iRe^{i\theta} d\theta\right| =0$$ Could someone give me some pointers? A ...
3
votes
1answer
71 views

If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $[0,1]$, then $|f'(1/2)|\le 1/4$

Let $f : [0,1] \rightarrow \mathbb{R}$ be a function whose second order derivative $f''(x)$ is continuous on $[0,1]$. Suppose that $f(0) = f(1)=0$ and that $|f''(x)| \leq 1$ for any $x \in [0,1]$. ...
0
votes
1answer
39 views

The minimum value of the expression [on hold]

Please help me with the problem for 9th grade pupils: Find the minimum value of the expression $\frac{1}{1+x^2}+\frac{1}{1+y^2}$ with $x\ge1, y\ge1$ and $xy=2014$. Thank you!
4
votes
0answers
128 views
+50

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
3
votes
0answers
99 views

Prove that $\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}$

Prove that $$\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}\quad\forall n\in\mathbb{N}\backslash\{1\}.$$ Please show all passages and what ...
0
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0answers
34 views

Prove that $\frac{ab} {a^2+b^2} \frac{cb} {c^2+b^2} \frac{ac} {a^2+c^2} $ [duplicate]

Let $a,b,c$ be positive real numbers suvh that $a+b+c=1$ Prove that $\frac{ab} {a^2+b^2} \frac{cb} {c^2+b^2} \frac{ac} {a^2+c^2} +\frac {1} {4}(\frac {1} {a} + \frac {1} {b} + \frac {1} {c} \ge ...
0
votes
0answers
69 views

Prove that: $\frac{1}{1-2x}+\frac{1}{1-2y}+2\ge 0$

Given: $x,y\in R$ : $x^4+y^4+4=\frac{6}{xy}$ Prove that: $\frac{1}{1-2x}+\frac{1}{1-2y}+2\ge 0$ Please help me !
1
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1answer
54 views

Proofs involving positive real numbers

I have two questions related to positive real numbers: If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner ...
1
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3answers
34 views

Quadratic formula in double inequalities

I have the double inequality: $-x^2 + x(2n+1) - 2n \leq u < -x^2 + x(2n-1)$ and I am trying to get it into the form $x \leq \text{ anything } < x+1$ Or at least solve for x as the ...
10
votes
2answers
119 views

Showing $\gamma < \sqrt{1/3}$ without a computer

In 1735 Euler gave the value of $\gamma$ as $0.577218.$ The constant is generally defined as the limit of the difference between the harmonic series and $\log n:~\gamma= ...
4
votes
2answers
120 views

Why use $(1+p)^n\geq 1+np$ to prove that the successive powers of a number $q^n$ with $-1<q<1$ approach zero as $n\rightarrow \infty$?

I'm reading Courant's What is Mathematics? In page $64$, he gives the example of limit of the successive powers of $q$. If $-1<q<1$ then the successive powers of $q$ will approach zero as $n$ ...
2
votes
2answers
49 views

Find the minimum possible value of $x(1-z)+y(1-x)+z(1-y)$

It is given that $$xyz=(1-x)(1-y)(1-z)$$ and $$x, y, z \epsilon (0,1)$$ Find the minimum possible value of the expression: $$x(1-z)+y(1-x)+z(1-y)$$ Using the AM-GM inequality concepts, I can write ...
9
votes
2answers
750 views

Find maximum without calculus

Let $f:(0,1]\rightarrow\mathbb{R}$ with $f(x)=2x(1+\sqrt{1-x^2})$. Is it possible to find the maximum of this function without calculus? Possibility through some series of inequalities?
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votes
2answers
20 views

Problem on CR inequality on finite sum [closed]

Let $f$ be a function from {1,2,3,....,10} to R, s. t. $(\sum_{i=1}^{10}|f(i)|/2^i)^2=(\sum_{i=1}^{10} |f(i)|^2)(\sum_{i=1}^{10}1/4^i)$ mark the correct statement. A. there are uncountably ...
1
vote
0answers
28 views

Find a Liapunov function to show asymptotically stable

Consider the system: \begin{cases} \dfrac{dx}{dt} = y \\[12pt] \dfrac{dy}{dt} = -(1+x^{2})\,y-\sin(x) \end{cases} $(0,0)$ is a critical point of this system and I need to show that it is ...
2
votes
1answer
39 views

trace inequality of positive definite matrices.

