Questions on proving, manipulating and applying inequalities.

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5
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2answers
2k views

Proving the AM:GM inequality [duplicate]

I am doing past exam papers preparing for the finals and I came across this questions about three times: Prove that: $$\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\geq \sqrt[n]{a_{1}.a_{2}...a_{n}}$$ ...
1
vote
0answers
171 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
0
votes
2answers
133 views

A peculiar quadratic inequality--Has it been seen before?

In the course of working on a problem on normal numbers, I ran into the following inequality: Let $n$ be a positive integer, $p_i$, $1\le i \le n$, be real numbers, and $q_i$, $1\le i \le n$, be ...
16
votes
2answers
535 views

Comparing sums of surds without any aids

Without using a calculator, how would you determine if terms of the form $\sum b_i\sqrt{a_i} $ are positive? (You may assume that $a_i, b_i$ are integers, though that need not be the case) When there ...
0
votes
1answer
51 views

A doubt about a delta estimation problem

I am able to solve these delta estimation problems, but I don't grasp the idea behind them. Here goes the problem: We have $f(x)=x^2$ and $a=5$ Determine $\delta > 0$ where $ 0 < \left| x ...
5
votes
1answer
276 views

Jensen's inequality

I am using Jensen's inequality and conditional expectation to prove the following inequality: Let $\lambda_i$ be real for $i\in \{1,2,...,M\}$ and $\bar{\lambda}=\frac{\sum_{i=1}^M\lambda_i}{M}$. ...
7
votes
2answers
81 views

Sum of radicals greater than 1

Prove that for every $n,m \in \Bbb N $ $$ \frac{1}{\sqrt[n]{1+m}} + \frac{1}{\sqrt[m]{1+n}} \ge 1 $$
2
votes
1answer
63 views

$H^1$ norm estimation of an affine function

Let $v(x)=\alpha +(\beta-\alpha)x$ a function in $H^1(\Omega)$ with $\Omega=[0,1]$ and $\alpha$ and $\beta$ are constants. How do we prove that there exist an constant $M >0$ such that ...
10
votes
6answers
2k views

How to prove an inequality with square roots?

Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. How do I show that $\sqrt{x^2+1}+\sqrt{y^2+1}\ge \sqrt{5}$?
3
votes
1answer
53 views

Help with this inequality

I am given four numbers $a,b,c,d$, such that $c>a>b,c>d>b$ and $0 \le a,b,c,d\le 1$ Can the following two inequalities hold strictly $ad\le bc$ and $(1-a)(1-d)\le (1-b)(1-c)$.
1
vote
0answers
45 views

What are the best and most elementary bounds for $n!$?

What this question is looking for is bounds on $n!$ that are elementary in nature (I seem to have a fetish for these type of proofs). In general, as the results become more complicated, they also ...
36
votes
3answers
1k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
2
votes
2answers
114 views

How find this $3\sqrt{x^2+y^2}+5\sqrt{(x-1)^2+(y-1)^2}+\sqrt{5}(\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y-1)^2})$

find this follow minimum $$3\sqrt{x^2+y^2}+5\sqrt{(x-1)^2+(y-1)^2}+\sqrt{5}\left(\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y-1)^2}\right)$$ I guess This minimum is $6\sqrt{2}$ But I can't prove,Thank you
0
votes
1answer
139 views

Inequality related to modulus of complex numbers

How to show: for any $\alpha >0$, there is constant depends on $\alpha$, say $C=C(\alpha)$, such that, $$\mid \mid w \mid ^{\alpha} w - \mid z \mid ^{\alpha} z \mid \leq C (\mid w \mid ^{\alpha} ...
0
votes
1answer
47 views

Finding solutions to $\left( \frac{p}{c} \right) ^ c \left(\frac{1-p}{1-c}\right)^{1-c} \lt 1$

I wish to find solutions $(p,c)$ for the following inequality: $$\left( \frac{p}{c} \right) ^ c \left(\frac{1-p}{1-c}\right)^{1-c} \lt 1$$ Given that $0 \lt c \lt p \lt 1$ Is there any general good ...
1
vote
1answer
45 views

Estimate on differentiable functions on a bounded domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with $\partial \Omega \in C^1$ and $u \in C^{1, \alpha} (\overline \Omega)$, do we have $$|\frac{u(x) - u(y)}{x - y}| \leq ||Du||_{\infty}$$ for ...
2
votes
1answer
414 views

Obtain lower and upper bounds

How do I obtain upper and lower bounds for a summation function: $$ \sum_{i=1}^{25}i^4 $$ Somehow it involves an integral: $$ \int_{0}^{25}x^4\:\mathrm{d}x $$ If you solve it it gives $1953125$ (this ...
1
vote
1answer
169 views

How prove this $|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\frac{1}{20n^3}$

Prove that $$|\{n\sqrt{3}\}-\{n\sqrt{2}\}|>\dfrac{1}{20n^3}$$ let $t=\{n\sqrt{2}\}-\{n\sqrt{3}\}$ and $k=[n\sqrt{3}]-[n\sqrt{2}]$ then we have ...
0
votes
2answers
103 views

How to solve $|2x +1|< 1/4$?

