Questions on proving, manipulating and applying inequalities.

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2
votes
3answers
74 views

Prove $2^n\cdot n! ≤ (n+1)^n$ by induction.

An induction I'm struggling with. Prove $2^n\cdot n! ≤ (n+1)^n$ by induction. An idea was to show that $2^n\cdot n! ≤ 1+n^2$ since $1+n^2 ≤ (n+1)^n$ using Bernoulli. However the inequality is ...
7
votes
2answers
236 views

Unequal circles within circle with least possible radius?

It is the classical will-my-cables-fit-within-the-tube-problem which lead me to the interest of circle packing. So basically, I have 3 circles where r = 3 and 1 circle where r = 7 and I am trying to ...
1
vote
1answer
97 views

Does a schur-concave function needs to be symmetric?

Does a schur-concave function need to be symmetric? If not, how can we check schur-concavity for a non-symmetric function? I know this lemma (schur condition, this book) Let $\mathcal{I} \subset ...
4
votes
1answer
109 views

Inequality $ \left|x\sin\frac{1}{x}-y\sin\frac{1}{y}\right|\leq\sqrt{2|x-y|} $

$ \forall x,y>0 $, then $$ |x\sin\frac{1}{x}-y\sin\frac{1}{y}|\leq\sqrt 2\sqrt{|x-y|} $$ Is $\sqrt 2$ the minimum positive real number such that the inequality holds ? I try to apply Mean value ...
4
votes
2answers
104 views

If $a_1+a_2+\ldots+a_{2000}>a_1a_2\ldots a_{2000}$, prove that at least $1990$ of those numbers are equal to $1$.

If $a_1,a_2,\ldots,a_{2000}\in\mathbb N$ and$$a_1+a_2+\ldots+a_{2000}>a_1a_2\ldots a_{2000}$$ Prove that at least $1990$ of those numbers are equal to $1$. That's an unusual problem for me and I ...
2
votes
1answer
64 views

Prove or disprove an inequality involving statistics

Do we have any result in statistics like this: $$|\overline x - \mu_e| \leq \sigma$$ Here $\overline x$ denotes the usual mean of some given discrete observations, $\mu_e$ their median and ...
1
vote
1answer
36 views

Proving that $|\Phi_n(x)| > x-1$

Let $\Phi_n$ be the n-th cyclotomic polynomial. I'd like to prove that $$\forall n \geq 2, \forall x \in [2, \infty[, |\Phi_n(x)| > x-1$$ The result is clear when $n$ is prime, but I'm struggling ...
0
votes
2answers
58 views

Is convergence in the norm equivalent to convergence of norms?

If $\| \cdot \|$ is a norm on some space. Does the equivalence go both ways? $$\| f_n-f \| \to 0 \Longleftrightarrow \| f_n\| \to \| f\| $$ The $\implies$ direction is obvious since $\| f_n-f \| ...
0
votes
2answers
50 views

How do I get this inequality

$2d/(1+d^2)\leq 2/d$ How did he get from the thing on the left to the thing on the right? <= is less than or equal to. And also, where do I learn how to typ mathematics symbol on the ...
12
votes
3answers
338 views

Prove that $\,\,\displaystyle\inf_{n\in\mathbb N}\sum_{k=0}^{p}\lvert\sin{(n+k)^p}\rvert>0$

For any positive integer number $p$, show that $$\inf\left\{ {\left\vert\sin{(n^p)}\right\vert+\left\vert\sin{(n+1)^p}\right\vert+\cdots+ \left\vert\,\sin{(n+p)^p}\right\vert\, :\,n\in ...
10
votes
4answers
372 views

How to prove $\left(\frac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\frac{n}{n+1}\right)^n$

Show that: $$\left(\dfrac{n}{n+1}\right)^{n+1}<\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}<\left(\dfrac{n}{n+1}\right)^n$$ where $n\in \Bbb N^{+}.$ If this inequality can be proved, then we have ...
2
votes
1answer
83 views

Need help filling in the details of proof of Jensen's Inequality

In a book on PDEs that I'm reading, I am trying to fill in the details of the proof of Jensen's Theorem, and am having a little trouble with the algebra. Here is the statement of Jensen's Theorem in ...
1
vote
1answer
51 views

Outputting inequality with $e^x$

I many books I can find inequality which estimates $e$: $$\left(1+\frac{1}{n}\right)^n \lt e \lt \left(1+\frac{1}{n}\right)^{n+1}$$ I am wondering if correct is also to write: ...
1
vote
2answers
60 views

Probability inequality proof

I'm stuck on a homework question and don't even know where to start. Here it goes: If A and B are two events which are not impossible, prove that $$P(A\land B)\times P(A\lor B)\le P(A)\times P(B)$$
6
votes
4answers
254 views

Prove the given inequality

$$\sin^{2}A(\tan(B-C))>\sin^{2}B(\tan(A-C)) $$ $$\implies \frac{\sin^2 A}{\sin^2 B} > \frac{\tan(A-C)}{\tan(B-C)}$$ Given if $A>B>C$ and $A+B+C=180^\circ$ Is that implication correct if ...
3
votes
1answer
88 views

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$.

