Questions on proving and manipulating inequalities.

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Inequality.such as Nesbitt

Let $a,b,c >0 $ , prove that: $$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b} \leq \dfrac{a^2}{b^2+c^2}+\dfrac{b^2}{c^2+a^2}+\dfrac{c^2}{a^2+b^2}$$
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3answers
37 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 $$(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}.$$ However, I can't do this using straight up induction, strong or weak. Can ...
2
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2answers
52 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 ...
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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?
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0answers
22 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, ...
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1answer
44 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 ...
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1answer
41 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?
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0answers
26 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 ...
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1answer
28 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 ...
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1answer
32 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 ...
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1answer
39 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 ...
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2answers
18 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 ?
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2answers
90 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 ...
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0answers
15 views

Entropy proerty

Let $a,b,c>0$ be distinct postive reals. Define four different probability distributions: $$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$ ...
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1answer
14 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 ...
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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 ...
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1answer
41 views

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

How can I rearrange $|a-b|<|b|/2$ to get $a^2>(b^2)/4 $?
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0answers
30 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( ...
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1answer
27 views

If $X\ge 0$ and $a\ge E[X]$, then $P(X\gt a)\ge (E(X)-a)^2/ E(X^2)$ [on hold]

I need help with this problem. Prove that if $X\ge0$ and $E[X^2]<\infty$ then for all $a\neq0$, $E[X]\le a$, we have $$P(X\gt a)\ge\frac {(E(X)-a)^2}{E(X^2)}$$ Progress I have my doubts if ...
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1answer
41 views

How to prove that $(a+b-ab)^n+(1-a^n)(1-b^n) \geq n $ for $a,b\in [0,1]$ and $n\in\mathbb{N}$?

Let $a,b\in [0,1]$ and $n\in\mathbb{N}$. Prove the following inequality: $$(a+b-ab)^n+(1-a^n)(1-b^n) \geq n $$ I thought on using M Induction: Assuming that the inequality holds for $n=k,$ ...
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1answer
28 views

$k_{n+1}\le (1+2\varepsilon)k_n$ for $k_n:=\lfloor(1+\varepsilon)^n\rfloor$ and $\varepsilon>0$

Let $$k_n:=\lfloor(1+\varepsilon)^n\rfloor\stackrel{\text{def}}{=}\max\left\{k\in\mathbb{Z}:k\le(1+\varepsilon)^n\right\}\;\;\;\text{for }n\in\mathbb{N}$$ How can we prove $k_{n+1}\le ...
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0answers
34 views

What bounds can we put on the largest root of a polynomial?

For a polynomial $p(x)=x^{n+1}+a_{n} x^{n} + \cdots + a_1$ with roots $|x_1| < \cdots < |x_n|$ can we find relatively simple function $M(a_1, \dots, a_n)$ such that $$|x_i| \leq M(a_1, \dots, ...
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7answers
126 views

Prove $a^{2}+2ab+3b^{2}+2a+6b+3 \geq 0$ [on hold]

If $a,b\in\mathbb{R}$ prove that the following inequality holds: $a^{2}+2ab+3b^{2}+2a+6b+3 \geq 0$
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3answers
47 views

How to solve this inequality? $2\cos(x+1)>0$ [on hold]

Please help me answer this question. How can I solve the following inequality? Solve the following inequality: $2\cos(x+1)>0$. Thank you.
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2answers
69 views

What is the largest $k$ such that $ \frac { k(abc) }{ a+b+c } \le \left( a+b \right) ^{ 2 }+\left( a+b+4c \right) ^{ 2 } $?

Find the largest value of $ k $ such that $ \frac { k(abc) }{ a+b+c } \le \left( a+b \right) ^{ 2 }+\left( a+b+4c \right) ^{ 2 } $
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1answer
57 views

Inequality on matrix norm: $ \lVert A^n \rVert \leq \lVert A \rVert^n $ [on hold]

If $A$ is a $n \times n$ matrix and assume we have a matrix norm $\lVert \cdot \rVert$. In a proof I need the following property: $ \lVert A^n \rVert \leq \lVert A \rVert^n $. I don't know how to ...
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0answers
20 views

Trace of matrix with orthogonal matrix

Let $H$ be an orthogonal matrix. Let $\Phi$ be given matrix. What is $Tr(H\Phi)$? Any lower bounds that will be useful? What if $H$ is also symmetric?
5
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2answers
118 views

How prove this complex inequality with same as (2014 china CMO) Cauchy-Schwarz inequality

let $r$ is give numbers,let $z_{1},z_{2},\cdots,z_{n}$ such $|z_{i}-1|\le r,i=1,2,\cdots,n,r\in(0,1)$ show that ...
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0answers
29 views

Variational Inequalities and how they are used?

I am doing undergrad research in this field next semester and I have never heard of this topic before. I tried wikipedia and reddit for help but nothing seems to help. I just want to know what I'm up ...
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2answers
67 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
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0answers
99 views

Prove the Schwarz inequality using $ 2xy \leq x^2 + y^2 $

Im really bad at analysis and this problem was recommend to me to help me grasp some basics of $\epsilon $ $\delta $ So im doing a problem ( though its like 12 pieces ) this is i guess the fourth ...
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1answer
61 views

What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications. What I want is a collection of questions/problems which look very complex but get ...
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2answers
39 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 ...
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1answer
67 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 ...
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0answers
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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 ...
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0answers
28 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 ...
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1answer
56 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 ...
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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 ...
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1answer
24 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 ...
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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 ...
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1answer
43 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 ...
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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 = ...
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1answer
19 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 $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
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0answers
33 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 $$\begin{cases} |z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2=1\\ |z_{i}|\le\dfrac{1}{10},i=1,2,\cdots,n \end{cases}$$ ...
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1answer
58 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
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1answer
58 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
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1answer
62 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 ...
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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 ...
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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 ...
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2answers
39 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, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...