Questions on proving and manipulating inequalities.

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5
votes
1answer
63 views

If $f(0)=0$ and $f(1)=1$, prove that $\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If $f'$ is continuous, prove that $$\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$$ Progress I let ...
0
votes
1answer
19 views

$\left|(1+R^2e^{2i\theta})^2\right| \geqslant (R^2-1)^2$ in complex integration

I need to prove: $$\lim_{R\to +\infty} \left|\int_0^\pi \frac{e^{iaR(\cos\theta+i\sin\theta)}}{(1+R^2e^{2i\theta})^2}iRe^{i\theta} d\theta\right| =0$$ Could someone give me some pointers? A ...
0
votes
1answer
19 views

Proof if $n_k < n_{k+1}$ for all $k \in \mathbb{N}$, then $n_k \geq k$ for all $k \in \mathbb{N}$.

So if we proceed by induction on $k$, the base case $k = 1$ works since $n_1 \geq 1$ is true because $1$ is the smallest integer in $\mathbb{N}$. For the induction hypothesis, we have that $n_k \geq ...
2
votes
2answers
24 views

Why this power inequality for sums of real numbers holds?

$$\left|\sum_{i=1}^nx_i\right|^p \leq \begin{cases} \sum_{i=1}^n|x_i|^p & p\in(0,1]\\ n^{p-1}\sum_{i=1}^n|x_i|^p & p>1 \end{cases}$$ Can it be generalized for arbitrary sequences ...
0
votes
1answer
40 views

The minimum value of the expression [closed]

Please help me with the problem for 9th grade pupils: Find the minimum value of the expression $\frac{1}{1+x^2}+\frac{1}{1+y^2}$ with $x\ge1, y\ge1$ and $xy=2014$. Thank you!
2
votes
6answers
104 views

$2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$

If $2x^2+ 3y^2+4z^2 =1$ find the maximum of $4x+3y+2z$. This is a question from a regional math olympiad and thus there must exist solutions without application of calculus. I have no idea how to ...
0
votes
0answers
34 views

Prove that $\frac{ab} {a^2+b^2} \frac{cb} {c^2+b^2} \frac{ac} {a^2+c^2} $ [duplicate]

Let $a,b,c$ be positive real numbers suvh that $a+b+c=1$ Prove that $\frac{ab} {a^2+b^2} \frac{cb} {c^2+b^2} \frac{ac} {a^2+c^2} +\frac {1} {4}(\frac {1} {a} + \frac {1} {b} + \frac {1} {c} \ge ...
3
votes
1answer
72 views

If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $[0,1]$, then $|f'(1/2)|\le 1/4$

Let $f : [0,1] \rightarrow \mathbb{R}$ be a function whose second order derivative $f''(x)$ is continuous on $[0,1]$. Suppose that $f(0) = f(1)=0$ and that $|f''(x)| \leq 1$ for any $x \in [0,1]$. ...
9
votes
0answers
64 views

Summation of cosine terms

I got stuck on the following problem: Let $q\in \mathbb{N}$ be a fixed odd number and $k,n \in \{ 1,…,\frac{q-1}{2}\}$. I want to show that $$ \left|1 + 2\sum_{j=1}^k \cos (\frac{2\pi n}{q}j) \right| ...
0
votes
0answers
69 views

Prove that: $\frac{1}{1-2x}+\frac{1}{1-2y}+2\ge 0$

Given: $x,y\in R$ : $x^4+y^4+4=\frac{6}{xy}$ Prove that: $\frac{1}{1-2x}+\frac{1}{1-2y}+2\ge 0$ Please help me !
1
vote
3answers
34 views

Quadratic formula in double inequalities

I have the double inequality: $-x^2 + x(2n+1) - 2n \leq u < -x^2 + x(2n-1)$ and I am trying to get it into the form $x \leq \text{ anything } < x+1$ Or at least solve for x as the ...
11
votes
2answers
121 views

Showing $\gamma < \sqrt{1/3}$ without a computer

In 1735 Euler gave the value of $\gamma$ as $0.577218.$ The constant is generally defined as the limit of the difference between the harmonic series and $\log n:~\gamma= ...
1
vote
1answer
56 views

Proofs involving positive real numbers

I have two questions related to positive real numbers: If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner ...
9
votes
2answers
752 views

Find maximum without calculus

Let $f:(0,1]\rightarrow\mathbb{R}$ with $f(x)=2x(1+\sqrt{1-x^2})$. Is it possible to find the maximum of this function without calculus? Possibility through some series of inequalities?
-3
votes
2answers
20 views

Problem on CR inequality on finite sum [closed]

Let $f$ be a function from {1,2,3,....,10} to R, s. t. $(\sum_{i=1}^{10}|f(i)|/2^i)^2=(\sum_{i=1}^{10} |f(i)|^2)(\sum_{i=1}^{10}1/4^i)$ mark the correct statement. A. there are uncountably ...
1
vote
0answers
28 views

Find a Liapunov function to show asymptotically stable

Consider the system: \begin{cases} \dfrac{dx}{dt} = y \\[12pt] \dfrac{dy}{dt} = -(1+x^{2})\,y-\sin(x) \end{cases} $(0,0)$ is a critical point of this system and I need to show that it is ...
2
votes
1answer
39 views

trace inequality of positive definite matrices.

