Questions on proving, manipulating and applying inequalities.

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0
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1answer
44 views

Prove Basic Complex Number Inequalities

Let $$z_1 = a_1 + b_1i$$ $$z_2 = a_2 + b_2i$$ where $$|z_j| = \sqrt{a_j^2 + b_j^2}$$ Prove $$|z_1 + z_2| \le |z_1| + |z_2|$$ $$|z_1 + z_2| \ge |z_1| - |z_2|$$ $$|z_1 - z_2| \ge |z_1| - |z_2|$$ $$...
0
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4answers
45 views

Prove $|z1/z2| = |z1|/|z2|$ without using polar

Prove $|z1/z2| = |z1|/|z2|$ where $$z_1 = a_1+b_1i$$ $$z_2 = a_2+b_2i$$ $$|z_1| = \sqrt{a_1^2+b_1^2}$$ $$|z_2| = \sqrt{a_2^2+b_2^2}$$ $$RHS = \frac{|z_1|}{|z_2|} = \frac{\sqrt{a_1^2+b_1^2}}{\sqrt{...
1
vote
0answers
17 views

Can I presume that this inequality is a good aproximation for a divisor function?

I've used the Lemma 7.9 from page 73 from Krizek, Luca and Somer, 17 Lectures on Fermat Numbers From Number Theory to Geometry Springer CMS (2001) (you can see this page as a Google Book, type here ...
3
votes
1answer
85 views

Etemadi's inequality

In another post an inequality referred to as "Etemadi's Inequality" is mentioned twice - in the original post as well as in the answer. However, the contexts of usage are such as to raise the question ...
2
votes
1answer
60 views

inequality with min on both sides

Let $r\in[0,1]$ and $z\in\mathbb C,\mathrm{Im}z> 0.$ I am struggling to prove the following inequality: $$\frac{2(1+|z|)r}{\sqrt{|z^2-4|}}\wedge\sqrt{(1+|z|)r}\leq3\left(\left[\left(1+\frac{1}{\...
1
vote
3answers
33 views

Verify $\frac{(z_1+z_2)^2}{z_1\times z_2} \geq 0$

I have two complex numbers, $z_1$ and $z_2$, that both have the modules equal to 1 and their arguments are $\theta_1$ and $\theta_2$, respectively. I'd like to verify that $$\frac{(z_1+z_2)^2}{z_1\...
1
vote
0answers
21 views

$|\mathcal{R}((2a+ib)^{2n+1})|\neq b$ for coprime $2a,b$ and $n>1$

Assume $n>1$ is natural and set $f_n(a,b):=\mathcal{R}((2a+ib)^{2n+1})$ Prove that for every coprime pair $2a,b\in\mathbb{Z}$: $|f_n(a,b)|>b$. Note that we have $b|f_n(a,b)$ so the only thing ...
1
vote
2answers
45 views

Maximum value of $\sum_{i\neq j}a_ia_j$ subject to $\sum_{i=1}^n a_i=1$

Find the maximum value of $\sum_{i\neq j }a_ia_j$ subject to $\sum_{i=1}^n a_i=1$. Here, $a_i\in\mathbb R$, for all $i$. Can I take $a_i>0$ for all $i$? If yes, then I can actually use AM-GM and ...
1
vote
2answers
23 views

Solve the inequality $\frac {(\frac 2 3)^{x-1}-1}{\sqrt2-\sqrt[3]{2^{x-1}}} < 0$

I'd like to solve the following inequality: $$\frac {(\frac 2 3)^{x-1}-1}{\sqrt2-\sqrt[3]{2^{x-1}}} < 0$$ I made it so that $$z = 2^{x-1}$$ This is what the inequality now looks like: $$\frac ...
3
votes
4answers
110 views

Prove: $x^3+y^3\geq \frac{1}{4}(x+y)^3$

Prove: $x^3+y^3\geq \frac{1}{4}(x+y)^3$ for all $x,y$ positive. Let's look at $$\begin{split} &(x-y)^2(x+y)\geq 0 \\ \iff &(x-y)(x+y)(x-y)\geq 0\\ \iff& (x-y)(x^2-y^2)\geq 0 \\ \iff &...
0
votes
2answers
68 views

A **proof** for $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ [duplicate]

I need a proof for the inequality: $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$ for all natural numbers $t \geq 2$. For $t=2$ both sides are equal. Can someone find a proof for all $t$? maybe ...
3
votes
1answer
64 views

Is this number positive?

