Questions on proving and manipulating inequalities.

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“Sharp” Inequalities

When we say that an inequality is sharp, does it mean that it is "the best" inequality we can get between the two quantities involved? For example, I read that we would say that the inequality $$ ...
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1answer
30 views

proof some inequality by induction

I got to proof the following in-equality by induction for an assignment but having a hard time. $$ \frac{2n}{(a+b)^n} \leq \frac{1}{a^n} + \frac{1}{b^n} $$ $a,b > 0$ Thanks in adavance!
1
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1answer
45 views

Highschool Inequality

For all $a>0$ find a $b>0$ such that $|t-2|<b\Longrightarrow\sqrt{(t^2-4)^2+(3t-6)^2}<a$ $\textbf{Proof:}$ Let us square both sides of the second inequality, then we have ...
8
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4answers
173 views

Proving that $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{13}{24}$ by induction. Where am I going wrong?

I have to prove that $$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n}>\frac{13}{24}$$ for every positive integer $n$. After I check the special cases $n=1,2$, I have to prove that the given ...
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0answers
32 views

Prove, by induction that $a_{n} > \frac{1}{2^n}$ for all $n \in \mathbb{N}$ [on hold]

$$a_{1} = \frac{5}{2} $$ $$a_{n+1} = \frac{1}{2}(a_{n} +2)~\forall~n\in\mathbb{N} $$ Prove, by induction that, $$a_{n}\gt \frac{1}{2^n}~\forall~n\in\mathbb{N}$$
1
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1answer
58 views

How prove $a + b +c + \frac{1}{abc} \geq 4\sqrt{3}$? [on hold]

Let $a, b, c >0$ and $a^2+b^2+c^2 =1$ How prove $a + b +c + \frac{1}{abc} \geq 4\sqrt{3}$?
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1answer
37 views

Largest possible subset primes

Let $q$ be a Sophie Germain prime number, i.e. $2q+1=p$ is prime. Consider the set $\{1,2,3,\ldots,p-1\}$. Then what is the maximum size of a subset of this set, such that the subset contains no two ...
3
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3answers
27 views

$||x-2|-3| >1$, then $x$ belongs to which interval

$||x-2|-3| >1$, then $x$ belongs to: (a) $(-\infty, -2) \cup (0,4) \cup(6,\infty)$ (b) $(-1,1)$ (c) $(-\infty, 1)\cup (1, > \infty)$ (d) $(-2,2)$ Answer:(a) My solution: let $|x-2| = p,$ ...
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2answers
62 views

How can I prove this inequality? $(a^2+1)(b^2+1)>a(b^2+1)+b(a^2+1)$ [on hold]

Given that $a,b$ are real numbers. How can one show that $(a^2+1)(b^2+1)>a(b^2+1)+b(a^2+1)$ ?!!
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2answers
60 views

Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
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2answers
48 views

Proving that $-(2n+1/n+1) \leq 0$ for all n a natural number.

I was just wondering if someone can help me with a real basic proof. Prove that $-\frac{2n+1}{n+1} \leq 0 \forall n \in \mathbb N$. Is it just enough to show that $-\frac{2n+1}{n+1} > 0$ cannot ...
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1answer
47 views

How to show $\int_{x_0-\delta}^{x_0+\delta} g(x) > 0$ if $g(x_0)>0$? [on hold]

(a) Let $f$ and $g$ be Riemann integrable functions on $[a,b]$. Prove that if $f(x)\le g(x)$ for all $x\in[a,b]$, then $$\int_a^b f(x) dx \le \int_a^b g(x) dx.$$ (b) Prove that if $g$ is ...
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1answer
12 views

for $1 \geq x \geq 0: {2x^2\over{(2+x)}} \leq y \Rightarrow x \leq \left(\frac{3}{2} y \right)^{1/2}$

So what I did is prove that $f(x) := {2x^2\over{2+x}}$ is increasing and then invert $f$ on $[0,\infty]$ this yields $(f\restriction_{[0,\infty[})^{-1}(y) = \frac{1}{4}(y+\sqrt{y}\sqrt{y+16})$ and ...
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1answer
34 views

Is there any upper bound of this sum?

