Questions on proving, manipulating and applying inequalities.

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2
votes
0answers
33 views

Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$. [duplicate]

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it? I see that if we can show $P[1 \le X - Y \le 2] \le P[|X ...
3
votes
4answers
41 views

Show that the $C_n \geq 4^{n-1}/2^{n}$ where $C_n$ is the Catalan number

I write $C_n=\frac{1}{n+1} {2n\choose n}$ and try to prove this claim by induction. But it didn't quite work out. Any idea how to do this without much computation?
5
votes
5answers
94 views

For integer $n>2$, $(n!)^2 > n^n$

Problem: For integer $n>2$, show that $(n!)^2 > n^n$ My attempt: I tried using induction. For $n=3$, the given condition is satisfied. Let us suppose $k!^2>k^k$ for some $k\geq3$. Then, ...
5
votes
1answer
96 views

Transformation that preserves an increasing ratio between vectors

Consider two vectors $x=(x_1,x_2,\ldots,x_n)$, $y= (y_1,y_2,\ldots,y_n)$ such that all $x_i,y_i>0$ and \begin{align} \frac{y_1}{x_1}\le \frac{y_2}{x_2}\le\cdots\le \frac{y_n}{x_n} \end{align} Now ...
0
votes
1answer
23 views

$\left | \sum_{n\in \mathbb N} a_n b_n z^{n} \right | \leq C \left | \sum_{n\in \mathbb Z} b_n z^n \right | (z\in \mathbb C)$?

Let $ a_n , b_n \in \mathbb C$ for all $n\in \mathbb N.$ And there is $M>0$ such that $|a_n| \leq M$ for all $n\in \mathbb N.$ Can we expect $\left | \sum_{n\in \mathbb N} a_n b_n z^{n} \right | ...
0
votes
0answers
27 views

An inequality of first order partial derivatives.

Suppose $f:\mathbb R^2\to \mathbb C$ is $C^2$ with compact support. Show that $$\left\|\frac{\partial f}{\partial x_1}\right\|_p+\left\|\frac{\partial f}{\partial x_2}\right\|_p\le ...
15
votes
2answers
170 views

if $x^y=y^x$ show that $x+y>2e$

Let $0<x<y$, such that $$x^y=y^x$$ show that $$x+y>2e$$ Since $$y\ln{x}=x\ln{y}\Longrightarrow \dfrac{\ln{y}}{y}=\dfrac{\ln{x}}{x}$$ Let $$f(x)=\dfrac{\ln{x}}{x}\Longrightarrow ...
0
votes
2answers
30 views

Inversion of the inequality sign when raising to a negative power

How come since $e>1 \implies e^{-1/2} < 1^{-1/2}$. I know that one reverses the inequality signs when we take reciprocal of both sides or multiplies by a negative number. I have never seen ...
0
votes
0answers
37 views

Cauchy Schwarz look alike

Let $0<r<1<R$ be two fixed numbers. Suppose that there exist real numbers $x_1$, $y_1$, $z_1$, $x_2$, $y_2$, $z_2$, such that $x_i^2+y_i^2+(z_i-R)^2=r^2$ and $z_i\ge\frac{R^2+r-r^2}{R}$ for ...
1
vote
2answers
60 views

Proving that $\left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3$, given $1\leq a<b$

If $1\leq a<b$, then $$ \left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3.$$ Proceeding by integration by parts; let $u(x)=\sin(x)$ and $dv(x)=1/x$, then $u'=\cos(x)$ & $v(x)=\log(x)$. We ...
0
votes
0answers
35 views

If $f(x)\le 1$ implies $f(x)\le 1/2$ then $f(x+\delta)\le 1$?

Let $f(x)$ be a nonnegative continuous function of $x\in [0,K)$ with $f(0)\le 1/2$, and satisfies "$f(x)\le 1$ implies $f(x)\le 1/2$". Let $x_0\in[0,K)$ (so that $f(x_0)\le 1$ implies $f(x_0)\le ...
1
vote
6answers
64 views

Prove algebraically that, if $x^2 \leq x$ then $0 \leq x \leq 1$

It's easy to just look at the graphs and see that $0 \leq x \leq 1$ satisfies $x^2 \leq x$, but how do I prove it using only the axioms from inequalities? (I mean: trichotomy and given two positive ...
2
votes
0answers
57 views

Inequality involving fourth powers .