How do you solve $$|2x +1|< \frac{1}4$$
0
votes
3answers
2k views

Solution to cubic inequality

How do I find the solutions to this equation? $$(3x^3)-(14x^2)-(5x) \leq 0$$
0
votes
1answer
52 views

expected value involving probability of inequality random variables.

I have a question, not sure if this can be solved by calculation or Monte Carlo method For random variable G2+G2*min(2/G2-1,G1) where G1, G2 are indenpendt, G1~Lognormal(mu1,cigma1) ...
1
vote
2answers
100 views

Does this hold for three numbers [duplicate]

If $a\ge b\ge c\ge0$, does it hold that $\sqrt[3]{\left(a-b+c\right)^{2}}\ge\sqrt[3]{a^{2}}-\sqrt[3]{b^{2}}+\sqrt[3]{c^{2}}$? Thanks for any help.
1
vote
2answers
47 views

Evaluation Of Formula (Five Variables)

$a\geq 1$, $b\geq 1$, $c\geq 1$, $x\geq 1$, $y\geq 1$ Evaluate the range of $~$ $\dfrac{(a+1)(b+1)(c+1)(x+1)(y+1)}{a+b+c+x+y+1}$ I have no solution but hope that there exist one using no ...
5
votes
2answers
244 views

Proving $e^{\binom{n}{2}}>n!$

Prove that $$e^{\binom{n}{2}}>n!$$ $n \in \mathbb{Z_+}$ Sorry, couldn't attempt it.
5
votes
2answers
150 views

How prove this inequality $\dfrac{R}{r}\ge\dfrac{b}{c}+\dfrac{c}{b}$

in $\Delta ABC$,prove that $$\dfrac{R}{r}\ge\dfrac{b}{c}+\dfrac{c}{b}$$ where $R$ is the circumradius and $r$ is the inradius By the way.It is well konwn that Eluer inequality $$R\ge 2r$$ and it is ...
3
votes
1answer
119 views

Does this inequality involving differences between powers hold on a particular range?

Let $$f(x)=\left(1-\frac{2x}{x+c}\right)^{-n}-\left(1+\frac{x}{c}\right)^{2n}$$ and $$g(x)=\left(1-\frac{2x}{c}\right)^{-n}-\left(1-\frac{x}{c}\right)^{-2n}$$ where $c>0$ and $n>0$ are ...
4
votes
1answer
241 views

Proving $\binom{2n}{n}\ge\frac{2^{2n-1}}{\sqrt{n}}$

Prove that $$\binom{2n}{n}\ge\dfrac{2^{2n-1}}{\sqrt{n}}$$ By the way: I have see $$\binom{2n}{n}\ge\dfrac{4^n}{2n}=\dfrac{2^{2n-1}}{n}$$ proof: Applying the binomial theorem ...
2
votes
1answer
51 views

Maneuvering normal distribution inequality

I know $\Pr[N(\mu, \sigma^2) \geq \mu + k \sigma] = \Pr[N(0, 1) \geq k]$. Say I am given, $\Pr[N(\mu, \sigma^2) \geq \mu + q \sigma] = \Pr[N(0, 1) \leq k]$. How can I find a relation between $q$ and ...
4
votes
1answer
87 views

If $a\ge b\ge-c\ge0$, is $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?

Let $a\ge b\ge-c\ge0$. Is it true that $\sqrt[3]{a-b-c}\ge\sqrt[3]{a}-\sqrt[3]{b}-\sqrt[3]{c}$?
1
vote
1answer
67 views

Proof for an inequality(Trigonometric functions)

I would appreciate if somebody could help me with the following problem Q: How to proof $$\sqrt{1+a^2-2a\cos t}\leq (1-a)+2a\sin\frac{t}{2}(\text{where}~~ 0\leq a\leq 1, 0\leq t\leq 2\pi)$$
2
votes
2answers
83 views

we can prove $b_{1}=b_{3}=0$?

for any reanl numbers $t\in[-\sqrt{2},\sqrt{2}]$,then have $$-\dfrac{1}{2}\le t^4+b_{3}t^3+b_{2}t^2+b_{1}t+b_{0}\le\dfrac{1}{2}$$ prove or disprove $$b_{1}=b_{3}=0$$ my idea: I can only prove that ...
1
vote
1answer
30 views

Prove that $F_{\sum_{i=1}^ka_i}\geq \prod_{i=1}^kF_{a_i}\forall a_i,k \geq 1$

Is there an elegant way to do this? I don't think it's particularly difficult, since $F_n \sim \frac{\phi^n}{\sqrt{5}}$, so we expect that $F_{\sum_{i=1}^ka_i} \sim ...
1
vote
1answer
58 views

Inequality relating harmonic numbers

I'm stuck at proving the following inequality: For any m,n positive integers with $m \leq2^n$ we have $$ \sqrt{\frac{2^n}{1}} + ...
1
vote
1answer
249 views

Solving inequalities at powers greater than $2$

Usually when I see an inequality like $x^2 - 6x - 16 < 0$, I know that the answer is $-2 < x < 8$ because I can picture where the graph would lie below zero. However, for a problem like ...
4
votes
4answers
560 views

Why does the Cauchy-Schwarz inequality hold in any inner product space?