Assume $A,B \in M_n(\Bbb{R})$ are positive definite matrices, show that $$\text{Tr}(AB)\leq \text{Tr}(A)\text{Tr}(B) $$ I only prove it for $n=2$, it is straightforward calculate.but when $n \geq ...
7
votes
2answers
185 views

Find Minimum value of $P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$

Given: $x,y\in (-\sqrt2;\sqrt2)$ and $x^4+y^4+4=\dfrac{6}{xy}$ Find Minimum value Of $$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$$ Could someone help me ?
0
votes
1answer
23 views

Rational number inequality proof

Show that if $x > 1$ is a real number and if $a < b$ are rational numbers, then $0\le x^a \le x^b$. My professor told me that I'm supposed to use some $x^c$, such that $c$ $\epsilon$ $Q$ > $0$. ...
0
votes
1answer
18 views

inequality for real-valued Gaussian sums

I saw the following Lemma in an article: Let $\mathbf{b}\in \mathbb{R}^N$ be fixed, and let $\mathbf{\epsilon}\in \mathbb{R}^N$ be a random vector whose N entries are i.i.d. random variables drawn ...
2
votes
1answer
54 views

Prove: $ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $

Let $a,b,c>0$ such that $a+b+c=1$ Prove: $ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $ I don't have any idea. You guy have any idea??
0
votes
2answers
14 views

Variable intervals from system of inequalities

I have this system of inequalities: and I need to find possible intervals of i and j. Looking at the graph output from ...
3
votes
0answers
99 views

Prove that $\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}$

Prove that $$\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}\quad\forall n\in\mathbb{N}\backslash\{1\}.$$ Please show all passages and what ...
-2
votes
2answers
40 views

Prove this absolute value related inequality [closed]

$\left | |a+b|-|a|-|b| \right | \leq 2|b|$, $\forall a, b \in \mathbb{R}$. How can I prove it?
1
vote
1answer
19 views

Using the triangle inequality to show that if $|x| < 4$ then $|x^2-2x+3| < 27$

I'm starting school soon and doing some review problems to prep for Calculus. I'm a bit stuck on this problem: Show that if $|x| < 4$ then $|x^2-2x+3| < 27$. I know that I have to use the ...
0
votes
1answer
19 views

Triangular inequality in weighted graphs

In a finite directed complete graph $G ( V, E )$, if all edges have weight either $1$ or $2$, how to show that weights of edges of $G$ satisfies "Triangular Inequality"? Edited Where triangular ...
0
votes
1answer
62 views

Proving that $(1+m)^{-1/n} + (1+n)^{-1/m} \ge 1$

I need to prove the following inequality: $$ (1+m)^{-1/n} + (1+n)^{-1/m} \ge 1 $$ for every natural $m,n$. there shouldn't be any complicated math here, as this question if from a first semester ...
2
votes
7answers
281 views

Prove that $1+ \frac{1}{x^4} \geq \frac{1}{x} + \frac{1}{x^3}$

Prove That $$1+ \frac{1}{x^4} \geq \frac{1}{x} + \frac{1}{x^3}$$ where $x \in \mathbb Z^{+}$
1
vote
3answers
58 views

Prove $1+ (\frac{1}{x}) \geq (\frac{1}{x^4}) +(\frac{1}{x^3})$ [closed]

Prove That $$1+ \frac{1}{x} \geq \frac{1}{x^4} + \frac{1}{x^3}$$ where $x \in \mathbb Z^{+}$
-4
votes
2answers
52 views

Is this function defined?

Let a function Let a function $g(f)= \parallel \bigtriangledown f\parallel / sin \parallel f \parallel $ Is $g $ defined for $\left \| f \right \| \leq $ 1? $\left| \left|. \right|\right|$ ...
2
votes
3answers
28 views

Inequality proof using real numbers

If $x,y,z,w$ are positive real numbers such that $x < y$ and $z < w$, show that $xz < yw$. Show the converse and prove it or provide a counterexample. So I know that the proof is true, I ...
1
vote
0answers
31 views

Inequality of finite sequences of real numbers .

Is the following inequality true for real numbers $\lambda_{i}$ and $\mu_{i}$ $$\dfrac{\sum_{i=1}^{n}\lambda_{i}\mu_{i}^{2}}{\sum_{i=1}^{n}\lambda_{i}}\times \dfrac{1}{1+\sum_{i=1}^{n}\mu_{i}^{2}} ...
0
votes
2answers
51 views

Verifying Property of Stochastic Integral

I am trying to verify this simple property for a stochastic integral. Given that f(t,w) is a bounded, nonanticipating function for a given Wiener process $W_t$ show that $E((\int_{0}^{T} f(s,w) ...
30
votes
10answers
3k views

Old oxford scholarship question: $a^ab^b \ge a^bb^a$

Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
5
votes
2answers
72 views

Proving that $\frac{u^p}{p}+\frac{v^q}{q}\ge uv$ under the condition $\frac{1}{p}+\frac{1}{q}=1$