I have been into inequalities lately and I am stuck with this. I used a famous inequality at first $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \ge 3 (\frac{a^4+b^4+c^4}{3})^{\frac{1}{4}}$. From this ...
4
votes
1answer
63 views

Prove that : $\frac{a+b+c+d}{a+b+c+d+f+g}+\frac{c+d+e+f}{c+d+e+f+b+g}>\frac{e+f+a+b}{e+f+a+b+d+g}$

Prove inequality for positive numbers: $$\frac{a+b+c+d}{a+b+c+d+f+g}+\frac{c+d+e+f}{c+d+e+f+b+g}>\frac{e+f+a+b}{e+f+a+b+d+g}$$ My work so far: Lemma: If $x>y>0, t>z>0$, then ...
1
vote
1answer
51 views

Prove $(1+x)^p+(1-x)^p \ge 2(1+x^p)$ for $0\le x\le1$ and real number $p\ge2$.

I don't know how to prove the following questions: If $p\ge2$ is real, then $$ (1+x)^p+(1-x)^p \ge 2(1+x^p) \quad \text{for } 0\le x\le1; $$ if $1\le p<2$, then opposite direction of the inequality ...
1
vote
2answers
40 views

If $a,b,c>0$ and $abc=1\;,$ Then minimum value of Expression.

If $a,b,c>0$ and $abc=1\;,$ Then minimum value of $$\frac{a^2}{a^2+2}+\frac{b^2}{b^2+2}+\frac{c^2}{c^2+2}$$ $\bf{My\; Try::}$ Using $\bf{Cauchy\; Schwarz}$ Inequality ...
-4
votes
1answer
46 views

Inequality that just won't go up!

I've gotten this result in an exam question on economics, and I can't seem to get this to make sense. Here, $Y$ is unknown. So, how do we know that this is true? $$(1-Y)(1-C) < (1+Y)(1-P), \quad P ...
-1
votes
1answer
42 views

I want to show that $x^2 - x + C\epsilon\ge 0$ under some assumption. [closed]

Let $x\ge 0$. For sufficiently small $\epsilon>0$, assume that the property $x\le \sqrt\epsilon$ implies $x\le \frac{1}{2}\sqrt{\epsilon}$. Then I want to show that $$x^2 - x + C\epsilon\ge 0 $$ ...
1
vote
1answer
45 views

Inequality with sum of numbers

A have found a very interesting inequality in a Romanian magazine which I use to prepare for the Lithuanian Mathematical Olympiad. Let $a_1,a_2,...,a_n$ be positive real numbers such that $$\frac {1} ...
1
vote
1answer
53 views

A polynomial inequality

Let $f(x)\in\mathbb{R}[x]$ be a polynomial of degree $n$, which has only real zeros. I would like to show that $$(n − 1)(f'(x))^2 \geq nf(x)f''(x),$$ where $f'$ and $f''$ denote the first and second ...
2
votes
3answers
113 views

Prove inequality with $e^x$ and $\ln$ on the same side [duplicate]

The problem is to prove the following inequality: $$ (e^x - 1) \ln(1+x) > x^2 , \quad\text{ for } x >0 $$ Let me introduce notation $f(x) > g(x)$. At $x=0$ both sides are equal to $0$. So, ...
2
votes
1answer
65 views

Inequality about sums and products

Let $x_1,x_2,...,x_n$ be positive real numbers. Show that $$\frac {1} {2^n \times \sqrt {x_1 \times x_2 \times ... \times x_n} } + \sum_ {k=1}^n \frac {x_k} {(x_1+1)(x_2+1)...(x_k+1)} \ge 1.$$ I tried ...
1
vote
0answers
48 views

$1+x^4\leq 2(y-z)^2$ and switching of $x,y,z$

Find all triples of real numbers $x,y,z$ such that $1+x^4\leq 2(y-z)^2$, $1+y^4\leq 2(z-x)^2$, and $1+z^4\leq 2(x-y)^2$. Beside $(1,0,-1)$ and permutations, I can't find any others. We cannot have ...
1
vote
5answers
58 views

Prove that $\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$

Is it possible to prove that $$\frac{\pi^3}{48} \le \int_0^{\pi/2}\frac{x^2}{2-\sin(x)}\,dx \le \frac{\pi^3}{24}$$ without evaluating the integral?
0
votes
0answers
17 views