Let $(a_{ij})$ be a collection of non-negative numbers indexed by integers $1\le i,j \le N$ where $N$ is some fixed integer. Let $(c_{ij})$ be another collection of real numbers also indexed by ...
2
votes
3answers
105 views

A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$

I'm struggling with proving the following inequality: $$\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$$ for all $t \geq 2$. I think it is monotonic non-increasing in $t$, which would suffice. ...
-1
votes
1answer
44 views

How prove this inequality with $3(x^2-x+1)(y^2-y+1)(z^2-z+1)$

Let $x,y,z$ be real numbers, prove that $$3(x^2-x+1)(y^2-y+1)(z^2-z+1)\ge (xyz)^2+x^2(y+z)+y^2(z+x)+z^2(x+y)-5xyz+1$$ with equality if $x=0,y=z=1$ .
4
votes
2answers
68 views

Find the smallest $\alpha$ such that, for all $x,y,z$, $\alpha\,\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge(xyz)^2+|xyz|+1$.

Find the smallest $\alpha\in\mathbb{R}$ such that, for all $x,y,z\in\mathbb{R}$, the following inequality holds $$\alpha\,\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge(xyz)^2+|xyz|+1\...
0
votes
2answers
77 views

Factorial sum estime

Prove that: $$\displaystyle \sum_{n=m+1}^\infty \dfrac{1}{n!} \le \dfrac{1}{m\cdot m!}$$ I have tried induction on $m$ but it does not work very well. Any suggestion?
1
vote
1answer
45 views

Proof verification: $f(x) \le g(x) \implies \lim_{x\to a}f(x) \le \lim_{x\to a}g(x)$

I am self-teaching calculus using Spivak's book, and it's hard for me to know whether my proofs are correct, if they are different from the proofs that Spivak gives. Could you help me to check whether ...
0
votes
1answer
27 views

Question about proof of triangle inequality in $\mathbb{C}$

I am having some trouble with one inequality used in the proof of the triangle inequality in $\mathbb{C}$. The main issue is realizing that for $z,w \in \mathbb{C}$, we have that $2 \cdot Re(z\...
0
votes
5answers
84 views

If $a+b=c+d$ and $0<a<b<c$ then is it true $ab>cd$?

If $a+b=c+d$ and $0<a<b<c$ then is it true $ab>cd$? This is the only thing I know: $\text{min}(a,b,c,d)=d$ as $d=a+(b-c)<a$. So it might be true that the inequality holds. I've ...
0
votes
2answers
36 views

What would be the result of taking the square root of this inequality?

Here is the inequality: $(x+3)^2\leq(y+3)^2$ I'm not sure what the result would be if I took the square root of both sides. I think that in this case: $(x+3)^2=(y+3)^2$ it would be: $(x+3)=\pm(y+...
0
votes
1answer
17 views

Prove an inequality by Second partial derivative test?

Let $a$, $b$ and $c$ be real positive numbers. Define a function $f$ on the domain $D = \{(x,~y) \in \Bbb R ^2 | x>0,~y>0 \}$ as follows: $$ f(x,y)=ax^{a+b+c}+by^{a+b+c}+c-(a+b+c)x^ay^b. $$ ...
1
vote
1answer
59 views

A general version of Gronwall's inequality

For the following $$|u(t)|^p\le C_1 \int_0^t |u(s)|^p\,ds+C_2$$ using Gronwall inequality, we have $$|u(t)|^p\le C_2(1+C_1 te^{C_1 t})$$ Now, my question is, for $$|u(t)|^p\le K_1 \int_0^t(1+|u(s)|^2)...
0
votes
1answer
25 views

Inequality involving the length of a vector and its components

Given a vector $\overrightarrow{v}$ of length $|\overrightarrow{v}|$ with components $x,y,z\gt 0$ how to prove the following inequality? $$|\overrightarrow{v}|^2\ge\sqrt 2(xy+yz)$$ Thanks
0
votes
1answer
108 views

