Questions on proving and manipulating inequalities.

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3
votes
1answer
210 views

Inequalities involving arithmetic, geometric and harmonic means

Let $A$, $G$ and $H$ denote the arithmetic, geometric and harmonic means of a set of $n$ values. It is well-known that $A$, $G$, and $H$ satisfy $$ A \ge G \ge H$$ regardless of the value $n$. ...
0
votes
1answer
18 views

Do there exist two vectors in a Hilbert space such that $(x,y)\geqslant k\|x-y\|^{-2}$,?

Let $H$ be a Hilbert space, $(x,y)$ denote the scalar product of the elements $x,y\in H$, $\|x\|$ denote the norm of $x\in H$, and $k>0$. Do there exist such $x,y\in H$ that $$ ...
3
votes
2answers
76 views

Minimum of $|\det(X+iC)|$

Let $C$ be a fixed real $n\times n$ matrix, $X$ be an arbitrary real $n\times n$ matrix. Find the minimum value of: $$|\det(X+iC)|=\sqrt{\det(X+iC)\det(X-iC)}$$ When $n=1$ it's clear that the ...
4
votes
2answers
38 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
0
votes
1answer
9 views

Relation between the mean value inequality over an area and over a surface

Suppose that $f$ is a locally integrable function on $\mathbb{R}^{N}$ $(N\geq2)$ such that for all $x$ in $\mathbb{R}^{N}$ and all positive real number $r$ we have \begin{equation} f(x)\leq ...
1
vote
1answer
27 views

An angular inequality

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on ...
1
vote
3answers
104 views

Is $-|x|\le\sin x\le|x|$ for all $x$ true?

I have seen in Thomas' Calculus that says to prove $\lim_{x\rightarrow0}\sin x=0$, use the Sandwich Theorem and the inequality $-|x|\le\sin x\le|x|$ for all $x$. My question is how could the ...
0
votes
2answers
32 views

Is $3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+1}-1)$?

Let $k$ be an integer. I need to prove that: $$3(2k+1)(2^{2k+1}-1)>(2^{k+3}-1)(2^{k+2}-1)$$ where $k>a$ for a suitable $a$. thanks in advance.
-1
votes
0answers
11 views

How find this minimum of the possible value $\max_{1\le j\le n}{\left(\displaystyle\sum_{i=1}^{35}x_{i,j}\right)}$

give the positive integer $n\in N^{+}$, and $x_{i,j}(1\le i\le 35,1\le j\le n)$, such $$\begin{cases} x_{i,j}=0 (or )1&1\le i\le 35,1\le j\le n\\ \displaystyle\sum_{j=1}^{n}x_{i,j}=27& 1\le ...
-1
votes
1answer
32 views

Proving an inequality about a $\sin x$ and $\exp x$ [on hold]

Show that $$\frac{\sqrt 2}{2}\ \sin x\geq e^x-1$$ for any $x\leq 0$.
3
votes
1answer
75 views

Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...
3
votes
0answers
80 views

Polynomial P(x) such that [on hold]

Let $P(x)$ be a real polynomial with degree $n$ such that $|P(x)| \lt 1$ for all $|x| \le 1$. Prove that $P(2) \lt 4^n$.
1
vote
1answer
20 views

Limits for expected value in a proof

I have a small step in a proof, that I'm not sure if I got it right. We have given the function $f(s):=\mathbb{E}[e^{\lambda S (s-1)}]$ where $S$ is a random variable such that: ...
5
votes
1answer
68 views

An inequality about a sequence

Let $(a_n)$ be a sequence such that $a_0=1 , a_1=2 , a_{n+1}=a_n+\dfrac {a_{n-1}}{1+ a_{n-1}^2} , \forall n \ge1 $ , then is it true that $52 < a_{1371} < 65$ ?
0
votes
1answer
28 views

Proof About the Product of Two Integers

I reading about of proof of the claim "If $a \ge 0$ and $b > 0$, then $a \le ab$. The proof the author is employing is inductive. I understand the basis case; however, I do not understand the proof ...
2
votes
3answers
56 views