Bounding the $q$-th moment of a Gaussian random variable

I have come across an inequality which confuses me: Suppose $X$ has a Normal$(0,\sigma^2)$ distribution. Then $$ (\mathbb{E}|X|^q)^{1/q} \leq \text{const.} \sqrt{q} \sigma $$ for $q\geq 1$. I ...
1
vote
1answer
23 views

How to combine inequalities

If $u,v$ are real numbers and $|u-3|<1/3$ and $|v-3|<2/3$ then show that $|v-u|<1$. I'm unsure about how to combine this inequalities and simplify. Thanks in advance.
14
votes
5answers
220 views

How to show $\frac{19}{7}<e$

How can I show $\dfrac{19}{7}<e$ without using a calculator and without knowing any digits of $e$? Using a calculator, it is easy to see that $\frac{19}{7}=2.7142857...$ and $e=2.71828...$ ...
1
vote
0answers
40 views

Prove that if $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ than $|f(x)| \leq c \left|\frac{1}{x}\right|^N$

The task is: Knowing that $\forall \lambda >0, x \neq 0$ $f(\lambda x) = \left(\frac{1}{\lambda}\right)^N f(x)$ prove that: $|f(x)| \leq c \left|\frac{1}{x}\right|^N$ I would really appreciate ...
0
votes
2answers
25 views

Matrix norm inequality proof - does this use Cauchy-Schwarz?

The matrix norm for $A : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (so $A$ is an $m \times n$ matrix) is given by $$\|A\| = \sup_{X \in \mathbb{R}^n \setminus \{0\}} \frac{|AX|}{|X|}$$ where $| \cdot |$ ...
1
vote
0answers
74 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...
2
votes
2answers
39 views

Prove by induction that $\sum_{k=1}^nk^p < (n+1)^{p+1}/(p+1), \quad n,p \in \mathbb{N}$

For $n=1$, we have at the left side $1^p$, and at the right side: $$ \frac{2^{p+1}}{p+1}\mathrm{~which~is } >1$$ so it holds for $n=1$, but how can we prove that $$ ...
2
votes
2answers
37 views

Help solving the inequality $2^n \leq (n+1)!$, n is integer

I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook: $2^n \leq (n+1)!$ Where $\mathbf{n}$ in an integer. I've tried applying lg to both ...
1
vote
0answers
28 views

Use Chebyshev’s inequality to choose $n$ such that $P(\bar{X} > 4) > 0.9$

Use Chebyshev’s inequality to choose n such that $$ P(\bar{X_n} > 4) > 0.9 $$ where $$ E[\bar{X_n}] = 5 \ \ \ \ \ Var[\bar{X_n}] = \frac{4}{n} $$ The problem I am having when using Chebyshev's ...
0
votes
2answers
52 views

Inequality involving square root exponents

Show that $$ 2^ {\frac {1} {\sqrt 2}} + 2^ {\frac {1} {\sqrt 3}} \gt 3.$$ I tried to use AM-GM inequality and Jensen's inequality, but I didn't get to any results.
0
votes
1answer
22 views

Does a sum of squares become smaller as the number of terms increases?

I am interested in the following question: Let $,kn$ be a positive integeres. Assume $\sum_{i=1}^{k} L_i=\sum_{i=1}^{k+1} \tilde L_i=n$, where $L_i,\tilde L_i$ are positive integers. Is it true ...
2
votes
2answers
78 views

How to properly find supremum of a function $f(x,y,z)$ on a cube $[0,1]^3$?

Solving an applied problem I was faced with the need to find supremum of the following function $$f(x,y,z)=\frac{(x-xyz)(y-xyz)(z-xyz)}{(1-xyz)^3}$$ where $f\colon\ [0,1]^3\backslash\{(1,1,1)\} ...
3
votes
0answers
38 views

Estimating $n!$ as $e \left(\frac ne \right)^n \le n! \le ne \left(\frac ne \right)^n$

I'm told that for $n \geq 2,$ $$\sum_{k=1}^{n-1} f(k) \leq \int_1^n f(x) \, dx \leq \sum_{k=2}^n f(k)$$ I am then asked to consider $\ln n! = \sum_{k=1}^n \ln k$ and show that for $n \geq 2$ $$n! ...
0
votes
1answer
26 views

