Questions on proving and manipulating inequalities.

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3
votes
4answers
189 views

Prove that, if $0 < x < 1$, then $(1+\frac{x}{n})^n < \frac1{1-x}$

More fully, if $n\ge 2$ is an integer and $0 < x < 1$, prove that $(1+\frac{x}{n})^n < \frac1{1-x}$. In addition, if $c > 1$ and $0 < x \le \frac{c-1}{c}$, prove that ...
1
vote
2answers
89 views

An inequality on $C^1$ periodic functions

Suppose $f \in C^1(\mathbb{R})$ and $f(x + 1) = f(x) \ \forall x \in \mathbb{R}$. Show that $$||f||_{\infty} \leq \int_0^1|f| + \int_0^1|f'|.$$ I have tried using techniques in Fourier Analysis such ...
2
votes
1answer
70 views

An inequality on holomorphic functions

Let $D := \{z \in \mathbb{C}: |z| < 1\}$ and $f\colon D \rightarrow \mathbb{C}$ be holomorphic. Suppose $\lvert f(z)\rvert \leq 1$ on $D$, show that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| ...
0
votes
2answers
33 views

Solving for $n$ in a geomtric progression

Given the general term of geometric sequence: $a_n = \dfrac{x}{2^n}$ I would like to solve for the value of n that makes $a_n =1$. My work so far: \begin{align*} a_n &= \frac{x}{2^n}\\ 2^n &= ...
2
votes
1answer
260 views

Proof by Contradiction: $100$ Balls & $9$ Boxes

Show, by giving a proof by contradiction, that if $100$ balls are placed in nine boxes, some box contains 12 or more balls. I would like to ask for a hint for this quesiton. Thank you.
1
vote
1answer
74 views

proving inequality $0 < x^4+2x^2-2x+1$ for $x>0$

How can I elegantly prove the inequality $0 < x^4+2x^2-2x+1$ for $x>0$. I have plotted this function in a Sage (an open source and free CAS) and I can see that there is a local min between $0$ ...
1
vote
1answer
130 views

Show $\sum_{i<j} |a_i a_j| b_{ij} \leq \sum_{i=1}^{n} a_i^2 \max_{1 \leq i \leq n} \sum_{j=1}^{n}b_{ij}$

I'm having trouble proving the inequality holds $$ \sum_{i<j} |a_i a_j| b_{ij} \leq \sum_{i=1}^{n} a_i^2 \max_{1 \leq i \leq n} \sum_{j=1}^{n}b_{ij} \quad b_{ij}\geq 0 $$ Thanks
3
votes
1answer
142 views

An inequality involving operator and trace norms

Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...
5
votes
6answers
6k views

Proving by induction: $2^n > n^3 $ for any natural number $n > 9$ [duplicate]

I need to prove that $$ 2^n > n^3\quad \forall n\in \mathbb N, \;n>9.$$ Now that is actually very easy if we prove it for real numbers using calculus. But I need a proof that uses mathematical ...
2
votes
1answer
79 views

Some statement about Cauchy product of sequences

Assume that we have two sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$ such that for each $l\in \mathbb N $ the sequence $\left(|n|^l a_n\right)_{n \in \mathbb Z}$ is bounded, there ...
10
votes
1answer
385 views

Inequality $\frac{1-3ab}{1-2ac}+\frac{1-3bc}{1-2ba}+\frac{1-3ca}{1-2cb}\geq 0$

Let $a\ne 0$, $b\ne 0$ and $c\ne 0$ such that $a^2+b^2+c^2=1$. Prove that: $$\dfrac{1-3ab}{1-2ac}+\dfrac{1-3bc}{1-2ba}+\dfrac{1-3ca}{1-2cb}\geq 0.$$ My attempt to the solution: We get that $ab ...
0
votes
1answer
107 views

Norm inequality (supper bound)

Do you think this inequality is correct? I try to prove it, but I cannot. Please hep me. Assume that $\|X\| < \|Y\|$, where $\|X\|, \|Y\|\in (0,1)$ and $\|Z\| \gg \|X\|,\|Z\| \gg \|Y||$. prove that ...
4
votes
1answer
229 views