Proving an inequality about a sequnce with Cauchy-Schwarz

show that $$\sum\limits_{i=1}^n \frac{x_i}{i^2} \geq \frac{1}{1} + \frac{1}{2} + \dots +\frac{1}{n}$$ where $x_1,x_2,\dots,x_n$ are natural numbers and all of them are different numbers(no such a ...
0
votes
1answer
38 views

cauchy schwarz inequality extreme

cauchy schwarz inequality states that: (case of real numbers) $$ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) $$ and we ...
0
votes
0answers
18 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
2
votes
1answer
37 views

Generalization of Bernoulli's Inequality

Is it possible to generalize Bernoulli's Inequality to $(x+y)^n \geq x + ny$ provided $x+y \geq 0 $ and $x \geq 1$ and $n$ is a positive natural number? I was thinking that the proof follows by ...
2
votes
0answers
54 views

Prove $\sum\limits_{\mathrm{cyc}}\sqrt{a^2+bc}\leq{3\over2}(a+b+c)$ with $a,b,c$ are nonnegative

Hope someone can help on this inequality using nonanalytical method (i.e. simple elementary method leveraging basic inequalities are prefered). Prove ...
3
votes
0answers
169 views

How prove this stronger Cauchy-Schwarz inequality for traces of compression matrices

Question: Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
0
votes
0answers
13 views

Max Earning Probability of the Portfolios [on hold]

I have a problem about deriving Chebyshev Inequality From A.D. Roy Safety first Portfolios model He would like to Min Prob(Rp≤RL)≤α It's meant u want min probability of Rp less equal than -5% ...
11
votes
2answers
215 views

Why is Volume^2 at most product of the 3 projections?

Is there a simple proof for $$ \text{Vol}^2(P)\le \prod_{i=x,y,z} \text{Area}(\text{Proj}_i(P)), $$ where $P\subset \mathbb R^3$ and $\text{Proj}_z(P)$ denotes the projection of $P$ to the $z=0$ ...
5
votes
3answers
197 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
3
votes
3answers
39 views

Prove that $2n+1 \leq 2^n$ for $n \geq 3$ using mathematical induction.

Question: $2n+1 \leq 2^n$, for all $n \geq 3$ I've tried: Basis: $P(3) = 7 \leq 8 $, so basis step is valid Pick an arbitrary value from the universe, $k \geq 3$ Inductive Step: $2k + 1 \leq ...
0
votes
1answer
52 views

determing constant in inequality with nonnegative numbers

Let $ r \geq 1$ be an integer. Prove that there exists a constant $ C_r = C(r)>0$ such that for any non-negative real numbers $ a_1, a_2, \cdots, a_n \in [0, \infty)$ the following inequality ...
0
votes
1answer
23 views

An application of Holder's inequality to show one norm is smaller than another

Let $p(s) = r(s) + m-1$ where $r:[0,T) \to [q,\infty)$ where $q \geq 2$ and $m > 1$ is fixed. Let $\text{Vol}(\Omega) = 1$. Then can we show that $$\lVert u \rVert_{L^{r(s)}(\Omega)} \leq ...
8
votes
1answer
515 views

How to prove $\frac{1}{4}(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a})\ge \sqrt[4]{\frac{a^4+b^4+c^4+d^4}{4}}$

Let $a,b,c,d>0$, show that $$\dfrac{1}{4}\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)\ge \sqrt[4]{\dfrac{a^4+b^4+c^4+d^4}{4}}$$ I know this is interesting ...
1
vote
2answers
232 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
2
votes
1answer
35 views

Given $a_{m*n} \leq a_m + a_n$, show that there exists $C$ such that $a_n \leq C log(n)$

Given $\{a_n\}$ is non decreasing, non negative and $$a_{m\cdot n} \leq a_m + a_n,$$ show that there exists $C$ such that $a_n \leq C \log(n)$ for $n\geq 2$. First taking $n=2^k$, we see that ...
6
votes
3answers
129 views

Asymptotic behaviour of the integral of the quadratic mean of the coordinates on the hypercube

I have to compute the limit $\lim_{n\to +\infty}I_n$, where: $$\qquad I_n=\int_{[0,1]^n}\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}\,d\mu.$$ I believe that its value is just $\frac{1}{\sqrt{3}}$, since the ...
4
votes
2answers
47 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
1
vote
2answers
52 views

How can I show that $n^{n+2}<(2n)!$ for any integer $n$.