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$. Restrict the values of $p$ in $[0,2\pi]$. The given equation has real roots if: $$\cos^2 ...
2
votes
2answers
35 views

Is the following statement true? And if so where can I find a proof

Is the following true for real numbers? If $x < a*b$ then there exists $c$ and $d$ such that $x=c*d$ and $a>c$ and $b>d$. Thanks...
2
votes
1answer
43 views

Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
1
vote
0answers
64 views

Increasing fraction of sum of binomial coeffients

Let $n$ be a positive integer. Show that the quantity $$ \displaystyle \frac{ \displaystyle \sum_{i=1}^n { n+k \choose i-1 } }{ \displaystyle \sum_{i=1}^n { n+k+1 \choose i } } $$ is ...
3
votes
1answer
56 views

How prove this $\prod_{cyc}(a^3+2b+\frac{2}{a^2+1})\ge 64$

let $a,b,c>0$ and such $$abc\ge 1$$ show that $$\left(a^3+2b+\dfrac{2}{a^2+1}\right)\left(b^3+2c+\dfrac{2}{b^2+1}\right)\left(c^3+2a+\dfrac{2}{c^2+1}\right)\ge 64$$ my try: ...
4
votes
1answer
93 views

A homogeneous but slightly asymmetric inequality involving $L_1,L_{p-1}$ and $L_p$ norms.

I need to prove or disprove the following inequality: for any $Z=(z_1,\ldots,z_l)\in\mathbb{C}^L$ and for any $p\geq 2$, $$\biggl|\biggl\|\sum_{j=1}^L z_j\biggr\|^p - ...
19
votes
4answers
401 views

Prove: $\sin (\tan x) \geq {x}$

I bumped into this question: Question: Prove that for $x\in \Bigl[0,\dfrac {\pi}{4}\Bigr]$, $$\sin (\tan x) \geq {x}$$ This seems to be an innocent inequality but I am already exhausted trying ...
0
votes
1answer
357 views

My proof of the inequality of arithmetic and geometric means

Doing the exercises from the Apostol's Calculus I give my proof that the geometric mean is less than or equal the arithmetic mean ($G \le M_1$). I followed the hints from the book and I think I've ...
0
votes
2answers
186 views

Prove that $py^{p-1}(x-y)<x^p - y^p<px^{p-1}(x-y), (0<y<x,p>1)$

I don't know how to proof the following question: $$py^{p-1}(x-y)<x^p - y^p<px^{p-1}(x-y),(0<y<x,p>1)$$ Thank you very much.
1
vote
1answer
69 views

Triangle Inequality Question

If $-2 \leq x \leq \pi/2$, show that $|2x^3 - 4x^2 + 3x - \sin x|\leq 39$ can someone help me with this question? i'm having difficulty trying to incorporate triangle inequality with it.
5
votes
3answers
641 views

Using roots and exponents when solving inequalities

This is my first question on the site and I'm sure there'll be many more. But for now to the point. I am really having trouble understanding the proper use of roots and exponents to try to solve ...
1
vote
0answers
74 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
0
votes
1answer
145 views

best constant in quasi triangle inequality for $L^p$ spaces with $0 < p \le 1$

Currently doing a problem that ask me to prove the best $K$ such that the quasi triangle inequality $||f+ g||_p \le K (||f||_p + ||g||_p)$ for $L^p $ spaces holds, where $0< q \le 1$, is $2^{1/p -1 ...
0
votes
4answers
49 views

Where is a flaw in these logical implications?

We have a theorem: If $a \le x < a + {\frac yn}$ for $y > 0$ and all natural $n \ge 1$ then $x = a$. Suppose I derive that $a < x$ and $x < a + {\frac yn}$ for all $n \ge 1$. In other ...
6
votes
2answers
142 views

Show that $f(x)=0$ for all $x\geq0$

I have been struggling with this problem.. Q. Let $f(x)$, $x\geq 0$, be a non-negative continuous function, and let $F(x)=\int_0^x f(t) dt$, $x\geq0$. If for some $c>0$, $f(x)\leq cF(x)$ for ...
3
votes
7answers
202 views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without ...
0
votes
1answer
74 views

Some questions on the proof of Hoelders inequality.