I am working through linear algebra problems in Apostol's Calculus, and he has numerous problems that seem to imply that Cauchy-Schwarz holds no matter how the inner product is defined. Then, he has ...
0
votes
0answers
67 views

Showing an inequality holds

I want to show that $$ 2a_2^2b_1^2+a_2^2b_3^2-4a_1a_1b_1b_2-2a_2a_3b_1b_2-2a_1a_2b_2b_3-2a_2a_3b_2b_3+a_3^2b_1^2+2a_1^2b_2^2+a_3^2b_2^2+2a_1a_3b_2^2+a_1^2b_3^2-2a_1a_3b_1b_3 \geq 0. $$ This is what I ...
7
votes
4answers
277 views

Looking for a slick way to prove this.

Suppose that $a_k \ge 0$, $k=0,1,2,3,…$ satisfies $a_k^2\le a_{k-1}a_{k+1}$ for all $k\ge 1$. Show that for all $k\in \{0,1,\dots, N\}$ we have $$ a_k \le a_0^{1-k/N} a_N^{k/N} $$ I am looking ...
0
votes
2answers
90 views

Prove $|x+1|\leq 4$ implies that $-4\leq x\leq 2$.

How do I prove that if $x$ is a real number, then $\lvert x+1 \rvert\leq 3$ implies that $-4\leq x\leq 2$. EDIT: $\lvert x+1 \rvert\leq 4$ should be $\lvert x+1 \rvert\leq 3$
0
votes
1answer
161 views

cardinality, set of values

For any given $a,b,c \in \mathbb{Z}$, $a \le b$, what is the cardinality $\left|[a,b] \cap c\mathbb{Z} \right|$ ? I pick at random an element $a$ from the finite set $\mathbb{Z}/n\mathbb{Z}$. I ...
-1
votes
3answers
124 views

Prove that $n! \geq 2^{n-1}$ for $ n\geq1$ [duplicate]

Mathematical Induction:-Prove that $n! \geq 2^{(n-1)}$ for $n\geq 1$. I tried mathematical induction but could not
0
votes
2answers
109 views

Generalization of Minkowski's inequality.

Minkowski's inequaity states that $$\displaystyle{(|x_1+y_1|^p + |x_2+y_2|^p +\dots +|x_n+y_n|^p)^{\frac{1}{p}}\leq (|x_1|^p + |x_2|^p +\dots +|x_n|^p)^{\frac{1}{p}}+(|y_1|^p + |y_2|^p +\dots ...
3
votes
1answer
216 views

Where is the mistake in this argument that $(\sqrt8)^{\sqrt 7} >(\sqrt7)^{\sqrt 8}$?

I posted an answer in this question to prove that $(\sqrt8)^{\sqrt 7}<(\sqrt7)^{\sqrt 8}$ I started with $$f(x)=\frac{\ln x}{x}$$ $$f'(x)=\frac{1-\ln x}{x^2}$$ $$f'(x)>0 : x\in)0,e($$ ...
1
vote
2answers
196 views

Does the following inequality hold if and only if $N$ is an odd deficient number?

Let $N \in \mathbb{N}$. (That is, let $N$ be a positive integer.) This is in reference to two of my earlier questions here at MSE: Does the following inequality hold true, in general? Does this ...
0
votes
2answers
144 views

How to show that we can always choose a smaller number?

Suppose $A$ consists of all numbers $x \ge 0$ such that $x^2 \lt 2$. If the number $\sqrt{2}$ did not exist, there would not be a least number greater than all the numbers of $A$; for any $y > ...
0
votes
2answers
211 views

Conditional expectation and sum of random variables

Let $Y, X_1, . . . , X_n$ be continuous random variables (not necessarily independent) with non negative range, i.e. $P(Y < 0) = 0$ and $P(X_i < 0) = 0$ for $i = 1 \ldots n$, verifying the ...
1
vote
1answer
943 views

Proving Jensen's inequality.

I've been wondering about this problem for some time. This is in a way related to the proof of Jensen's inequality. Say we have a convex function $f$. Can we prove $$\frac{f(a)+f(b)+f(c)}{3}\geq ...
2
votes
4answers
178 views

Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
1
vote
0answers
29 views

How find this $m$ Value range

let $a\ge\dfrac{2^{m-1}-1}{m-1}$and such $$\left(\dfrac{\dfrac{3}{4}(\dfrac{3}{4}+a)(\dfrac{3}{4}+2a)\cdots(\dfrac{3}{4}+(m-1)a)}{(1+a)(1+2a)\cdots ...
1
vote
2answers
139 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
2
votes
2answers
180 views

Apostol question on alternative definition of dot product

The problem says: Suppose we define the dot product by $A\cdot B = \sum_{k=1}^n |a_kb_k|$. Which of the following properties hold with this new defition? Does the Cauchy-Schwarz inequality still ...