How prove this $\left(\sum_{k=1}^{n}a_{k}\right)^2\ge\sum_{k=1}^{n}(a_k)^3$

let $a_{n}\ge a_{n-1}\ge\cdots\ge a_{0}= 0$,and for any $i,j\in\{0,1,2\dots,n\},j>i$,then have $$a_{j}-a_{i}\le j-i$$ show that $$\left(\sum_{k=1}^n a_k \right)^2\ge\sum_{k=1}^n (a_k)^3$$ my idea ...
0
votes
1answer
80 views

What is the speed of $x^2/\sin^2x$ tends to $1$?

$$1+\frac{x^2}{3}\leq\frac{x^2}{\sin^2x}\leq 1+(1-\frac{4}{\pi^2})x^2,x\in [0,\pi/2]?$$ How to prove this? I know just the $x^2$, but $1/3$, and $1-4/\pi^2$ is not known. And it is the finest ...
0
votes
1answer
73 views

A simple inequality in about integer part of numbers?

This question follows A simple inequality in calculus?. I have to solve this inequality in about $s$: $$\left(\left[\dfrac{r}{s}\right] + 1 \right) s \le 1,$$ ...
1
vote
2answers
84 views

A simple inequality in calculus?

I have to solve this inequality: $$\left(\left[\dfrac{1}{s}\right] + 1 \right) s < 1,$$ where $ 0 < s < 1 $. I guess that $s$ must be in this range: $\left(0,\dfrac{1}{2}\right]$.But I ...
7
votes
6answers
455 views

Prove that: $a^2+b^2+(1-a-b)^2\ge \frac {1}{3}$

Where $a$ and $b$ are any given real number. I have tried solving it using partial derivative. $$ s=a^2+b^2+(1-a-b)^2$$ $$\frac{\partial s}{\partial a}=2a-2(1-a-b) \tag{1}$$ $$\frac{\partial ...
2
votes
1answer
66 views

Inequality with sum of rows of symmetric matrices

Let S be a symetric matrix, with coefficients positive or zero, and T its square $$T=S^2$$ Let $S_i$ and $T_i$ be the sum of the $i$-th row (or column) of $S$ and $T$ respectively. I noticed that ...
2
votes
2answers
137 views

Prove an inequality concerning $\sqrt[3]{4a^3+4b^3}+\sqrt[3]{4b^3+4c^3}+\sqrt[3]{4c^3+4a^3}$

Let $a,b,c$ be positive. I need to prove $\sqrt[3]{4a^3+4b^3}+\sqrt[3]{4b^3+4c^3}+\sqrt[3]{4c^3+4a^3}\leq \dfrac{4a^2}{a+b}+\dfrac{4b^2}{b+c}+\dfrac{4c^2}{c+a}$ Thanks!
2
votes
3answers
113 views

How prove this inequality for $a,b,c,d$ are real numbers

let $a,b,c,d$ are real numbers,show that $$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}\ge6\sqrt{|ad-bc|}$$ This problem is creat by China's ...
2
votes
0answers
80 views

Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
1
vote
1answer
143 views

Find the minimum of this expression

This is a problem in my exam and I can't find the solution using elementary inequality knowledge. Can anyone here help me solve this. Thanks $a,b,c $ are positive real numbers which satisfy ...
10
votes
3answers
261 views

Show that $\frac {a+b+c} 3\geq\sqrt[27]{\frac{a^3+b^3+c^3}3}$.

Given $a,b,c>0$ and $(a+b)(b+c)(c+a)=8$. Show that $\displaystyle \frac {a+b+c} 3\geq\sqrt[27]{\frac{a^3+b^3+c^3}3}$. Obviously, AM-GM seems to be suitable for LHS. For RHS, ...
5
votes
3answers
198 views

if $f(k)=\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{\infty}\dfrac{1}{t^2}$ then $f(k+1)>f(k)$

let $$f(k)=\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{\infty}\dfrac{1}{t^2}$$ prove $$f(k+1)>f(k)$$ my idea: ...
1
vote
1answer
70 views

Find the maximum of $xy(72-3x-4y)$?