Assume $A,B \in M_n(\Bbb{R})$ are positive definite matrices, show that $$\text{Tr}(AB)\leq \text{Tr}(A)\text{Tr}(B) $$ I only prove it for $n=2$, it is straightforward calculate.but when $n \geq ...
0
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1answer
23 views

Rational number inequality proof

Show that if $x > 1$ is a real number and if $a < b$ are rational numbers, then $0\le x^a \le x^b$. My professor told me that I'm supposed to use some $x^c$, such that $c$ $\epsilon$ $Q$ > $0$. ...
18
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5answers
1k views

proving $\mathrm e <3$

Well I am just asking myself if there's a more elegant way of proving $$2<\exp(1)=\mathrm e<3$$ than doing it by induction and using the fact of ...
0
votes
1answer
17 views

inequality for real-valued Gaussian sums

I saw the following Lemma in an article: Let $\mathbf{b}\in \mathbb{R}^N$ be fixed, and let $\mathbf{\epsilon}\in \mathbb{R}^N$ be a random vector whose N entries are i.i.d. random variables drawn ...
0
votes
1answer
735 views

generalized inequalities defined by proper cones

The generalized inequality defined by a proper cone $K$ is that $x \ge_{K} y$ if $x-y \in K$ for $x,y \in K$. Does this means that for any $x \in K$, we have $x \ge_{K} 0$ since $x - 0 = x \in K$ ? ...
2
votes
1answer
54 views

Prove: $ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $

Let $a,b,c>0$ such that $a+b+c=1$ Prove: $ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $ I don't have any idea. You guy have any idea??
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2answers
14 views

Variable intervals from system of inequalities

I have this system of inequalities: and I need to find possible intervals of i and j. Looking at the graph output from ...
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2answers
40 views

Prove this absolute value related inequality [closed]

$\left | |a+b|-|a|-|b| \right | \leq 2|b|$, $\forall a, b \in \mathbb{R}$. How can I prove it?
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0answers
86 views

Prove $\|x-y\|\|x+y\|\le\|x\|^2+\|y\|^2$ for all $\mathbb{R}^n$

Prove $\|x-y\|\|x+y\|\le\|x\|^2+\|y\|^2$ for all $\mathbb{R}^n$ I've been struggling with this for a while and haven't figured out a way to do it either geometrically or algebraically.
1
vote
1answer
19 views

Using the triangle inequality to show that if $|x| < 4$ then $|x^2-2x+3| < 27$

I'm starting school soon and doing some review problems to prep for Calculus. I'm a bit stuck on this problem: Show that if $|x| < 4$ then $|x^2-2x+3| < 27$. I know that I have to use the ...
2
votes
1answer
64 views

Inequality in tetrahedron

You are given a tetrahedron $ABCD$. $ACB = ADB = 90^\circ$. $AC = CD = DB$. Prove that $AB < 2 * CD$. I know that $AD = CB$ and $CBD = DCB = ADC = CAD$.
2
votes
7answers
281 views

Prove that $1+ \frac{1}{x^4} \geq \frac{1}{x} + \frac{1}{x^3}$

Prove That $$1+ \frac{1}{x^4} \geq \frac{1}{x} + \frac{1}{x^3}$$ where $x \in \mathbb Z^{+}$
0
votes
1answer
19 views

Triangular inequality in weighted graphs

In a finite directed complete graph $G ( V, E )$, if all edges have weight either $1$ or $2$, how to show that weights of edges of $G$ satisfies "Triangular Inequality"? Edited Where triangular ...
0
votes
1answer
23 views

Summation of quotient and quotient of summation

I have $P_1, P_2, P_3, \dotsc, P_n, S_1, S_2, S_3, \dotsc, S_n$. Is it always true that: $$ \frac{P_1+P_2+P_3+\dotsb+P_n}{S_1+S_2+S_3+\dotsb+S_n} \leq ...
0
votes
1answer
61 views

Proving that $(1+m)^{-1/n} + (1+n)^{-1/m} \ge 1$

I need to prove the following inequality: $$ (1+m)^{-1/n} + (1+n)^{-1/m} \ge 1 $$ for every natural $m,n$. there shouldn't be any complicated math here, as this question if from a first semester ...
1
vote
3answers
58 views

Prove $1+ (\frac{1}{x}) \geq (\frac{1}{x^4}) +(\frac{1}{x^3})$ [closed]

Prove That $$1+ \frac{1}{x} \geq \frac{1}{x^4} + \frac{1}{x^3}$$ where $x \in \mathbb Z^{+}$
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votes
2answers
52 views

Is this function defined?

Let a function Let a function $g(f)= \parallel \bigtriangledown f\parallel / sin \parallel f \parallel $ Is $g $ defined for $\left \| f \right \| \leq $ 1? $\left| \left|. \right|\right|$ ...
0
votes
0answers
15 views

Are the questions in each of the following sets a family? [closed]

$$Y=(3X+1)(2X-1)(X+3)(X-2)$$ $$Y=2(3X+1)(2X-1)(X+3)(X-2)$$ $$Y=3(3X+1)(2X-1)(X+3)(X-2)+1$$ $$Y=4(3X+1)(2X-1)(X+3)(X-2)+2$$ I am having trouble determining if these functions are all a family. If so, ...