The following is a problem (6.10) from Rudin's principles of Mathematical analysis. Let $p$ and $q$ be positive real numbers such that $$\frac{1}{p}+\frac{1}{q}=1.$$ Prove that if $u\ge 0$ and ...
0
votes
1answer
29 views

How is the area of this triangle calculated

I was reading "Problems of Calculus in one variable" by I A MARON, and came across this solved example in first chapter which I am unable to comprehend, please help me understand this. Scan of the ...
0
votes
1answer
23 views

Summation of quotient and quotient of summation

I have $P_1, P_2, P_3, \dotsc, P_n, S_1, S_2, S_3, \dotsc, S_n$. Is it always true that: $$ \frac{P_1+P_2+P_3+\dotsb+P_n}{S_1+S_2+S_3+\dotsb+S_n} \leq ...
2
votes
0answers
60 views

How prove this $\int_{0}^{1}P(x)dx>C_{n}$

Question: let the Polynomials $$P(x)=x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_{1}x+a_{0}$$ show that: there exsit $C_{n}$( only dependent on $n$ )such $$\int_{0}^{1}P(x)dx>C_{n}$$ ...
0
votes
2answers
20 views

What are sets of values in inequality?

I have this question on my study guide: Bill's bank account has less than 7 dollars in it. Which set of values makes the inequality $b < 7$ true? What does the "set of values" mean here, ...
5
votes
1answer
57 views

Lower bound of Euler phi function times sum of divisors

After some work, I got this nice inequality: $$ \frac{n^2}{2} < \phi(n)\cdot \sigma(n) $$ where $\phi(n)$ is Euler's phi function and $\sigma(n)= \sum_{d|n} d$. I know this is true because I'm ...
0
votes
1answer
23 views

Is it possible to obtain the following inequality?

Let $X,Y$ be two random variables such that $E\|X\|^{2n}\leq c_1^n,n\in\mathbb{N}$ and $E\|Y\|^{2n}\leq c_2^n,n\in\mathbb{N}$. Clearly, we have then $$E\|X+Y\|^2 \leq 2E\|X\|^2 + 2E\|Y\|^2 \leq ...
0
votes
1answer
16 views

Inequality problem with two modulus

Solving the inequality $|x -3|+|x + 2|<11$. I am used to solving these with one modulus function, with two I have not been successful in reaching a correct answer.I attempted squaring both sides ...
0
votes
0answers
23 views

Solving system of inequalities, with solution in only natural numbers, with priority on variables

If I have the equations $27a+30b+33c+36c \geq x$ $a+b+c+d=4$ and want to solve them using only natural numbers (including 0) for both $x=131 $ and $x=142 $ preferably but not necessarily with ...
9
votes
0answers
173 views

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
3
votes
1answer
37 views

Proof of Bernoulli like inequality

I found this inequality and I would like to prove it: $$ (1+x)^n \leq 1 + \frac{nx}{(1-(n-1)x)} $$ with with $n>1$ and $-1<x<1/(r-1)$. Does anybody have an idea? Edit: I added the ...
1
vote
2answers
33 views

Variation of Jensen-Inequality

I just read a variation of Jensen's Inequality which states: If $f: \mathbb{R} \rightarrow \mathbb{R} $ is a convex function, $ \phi \in \mathcal{L}^1(\mathbb{R}^n)$ with $ \phi \geq 0$ and $ \int ...
1
vote
1answer
28 views

Bound for Outlyingness

Given a sample of $n$ data, $x_1, \dots, x_n$. Define the sample mean $$\bar x := \frac{1}{n}(x_1+\cdots+x_n),$$ and sample variance $$s^2 := \frac{1}{n-1} \sum_{i=1}^n (x_i-\bar x)^2.$$ To measure ...
6
votes
1answer
52 views

Prove that : $f(\sin x)+f(\cos x) \ge 196, \forall x\in\left(0;\frac{\pi}{2}\right)$

Given: $$f(\tan2x)=\tan^{4}x+\frac{1}{\tan^{4}x}, \forall x\in\left(0;\frac{\pi}{4}\right)$$ Prove that :$f(\sin x)+f(\cos x) \ge 196, \forall x\in\left(0;\frac{\pi}{2}\right)$ Could someone help me ...
5
votes
4answers
71 views

Prove $x + \frac{1}{x} \geq 2$ for $x>0$.

Proof that $x+\frac{1}{x}\geq2$ for $x>0$ Would this be correct? $x*(x+\frac{1}{x}\geq2)$ $x^2+1\geq2x$ $x^2-2x+1\geq2x-2x$ $x^2-2x+1\geq0$ Plug in 1 for x: $(1)^2-2(1)+1\geq0$ $1-2+1\geq0$ ...
3
votes
2answers
24 views

a<b<c $\implies$ |b|$\leqslant$max(|a|,|c|)?

Given a, b, c 3 real numbers. Prove that if a < b < c, then |b|$\leqslant$max (|a|,|c|). I orginally proved this by discussing four cases as 1) a,b,c<0 2) a<0, b,c>0 3) a,b<0, c>0 ...