Algebraic Inequality

If a,b,c are positive real numbers and $z = \frac{b^2 + c^2}{b+c} + \frac{c^2 + a^2}{a+c} + \frac{a^2+b^2}{a+b}$ then only one of the following statements is always true , which on is it ? a) ...
2
votes
1answer
70 views

Prove the inequality $\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\geq 4$

$a, b, c, d$ are positive reals. How would I prove the inequality $$\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a} \geq 4$$ I have tried using the rearrangement inequality with ...
0
votes
3answers
42 views

Inequality with a square root

If the inequality $ (x+2)^{\frac{1}{2}} > x $ is satisfied. what is the range of x ? My approach - I squared both the sides and proceeded on to solve the quadratic obtained in order to solve the ...
3
votes
1answer
27 views

Rational function between a constant and a third root

Is there a rational function $f(x)\in{\mathbb Q}(x)$ such that $\sqrt{2} \leq f(x) \leq \sqrt[3]{2x}$ for all $x\geq\sqrt{2}$ ? My thoughts : it is easy to find such an $f$ if we relax the ...
0
votes
2answers
20 views

Proof the inequalitiy for the two Matrices $A, B$

Let $ A,B \in C^{nxn}$ then $$ 1 \le || (\lambda I-B)^{-1})(A-B)||\le ||(\lambda I-B)^{-1}||*||(A-B)||$$ for any eigenvalue $\lambda $ of $ A $ which is not an eigenvalue of $ B$ and any operator ...
1
vote
1answer
62 views

Geometric inequality $180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}$

Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}\le 180^{\circ}$$ My work so far: ...
1
vote
2answers
51 views

Proving $\sqrt x\ge\log(x+1)$

What is a simple proof that $\sqrt x\ge\log(x+1)$ for $x\ge 0$? I'm trying to prove that $\sum_n\frac{\log n}{n(n-1)}$ converges, and my idea is to upper bound this with the telescoping sum ...
0
votes
2answers
26 views

Application of A.M. -G.M. inequality

Let x, y,z be positive numbers. The least value of $ \frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{.5}}$ is a) $\frac{9}{2^{.5}}$ b) 6 c) $\frac{1}{6^{.5}}$ d.) None of the above I tried applying the A.M. ...
1
vote
3answers
54 views

Prove when $abc=1$: $ \frac{a}{2+bc} + \frac{b}{2+ca}+\frac{c}{2+ab} \geq 1$

Question: Prove the following inequality which holds for all positive reals $a$, $b$ and $c$ such that $abc=1$: $$ \frac{a}{2+bc} + \frac{b}{2+ca}+\frac{c}{2+ab} \geq 1$$ My thoughts were ...
1
vote
2answers
53 views

Problem in Solving an Inequality

The problem is: $Prove$ $that$ $|\sin^2 (x)-\sin^2 (y)|\le |x-y|$ $ for $ $ all $ $ x,y>0$. $My$ $work$ : $$\sin^2 (x)\le|\sin x|\le|x|\le|x-y|+|y|$$ and so is $$|\sin^ 2 (x)-\sin^2 (y)|\le ...
0
votes
1answer
14 views

On the relationship between $\max(p_i)$ and $\omega(b)$, if $\sigma(b^2)/b^2$ is bounded above by a specific number $U$

Let $\omega(x)$ denote the number of distinct prime factors of $x$, and let $\sigma(x)$ be the sum of the divisors of $x$. Denote the abundancy index of $x$ by $I(x) = \sigma(x)/x$. Let the number ...
1
vote
1answer
18 views

Question about the validity of a proof involving the abundancy index

Let $\sigma(x)$ be the sum of divisors of $x$, and denote the abundancy index of $x$ by $I(x) = \sigma(x)/x$. Consider the number $y^2 \in \mathbb{N}$, and suppose that I know that $I(y^2) < 4/3$. ...
1
vote
1answer
20 views

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$?

Does this Diophantine inequality have any solutions for $p, q \in \mathbb{N}$? $$p^2 q^2 \geq 3 p^2 q + 3p^2 + 3pq^2 + 3pq + 3p + 3q^2 + 3q + 3$$ I tried to use Wolfram Alpha, and it says that ...