$x$ and $y$ are positive. I have been stuck on this problem for a while now, any hints please?
3
votes
0answers
77 views

$\pi^4+\pi^5 < e^6 $ [duplicate]

Any idea about this inequality: $$\large \pi^4+\pi^5<e^6$$ Any hints would be appreciated.
6
votes
1answer
158 views

How prove $\frac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\frac{1}{t^2}<e$

Let $k,n\in \mathbb{N},n\ge k$, prove that $$\dfrac{(k+1)^{k+1}}{k^k}\sum_{t=k+1}^{n}\dfrac{1}{t^2}<e.$$ I got the impression that this inequality is very sharp. My idea: ...
3
votes
1answer
146 views

Prove that $\frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2}\geq 1$

Let $abcd=1$ and $a,b,c$ and $d$ are all positive. Prove that $\dfrac{1}{(1+a)^2}+\dfrac{1}{(1+b)^2}+\dfrac{1}{(1+c)^2}+\dfrac{1}{(1+d)^2}\geq 1$ I am probably able to do this by assuming $a\geq ...
1
vote
2answers
1k views

show operator norm submultiplicative

We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and ...
0
votes
2answers
117 views

Rearranging an inequality

We are given that $x > y \geq 0$ and we know $y > x/2$. I'd like to show that $x -y < x/2$? Here's where I've attempted so far and I am getting stuck. $x > y > x/2$ $x - y > 0 ...
1
vote
1answer
77 views

How prove this $|a-b||c|\le|a-c||b|+|b-c||a|$ if $a,b,c$ are vectors? [duplicate]

Let $a,b,c$ are vector numbers,show that $$|a-b||c|\le|a-c||b|+|b-c||a|$$ my idea: $a_{i},b_{i},c_{i}\in \mathbb{R}$ let $$a=(a_{1},a_{2},\cdots,a_{n})$$ $$b=(b_{1},b_{2},\cdots,b_{n})$$ ...
3
votes
1answer
184 views

Help with proving the Fatou's Lemma for discrete functions

I need help to understand two steps in this proof for the Fatou's lemma in its discrete version. Let $f_k:\mathbb{N} \rightarrow [0, \infty), k\in\mathbb{N}$, be a sequence of functions. Then $$ ...
0
votes
0answers
86 views

When is the “inequality” approach to limits valid?

For example, let's say $\lim_{x\to \infty} [g(x)]^{f(x)}=1$ . If we know that as $x \to \infty$, $h(x)> g(x)$ , we can say that $\lim_{x\to \infty} [h(x)]^{f(x)}$ equals $\infty$ . However, I ...
4
votes
2answers
117 views

When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I'm trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$ Let $\|x+y\|=\|x\|+\|y\|$ Squaring both sides, $\langle ...
1
vote
0answers
132 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
6
votes
2answers
170 views

If a sequence of natural numbers satisfies $\gcd(a_{i+1},a_{i})>a_{i-1}$, then $a_{n}>2^n$

Given a sequence $\{a_{n}\}$ in $\mathbb{N}$ such that $\gcd(a_{i+1},a_{i})>a_{i-1},$ for any $i\ge 2$, show that $a_{n}>2^{n-1}$. Thank you everyone, my friend asked me about this problem, ...
2
votes
2answers
83 views

How to prove this inequality $(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}>2$

let $x>0$,show that $$(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}>2$$ Do you have any nice method? My idea $F(x)=(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}$ then we hvae $F'(x)=\cdots$ But it's ugly. ...
5
votes
2answers
311 views

Proving the inequality $\frac{a^3}{b^2-bc+c^2}+\frac{b^3}{a^2-ac+c^2}+\frac{c^3}{a^2-ab+b^2}\geq a+b+c$

I am trying to prove the following inequality For all positive numbers $a$, $b$ and $c$ we have $$\dfrac{a^3}{b^2-bc+c^2}+\dfrac{b^3}{a^2-ac+c^2}+\dfrac{c^3}{a^2-ab+b^2}\geq a+b+c$$ I can probably ...
0
votes
2answers
145 views

Solving inequalities containing rounding / integer math

Maybe I just have a mental block and this is really easy, but I'm having a hard time figuring out the following problem: Given an integer value $p$ and a scaling factor $f \in \mathbb{Q}$, i.e. $f = ...
2
votes
2answers
77 views

Showing that this Coercivity condition implies uniform boundedness of a minimising sequence.