How can we find minimum of $f(x,y,z)?$

Let $k\in\mathbb{N}$ and $x,y$ and $z$ are positive real number such that $x+y+z=1$. How can we find minimum of $f(x,y,z)$ where $$f(x,y,z)=\frac{x^{k+2}}{x^{k+1}+y^{k}+z^{k}}+\frac{y^{k+2}}{y^{k+1}+z^...
1
vote
2answers
113 views

Which one is bigger $\sqrt[1023]{1024}$ or $\sqrt[1024]{1023}$

Which one is bigger $\sqrt[1023]{1024}$ or $\sqrt[1024]{1023}$ I am really stuck with this one.My friend says that it can be solved by $AM-GM$ but I didn't succes.Any hints?
0
votes
3answers
101 views

How much is the minimum of $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$

$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$$ We have three unknowns if they were two it were easy but I have no idea when it becomes three unknowns any hints? note:There isn't any information about value ...
1
vote
2answers
105 views

Find the smallest number $\alpha$, such that for all $x,y,z$ $\alpha(x^2-x+1)(y^2-y+1)(z^2-z+1)\ge(xyz)^2-xyz+1$

Find a smallest number $\alpha$, such that for all $x,y,z$ (not all of which are positive) inequality $$\alpha(x^2-x+1)(y^2-y+1)(z^2-z+1)\ge(xyz)^2-xyz+1$$ My work so far: Let $f(t)=t^2-t+1$. ...
5
votes
4answers
135 views

Prove that $10^{340} < \dfrac{5^{496}}{1985}$

Prove that $10^{340} < \dfrac{5^{496}}{1985}$. I said since $2^{13} < 10^{4}$, we see that $5 = \dfrac{10}{2} > 10^{\frac{9}{13}}$ and so $10^{340} < \dfrac{10^{343.38}}{1985} <\dfrac{...
2
votes
2answers
162 views

Constant such that $\max\left(\frac{3}{3-2c},\frac{3a}{3-2d},\frac{3b}{3-2e}\right)\geq k\cdot\frac{2+3a+4b}{9-c-2d-3e}$

What is the greatest constant $k>0$ such that $$\max\left(\frac{3}{3-2c},\frac{3a}{3-2d},\frac{3b}{3-2e}\right)\geq k\cdot\frac{2+3a+4b}{9-c-2d-3e}$$ for any $0\leq b\leq a\leq 1$ and $0\leq c\...
0
votes
2answers
43 views

If $\left| x-2\right| < \frac{1}{100}$ Show That $\left| x^2 -4 \right| < \frac{1}{10}$

I am working through a problem involving absolute value inequalities, and I just can't seem to get on the right track. The problem is, if: $$\left| x-2 \right| < \frac{1}{100} \text{,} $$ show ...
3
votes
3answers
104 views

On the proof that $\left|\frac{z_1-z_2}{1-z_1\bar{z_2}}\right|\lt 1$

Question: Prove that $\left|\dfrac{z_1-z_2}{1-z_1\bar{z_2}}\right|\lt 1$ if $|z_1|\lt1$, $ |z_2|\lt 1$ My solution: I had no idea how to go about this one so instead I started simplifying the ...
0
votes
2answers
101 views

Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k} $

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
6
votes
1answer
247 views

(Tournament of towns 1994) Prove the inequality

Let $a_1,a_2,\ldots,a_n$ be real positive numbers. Prove that $$\left(1+\frac{a_1^2}{a_2}\right)\left(1+\frac{a_2^2}{a_3}\right) \cdots \left(1+\frac{a_n^2}{a_1}\right) \geq(1+a_1)(1+a_2) \cdots (1+...
1
vote
4answers
51 views

Solving the Inequality $\frac{14x}{x+1}<\frac{9x-30}{x-4}$

The question says to find all the integral values of x for which the inequality holds. the question is $$\frac{14x}{x+1}<\frac{9x-30}{x-4}$$ My Solution \begin{align} & \frac{14x}{x+1} < ...
1
vote
1answer
42 views

Is $\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}$ bounded independently of $a$?

Fix an $a\in \mathbb{R}$ and consider the sum $$\sum_{n \neq 0}\left(\frac{1+∣a∣}{1+|a-n|}\right)^{100}e^{-n^2}.$$ Is this sum bounded independent of $a$? I think the answer should be yes since for $...
1
vote
0answers
30 views

Calculate (or estimate) $S(t)=\sum_{j=0}^k \binom k j |t-j|^{k-1}$.