When I was try to show that the series $\sum_n \frac{n^n}{(2n)!}$ is convergent using comparison test, I stuck at the point $n^{n+2}<(2n)!$ I think it can be show using mathematical induction. If ...
4
votes
0answers
60 views

How to prove $\sqrt[n]{n}$ monotone decreases using inequality? there is a hint but I can't

How to prove $\sqrt[n]{n}>\sqrt[n+1]{n+1}$ ,$n\ge 3$monotone decreases by using this hint? I can solve it in other ways but I don't know how to solve it using this hint. hints:consider the ...
6
votes
3answers
140 views

Math Induction Proof: $(1+\frac1n)^n < n$

So I have to prove: For each natural number greater than or equal to 3, $$(1+\frac1n)^n<n$$ My work: Basis step: $n=3$ $$\left(1+\frac13\right)^3<3$$ $$\left(\frac43\right)^3<3$$ ...
8
votes
0answers
197 views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrix $A,B,C\in M_{n}(C)$ is Hermitian matrix and is Positive definite matrices ,such $$A+B+C=I_{n}$$show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge ...
2
votes
1answer
38 views

Sufficient Condition for Subadditivity

A function $f:\mathbb{R}^n \rightarrow [0,\infty)$ is said to be subadditive if $f(x + y) \leq f(x) + f(y), \quad\forall x,y \in \mathbb{R}^n$. Is there a sufficient condition for subadditivity in ...
2
votes
1answer
31 views

is my induction proof sufficient?

question; prove that $\forall\ n\ge4, n\in \mathbb{Z}, \ n!\gt n^2$. my work; let $n=4$ then $4!=24 \gt 4^2=16.$ true. now assume $n! \gt n^2$ is true for all $n\le k$ so now assume $k! \gt ...
0
votes
0answers
50 views

Help inequality with $O(\cdot)$ and $\Omega(\cdot)$

Suppose,$$f(T)\le O\left(\sqrt{\dfrac{\log( T/\delta)}{T}}\right).$$ If we let $\delta=\dfrac{1}{n^2}$ and $T\ge\Omega\left(n^2\log n\right)$, then: $$f(T)\le \dfrac{1}{n}.$$ Can anyone ...
6
votes
2answers
55 views

How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
0
votes
0answers
23 views

Prove natural log between two finite harmonic sums [duplicate]

Prove for n in the naturals we have: $$\sum_{k=2}^n 1/k \le \ln(n) \le \sum_{k=1}^{n-1} 1/k$$ Intuitively this makes sense to me but I can't for the life of me figure out how to start this proof.
0
votes
0answers
15 views

exponential inequality for sum of dependent random variables

I have proved an inequality for the expectation in the context of dependent random variables. Can you please confirm it and give me some feedbacks? If $X_1,X_2,X_3,\ldots,X_m$ are $m$ dependent mean ...
0
votes
0answers
29 views

Can we find some expressions for $p$ and $q$?

Let $f\colon\mathbb R\to\mathbb R$ be a real analytic function. Assume also that $f$ has a zero at $s=1$ of order $m$. Assume that there exists an integer $r$ such that ...
11
votes
3answers
306 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 ...
3
votes
4answers
234 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
3
votes
1answer
109 views

Find maximum of $P$

Let $$P = \frac{{{x^2}}}{{{x^2} + yz + x + 1}} + \frac{{y + z}}{{x + y + z + 1}} - \frac{{1 + yz}}{9}.$$ Find maximum of $P$ where $x, y,z$ are nonnegative real numbers such that ${x^2} + {y^2} + ...
0
votes
0answers
43 views

An extremal problem using AM-GM inequality

Let $x, y, z$ be nonnegative real numbers and such that $$ x^2+y^2+z^2=2. $$ Find the maximum value of $$ P=\frac{x^2}{x^2+yz+x+1}+\frac{y+z}{x+y+z+1}-\frac{1+yz}{9}. $$ My attempt. I guess that $P$ ...
1
vote
3answers
44 views

Positive integral everywhere implies positive function a.e

I would like to get feedback on my demonstration of this simple statement : Let $f$ be an integrable function on the measure space $(X,S,\mu)$. \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ...
0
votes
1answer
61 views

An inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}|f(x)-g(x)|dx$

Does there exist an inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}|f(x)-g(x)|dx$ or an inequality between $\int_{a}^{b}f(x)g(x)dx$ and $\int_{a}^{b}(f(x)-g(x))^2dx$ ? Thank you very ...
0
votes
1answer
471 views

Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...