I have some questions about the proof of Hoelder's inequality. Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in ...
7
votes
2answers
332 views

How prove this inequality $\sum_{cyc}\frac{a^2}{b(a^2-ab+b^2)}\ge\frac{9}{a+b+c}$

let $a,b,c>0$, show that $$\dfrac{a^2}{b(a^2-ab+b^2)}+\dfrac{b^2}{c(b^2-bc+c^2)}+\dfrac{c^2}{a(c^2-ca+a^2)}\ge\dfrac{9}{a+b+c}$$ My try: since this inequality is homogeneous ,without loss of ...
0
votes
2answers
164 views

Difference between two inequality symbols

I have come across the symbol $ \leqq$ in a paper I am reading. Is there any difference between this symbol and the symbol $ \leq $?
2
votes
4answers
120 views

Stuck while trying to prove $2k^3 \geq (k + 1)^3$…

how can I prove the following: $2k^3 \geq (k + 1)^3$ This is the final part of the elaborate proof for $2^n > n^3 $ give $ n \geq 10$ I have used induction and end up with: $ 2^{K+1} > 2k^3 $ ...
3
votes
2answers
82 views

How to solve this inequality.

Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. Show that $\sin\frac{\alpha}{2}.\sin\frac{\beta}{2}.\sin\frac{\gamma}{2}<{\frac{1}{4}}$
0
votes
1answer
95 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
0
votes
1answer
43 views

Functional inequality : bounded functions

Suppose $f''(x)$ exists ($f(x)$ can be differentiated two times) And the function and the second derivative is bounded : $\left|f(x)\right|\le P$, $\left|f''(x)\right|\le Q$ Then, how can I prove ...
5
votes
3answers
217 views

minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$

How to prove that the minimum value of $\cos(A-B)+\cos(B-C) +\cos(C-A)$ is $-3/2$
3
votes
1answer
106 views

How to show this crazy inequality of logarithms and constant number?

Is there any way to solve this inequality? I asked my friend for help, but he couldn't do it. I can't use even derivatives and his solution was including them. So, after many transformations i have to ...
1
vote
0answers
51 views

Generalized inequality with parameters $\alpha, \beta$

Let $d$ be a positive integer, and let $\alpha, \beta$ be positive real numbers such that $\alpha+\beta=1$. Consider the inequality in $k$ variables $x_1, x_2, …, x_k$, $$ \alpha \cdot \sum_{i=1}^k ...
2
votes
4answers
61 views

Proof of the inequality $(p+1)n^p<(n+1)^{p+1}-n^{p+1}<(p+1)(n+1)^p$

Question: Let $n, p \in \mathbb{N}$. Prove that $$(p+1)n^p<(n+1)^{p+1}-n^{p+1}<(p+1)(n+1)^p \tag{$\star$}$$ What I have noticed so far is, that with $f(x) := x^p$ I can conclude that ...
1
vote
1answer
126 views

A hard inequality with $a+b+c=3$

Let $$\displaystyle\left\{ \begin{array}{l} a,b,c>0\\ a+b+c=3 \end{array} \right.$$ Prove that: ...
1
vote
1answer
105 views

An inequality with $a,b,c>0$

Let $a,b,c>0$. Prove that: ...
1
vote
3answers
122 views

Show that $\sum_{i=1}^n \frac{1}{i^2} \le 2 - \frac{1}{n}$

So I am able to calculate the given problem and prove $P(K) \implies P(k + 1)$; it's been sometime since I did proofs and I perform my steps I get what Wolfram Alpha shows as an alternate solution. ...
0
votes
1answer
73 views

Which different probabilistic bounds/inequalities apply when we are given a lower bound on the sample size

Let m be the sample size and $X_i$ be a r.v. that we sample and define a new r.v. such that: $$M_m=\frac{1}{m}\sum^m_{i=1}{X_i}$$ My question is, what type of probabilistic inequalities require some ...
0
votes
1answer
42 views

Basic inequality involving sin

Show that $$ \frac{|t|}{\pi} \leq |\sin (\frac{t}{2})| , t \in [- \pi , \pi]$$ What I tried: I took $ f(t) = \sin^2 (\frac{t}{2}) - \frac{t^2}{\pi^2}$ and I am trying to study if the function ...
1
vote
0answers
41 views

Bound on the difference of two convergent infinite products

Let $(\alpha_n)$ and $(\beta_n)$ be two sequences of non-zero complex numbers such that the products $\prod_n \alpha_n$ and $\prod_n \beta_n$ are convergent. How to prove the following inequality? $$ ...
1
vote
0answers
159 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
3
votes
1answer
75 views

Minimum value of $P=\sqrt{2x^2+2y^2-2x+2y+1}+\sqrt{2x^2+2y^2+2x-2y+1}+\sqrt{2x^2+2y^2+4x+4y+4}$

Let $x,y∈R$ . Find the minimum value of this expression: $P=\sqrt{2x^2+2y^2-2x+2y+1}+\sqrt{2x^2+2y^2+2x-2y+1}+\sqrt{2x^2+2y^2+4x+4y+4}$ We have: ...