The following problem is in Dacorogna's book "Introduction to the Calculus of Variations": Let $\Omega\subset\mathbb{R}^n$ be open and bounded with a Lipschitz boundary. Let $f\in C(\mathbb{R}^n\times ...
3
votes
2answers
85 views

How prove this $|x_{p}-y_{q}|>0$

let $$x_{1}=\dfrac{1}{8},x_{n+1}=x_{n}+x^2_{n},y_{1}=\dfrac{1}{10},y_{n+1}=y_{n}+y^2_{n}$$ show that: for any $p,q\in N^{+}$ we have $$|x_{p}-y_{q}|>0$$
5
votes
1answer
137 views

Estimation on elliptic operator

Assume the strongly elliptic property, i.e. $$\sum_{|\alpha|= m}a_\alpha \xi^\alpha\neq0,\ \forall \xi \in \mathbb{R}^d\backslash\{0\},$$ and $$\sum_{|\alpha|\leq m}a_\alpha ...
2
votes
0answers
55 views

Search for a candidate function with specific properties?

Given the following expression: $$ \mathcal{F(p,c,r,s)} = \frac{c^2 p^2 \left(s f'(s)-2 f(s)\right)^2}{4 f(s) \left(c^2 f(s) \left(c^2 p^2 f(s)+s^2 \left(r^2-p^2\right)\right)+\left(-r^2-1\right) ...
8
votes
3answers
194 views

For real numbers $x$ and $y$, show that $\frac{x^2 + y^2}{4} < e^{x+y-2} $

Show that for $x$, $y$ real numbers, $0<x$ , $0<y$ $$\left(\frac{x^2 + y^2}{4}\right) < e^{x+y-2}. $$ Someone can help me with this please...
0
votes
2answers
76 views

Metric on a set

Can someone provide a hint for solving the following. Show that $d:(R^{\infty})^2\to R_+$ is a metric. $$d(x, y)=\sqrt{\sum_{i=0}^{\infty}{(x_i-y_i)^2}}$$ I need a hint for showing that $d$ ...
0
votes
2answers
148 views

solution of the equation with exponential function

Let $m, n$ be an integers. Let $b \in R$. Solve the following equation for $n$. $$ \exp(m-\frac{2}{\pi}n)=n^{b/{\pi}}. $$ Thank you.
0
votes
1answer
54 views

Proving that $\Pr[|X| > T\sqrt{n}/2] \geq \Pr[|X-\mathbb EX| < T\sqrt{n}/2]$

I am reading a paper Revealing information while preserving privacy and I am stuck in a step in the proof of Lemma 4. I'll write the relevant details below so you do not need to extract them from the ...
5
votes
2answers
87 views

Inequalities $x/y > 0$

I am sure I am just being stupid but I can't get the illogic in this problem: $$ \begin{align} \frac{x}{y} &> 0\\ \frac{x}{y}\cdot y &> 0\cdot y\\ x &> 0 \end{align} $$ ...
6
votes
1answer
151 views

Can this inequality proof be demystified?

At http://www.artofproblemsolving.com/Forum/download/file.php?id=44351 , one finds a short proof by Vasile Cirtoaje of the inequality $$ \sqrt{8(a^2+bc)+9}+\sqrt{8(b^2+ac)+9}+\sqrt{8(c^2+ab)+9} \geq ...
1
vote
1answer
83 views

How to prove this simple inequality?

Please help me to prove this inequality. Suppose $X$ and $Y$ are independent and $EX=EY=0$, then we must have $E(|X|) \leq E(|X+Y|)$. Thanks.