Let $t\in\mathbb R$. Calculate, or estimate from above and below, the following sum $$S(t)=\sum_{j=0}^k \binom k j |t-j|^{k-1}.$$ I have not any idea.
0
votes
0answers
32 views

Squaring both sides of an inequality

I found that I can square both sides of an inequality as long as both sides are non-negative. But if we do, for example, this inequality: ...
0
votes
2answers
42 views

Why does this inequality hold for $|y|\geq 1$?

My lecture notes use this inequality for a complex $z=x+iy$ with $|y|\geq 1$ $$|\cot(z)| \leq \frac{1+\exp(-2|y|)}{1-\exp(-2|y|)}.$$ How can I show it? My attempt: \begin{align*} |\cot(z)| &= \...
-1
votes
4answers
91 views

Prove $n^{n/2} < n!$ if $n \gt 2$ [duplicate]

Ive been stuck on this question for so long.How do i do it? $n^{n/2} < n!$ if $n \gt 2, n \in \mathbb{N}$. Please help guys.
3
votes
3answers
104 views

Solving the trig inequality $|\sin{x} + \cos{x}| > 1$

$|\sin{x} + \cos{x} |> 1$ How to solve this kind of question? Is there any websites to learn trigonometry inequalities? My teacher only taught us the simple question but not the complicated one. ...
1
vote
1answer
52 views

Number of divisors greater than a number [closed]

Given a number $x$, it is easy to count its total number of divisors by combinatorial method. Is there a way to efficiently determine the number of divisors of $x$ greater than a given number $y$?
1
vote
1answer
56 views

Give an example of an inequality with exactly 3 solutions. [closed]

I am helping a friend try and do their pre-algebra and cannot for the life of me figure this out. I'm pretty sure it's an error in the phrasing, but I'm not too sure. The only inequalities I can come ...
1
vote
1answer
51 views

Proving the inequality $\bigg|\int_a^b f(x) \, dx - \frac{b-a}{n} \sum_{k=1}^n f\big(a + \frac{2k-1}{2n}(b-a)\big) \bigg| < \frac{C}{n^2}$

Suppose $f$ is twice differentiable and $|f''(x)| < B,$ some constant. Using Taylor's theorem, it is easy to show that $$\big|2Af(0) - \int_{-A}^A f(x)\;dx\big| < \frac{A^3B}{3}.$$ I am ...
-2
votes
0answers
32 views

Verifying inequality in mathematica

I am trying to prove or disprove the flowing inequality : $0<a, b<1, a^2+b^2=1, 0<x<1/2, 0<y<1/2$ $\rightarrow$ $1-a^{2x-1}b^{2y-1}>0$?? For this I wrote ...
2
votes
1answer
24 views

If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$?

The title says it all. If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$? Here $I(x)$ is defined to be the ratio $$I(x) = \...
3
votes
1answer
49 views

Maximizing $\frac{x(1-f(x))}{3-f(x)}$

Let $f:[0,1]\rightarrow[0,1]$ be a nondecreasing function such that $f(0)=0$ and $f(1)=1$. Let $x_1\in[0,1]$ be the value maximizing $x(1-f(x))$. Let $x_2\in[0,1]$ be the value maximizing $\frac{x(...
0
votes
1answer
32 views

Using induction on modified inequalities.

Here's the original problem: Prove by induction that $\left(\frac{1}{2}\right) \left(\frac{3}{4}\right) \cdots \left(\frac{2n-1}{2n} \right) \leq \frac{1}{\sqrt{n+1}}$ for all $n \in \mathbb{N}$. ...
0
votes
2answers
54 views

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$

For every real positive n prove that $\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}$.Hence,or otherwise prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$. Wher [x] denotes the greatest integer not ...
6
votes
0answers
74 views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
0
votes
2answers
46 views

How to prove $a,b,c \in \mathbb{R} \mid a+b+c \geq abc \implies 3abc(a+b+c) \geq 3(abc)^2$?

I'm working through some proofs from Cvetkovski's "Inequalities", when I came across a more difficult one (for newbies like me). Given $a, b, c \in \mathbb{R} \mid a+b+c \geq abc$, how can we prove ...