Questions on proving and manipulating inequalities.

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positive Maclaurin polynomials

Consider even degree Maclaurin polynomials M[n;2k] for (1+x)^n where degree = 2k < n and n is a positive integer. Examples: (1) The quadratic M[3;2] = 1 + 3x + 3x^2 is clearly positive for all ...
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2answers
32 views

Prove the inequality $(n+1)^4 < 4n^4$ for $n\geq 3$ by induction

The inequality I'm concerned with is $(n+1)^4 < 4n^4,\ n\geq 3$. I'm not sure how induction is supposed to work here. If I assume $(k+1)^4<4k^4$, I cannot see how this helps show ...
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2answers
34 views

Factorial inequality $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n$ using induction

I want to show $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n.$ Assume that it holds for some positive integer $k\geq 1$ and we will prove, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
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1answer
27 views

Inequality $a^2b^2+2(a+b)\geq 4ab+1$

Let $a,b\geq 1/2$. Prove that $$a^2b^2+2(a+b)\geq 4ab+1.$$ We know that $(ab-1)^2\geq 0$ implies $a^2b^2+1\geq 2ab$, so the inequality reduces to $2(a+b)\geq 2ab+2$, or $a+b\geq ab+1$. But this is ...
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1answer
25 views

Determinantal inequality for block matrices: if $A=(B,C)$ is a square matrix, then $|A|^2\le |B^TB|\cdot |C^TC|$

Suppose $A=(B,C)$ is a $n\times n$ matrix, $B$ is a $n\times s$ matrix, $C$ is a $n\times (n-s)$ matrix. Show that $|A|^2\leq |B^TB|\cdot |C^TC|$. If $A$ is singular, then it is obvious. If $A$ is ...
2
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1answer
46 views

Inequality$\Big|\sum_{j=1}^n a_{1j} x_j \Big|^2 \leq \sum_{j=1}^n |a_{1j}|^2 \sum_{j=1}^n |x_j|^2$

Let ${\bf A}$ be a $m \times n$ matrix with entries $a_{ij}$, and ${\bf x}$ be a $n \times 1$ vector with entries $x_{i}$. Then how can I show $$ \left\vert\,\sum_{j\ =\ 1}^{n} ...
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1answer
42 views

prove this martingale inequality

The problem is like this: Let $Y_1,Y_2,\ldots$ be nonnegative i.i.d. random variables with $E(Y_m)=1$. Let $X_n=\prod_{m\leq n} Y_m$, show that $\lim_{n\rightarrow \infty}X_n=0$ if $P(Y_m=1)<1$. ...
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1answer
45 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
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1answer
56 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 ...
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1answer
17 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 ...
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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 ...
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2answers
23 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 ...
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1answer
34 views

The minimum value of the expression [on hold]

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!
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6answers
98 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 ...
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0answers
31 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 ...
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1answer
66 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 $x \in [0,1]$. ...
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0answers
61 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 !
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3answers
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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 ...
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2answers
107 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= ...
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1answer
54 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 ...
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2answers
743 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?
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2answers
19 views

Problem on CR inequality on finite sum [on hold]

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 ...
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0answers
25 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 ...
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1answer
38 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 ...
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1answer
66 views
+100

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 ?
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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 $c$ $\epsilon$ $Q$ > $0$. ...
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1answer
15 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 ...
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1answer
53 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??
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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 ...
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0answers
95 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 ...
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2answers
38 views

Prove this absolute value related inequality [on hold]

$\left | |a+b|-|a|-|b| \right | \leq 2|b|$, $\forall a, b \in \mathbb{R}$. How can I prove it?
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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 ...
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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 ...
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1answer
60 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 ...
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7answers
280 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 $x \in \mathbb Z^{+}$
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3answers
57 views

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

Prove That $$1+ \frac{1}{x} \geq \frac{1}{x^4} + \frac{1}{x^3}$$ where $x \in \mathbb Z^{+}$
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0answers
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Are the questions in each of the following sets a family? [closed]

$$Y=(3X+1)(2X-1)(X+3)(X-2)$$ $$Y=2(3X+1)(2X-1)(X+3)(X-2)$$ $$Y=3(3X+1)(2X-1)(X+3)(X-2)+1$$ $$Y=4(3X+1)(2X-1)(X+3)(X-2)+2$$ I am having trouble determining if these functions are all a family. If so, ...
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2answers
72 views

How to apply Chebyshev's inequality? [closed]

When a fair coin is tossed 16 times, the random number of tails, $X$, has mean $\mathbb{E}[X]=8$ and variance $\operatorname{Var}(X)=4$. Using Chebyshev's Inequality, determine a lower bound for ...
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0answers
42 views

A Integral Inequality [closed]

If $f\in C^1[a,b]$,prove that$$ \int_{a}^{b}\left|f(x)-\frac{1}{b-a}\int_{a}^{b}f(x)dx\right|dx\leq\frac{b-a}{2}\int_{a}^{b}|f'(x)|dx $$
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2answers
51 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? $\left| \left|. \right|\right|$ ...
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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 ...
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0answers
30 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}} ...
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2answers
63 views

How to solve inequalities with complex numbers such as $\operatorname{Im}(2/z)\geq13$? [closed]

How can you solve for $z$ (a complex number) an inequality such as $$\operatorname{Im}(2/z)\geq13 $$ and the second question is the inequality $$\frac{x^2-2}{2x+3}\geq x$$ I tried solving for both ...
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1answer
35 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) ...
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10answers
2k views
+450

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 ...
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1answer
28 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 ...
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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 $$\int_{0}^{1}P(x)dx>C_{n}$$ ...
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2answers
19 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, ...