Questions on proving and manipulating inequalities.

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0
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2answers
28 views

Proving that this function is negligible

Let $f(n) =\frac{1}{2^{\sqrt n}}$, where $n \in \mathbb{N}$. I want to prove that $\forall a \in N, a \ge1\; \exists k: f(n) \le n^{-a}, \forall n \ge k$ I attempted to solve the inequality ...
4
votes
1answer
52 views

The equality case of the Schwartz inequality

Question: The fact that $a^2 \geq 0$ $ \forall a \in \mathbb{R}$; elementary as it may seem, is nevertheless the fundamental idea upon which most important inequalities are ultimately based. The ...
1
vote
0answers
11 views

Inequality to bound $\sum_i a_i b_i - \sum_i c_i d_i$ (harmonic eigenfunction/graph) type sum with constraints

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...
-6
votes
0answers
26 views

Solving the inequality: $0.275k_1^2k_2(1-k_2)+0.05k_1k_2-0.5k_1 +1 <0$? [on hold]

where $k_1>0$, $0<k_2<1$ and $k_1k_2<2$ This question come from a stability analysis by Jury test. I would like to get an answer to identify k_1 could you help me to solve this ...
3
votes
1answer
49 views

Inequality $\frac{x^3+y^3}{x-y}>4$

Let $x>y>0$ and $xy\geq 1$. Prove that $$\frac{x^3+y^3}{x-y}>4.$$ Of course we can factor $(x^3+y^3)=(x+y)(x^2-xy+y^2)$, but it is not very useful. For fixed $x-y$, we can try to find the ...
-5
votes
1answer
40 views

is this systems of equations solvable

i saw this equation posted yesterday and i wanted to ask if i could have help helping this kid solve it. he is in pre-algebra and i am in algebra so not much of a difference. i have tried to solve for ...
1
vote
1answer
23 views

Trouble Understanding Proof Of Invariant Relationship

In part of a proof I am reading this is stated: $2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ) + (a_n + c_n )^2 + (b_n + d_n )^2 ≥ 2(a_n^2 + b_n^2 + c_n^2 + d_n^2 ).$ (1) From this invariant inequality ...
0
votes
2answers
15 views

Finding possible values for a function

After diving into simple inequalities, I've come across a particular exercise that requires me to find all possible value for $x$ for a given function. After searching about it, it seems to be simply ...
2
votes
1answer
41 views

Is this an upper bound or lower bound?

I came across a probability distribution function in my work, it is however difficult to find in closed form, therefore I am looking to either upper bound or lower bound it. Assuming $a,b,T$ are ...
0
votes
1answer
36 views

Why $\left\|\sum_{i=1}^nx_i\right\|^2\leq n\sum_{i=1}^n\|x_i\|^2$

Why $$\left\|\sum_{i=1}^nx_i\right\|^2\leq n\sum_{i=1}^n\|x_i\|^2$$ for arbitrary norm on the inner product space over the real field? My attempt $$\left\|\sum_{i=1}^nx_i\right\|^2 = ...
-1
votes
1answer
18 views

Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
0
votes
0answers
27 views

How prove this the number of ordered $n$-tuples $(\varepsilon_{1},\cdots,\varepsilon_{n})$such this following inequality is $2^{n-100}$

Interesting Question: for any complex numbers $z_{1},z_{2},\cdots,z_{n}$ such $$\begin{cases} |z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2=1\\ |z_{i}|\le\dfrac{1}{10},i=1,2,\cdots,n \end{cases}$$ ...
3
votes
1answer
55 views

Prove that $a^ab^bc^c\ge (abc)^{(a+b+c)/{3}}$ [duplicate]

Prove that $$a^ab^bc^c\ge (abc)^{(a+b+c)/3}$$ My attempt: $$a^ab^bc^c\ge (abc)^{(a+b+c)/{3}}\implies ...
8
votes
1answer
55 views

$(1-a)(1-b)(1-c)(1-d)\geq abcd$ for $a^2+b^2+c^2+d^2=1$

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Prove that $$(1-a)(1-b)(1-c)(1-d)\geq abcd.$$ I thought about substituting $a=\sqrt{w},b=\sqrt{x}$, etc. (assuming first that $a,b,c,d$ ...
2
votes
1answer
57 views

Proving inequality $(x^2+y^2)(y-1)+yx-y^2<0$

I have an inequality which came out of Lyapunov function for system of ODE's: $$(x^2+y^2)(y-1)+yx-y^2<0.$$ To prove stability of my solution, I have to prove that the inequalty is true in area ...
0
votes
0answers
13 views

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 ...
1
vote
2answers
37 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
2answers
37 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, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
3
votes
1answer
30 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 ...
2
votes
1answer
33 views

Determinantal inequality for block matrices: if $A=(B,C)$ is a square matrix, then $|A|^2\le |B^TB|\cdot |C^TC|$

Suppose $A=(B,C)$ is a $n\times n$ matrix, $B$ is a $n\times s$ matrix, $C$ is a $n\times (n-s)$ matrix. Show that $|A|^2\leq |B^TB|\cdot |C^TC|$. If $A$ is singular, then it is obvious. If $A$ is ...
2
votes
1answer
46 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} ...
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
46 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 ...
5
votes
1answer
59 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
18 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 ...
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 ...
2
votes
2answers
23 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 ...
0
votes
1answer
38 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!
2
votes
6answers
103 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 ...
0
votes
0answers
33 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 ...
3
votes
1answer
70 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 $x \in [0,1]$. ...
9
votes
0answers
62 views

Summation of cosine terms

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
0
votes
0answers
66 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
vote
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
113 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= ...
1
vote
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 ...
9
votes
2answers
749 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 [on hold]

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 ...
3
votes
1answer
79 views
+100

Find Minimum value of $P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$

Given: $x,y\in (-\sqrt2;\sqrt2)$ and $x^4+y^4+4=\dfrac{6}{xy}$ Find Minimum value Of $$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$$ Could someone help me ?
0
votes
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 $c$ $\epsilon$ $Q$ > $0$. ...
0
votes
1answer
15 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 ...
2
votes
1answer
53 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??
0
votes
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 ...
3
votes
0answers
98 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 ...
-2
votes
2answers
39 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?
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 ...
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
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 ...