$a_1,a_2,\ldots,a_n,k$ are all integers. Is there any upper bound of the following sum $$\sum_{a_1+a_2+\cdots+a_n=k\textrm{ and } a_1,a_2,\ldots,a_n\ge 0} \frac{1}{a_1!a_2!\cdots a_n!},$$ which is a ...
0
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1answer
63 views

Don't understand proof of why $\cos x$ is a contraction mapping on $[0, 1]$

I've read a couple proofs of why $\cos x$ is a contraction mapping on $[0,1]$ but none of them are clear enough for me to understand. What if we have something like $\lvert \cos x - \cos y \rvert = w ...
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2answers
56 views

Show that if a,b,c,d are positive then $\frac{ab}{a+b}+\frac{cd}{c+d}\le \frac{(a+c)(b+d)}{a+b+c+d}$

Show that if a,b,c,d are positive then $\frac{ab}{a+b}+\frac{cd}{c+d}\le \frac{(a+c)(b+d)}{a+b+c+d}$ I am stuck with this. Thanks in advance!
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1answer
175 views

How to prove or disprove this algebraic inequality?

How to prove or disprove the inequality (from math folklore) $$\sqrt{22(a^2+b^2+c^2)+5(ab+ac+bc)}\geq\sqrt{4a^2+ab+4b^2}+\sqrt{4b^2+bc+4c^2}+\sqrt{4c^2+ca+4a^2}$$ for nonnegative $a, b,$ and ...
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0answers
39 views

Find maximum width of a rectangle contained with another (diagonally)

It appears that the question of figuring out if a rectangle will fit inside of another, specifically at a diagonal, has been asked before. As such, utilizing the equation linked here, I was able to ...
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1answer
18 views

Holder's inequality. Proof using conditional extremums .Need help, can't see how one step is found.

Prove:$$\sum_{i=1}^{n}a_ix_i\leq (\sum_{i=1}^{n}a_i^p)^{1\over p}(\sum_{i=1}^{n}x_i^q)^{1\over q} $$ $(a_i\geq0,x_i\geq0,i=1,..n,p>1, {1 \over p}+{1\over q}=1)$ Let ...
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0answers
36 views
+50

An upper bound for $\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$.

Let us consider $a_n, \phi_n, K_{n}$ complex sequences. Let $$\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$$ where $\left|K_{n}\right|\leq \gamma$ and $\gamma>0$. Can we ...
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0answers
26 views

eigenvalues inequality finite differences

I have $x,y\in[0,1]^2$, $a\in[0,A]$ $t\in[0,T]$ and the mesh points $x_j = j \, \Delta x, j=0,\ldots,J$; $y_l = l \, \Delta y, l=0,\ldots,L$; $a_k = k\Delta a, k=0,...,K$ and $t_n = nh, n=0,\ldots,N$ ...
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2answers
27 views

How do I upperbound this expression?

With a given condition such as $$|x|^2 > |y|^2$$ Is there any way I can upper bound the following expression $$\log\left(1+\big||y|-x\big|^2\right) \leq \,\,\, ? $$ Thank you
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7answers
904 views

Inequality with (1-x) as denominator

How do I solve $\frac{1}{x-1}>0$ for $x$? If I multiply both sides with $x-1$ then becomes $1\gt 0$. I know it's wrong. How do I solve it?
5
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9answers
115 views

If $a_n = \frac{e^{n}}{e^{2n}-1}$ how do I show that $a_{n+1} \leq a_n$?

Let $$a_n = \frac{e^{n}}{e^{2n}-1}$$ How do I show that $a_{n+1} \leq a_n$? I don't know how to deal with the $-1$ in the denominator.
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0answers
36 views

Finding maxima of a 3-variable function.

Let $x,y,z$ be positive real number satisfy $x+y+z=3$ Find the maximum value of $P=\frac{2}{3+xy+yz+zx}+(\frac{xyz}{(x+1)(y+1)(z+1)})^\frac{1}{3}$
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0answers
11 views

Need a lower bound for a discrete monotonic distribution

I'm staring at the following expression: $$ \displaystyle \frac{\sum_{i=0}^{n}\sigma_i\left(\sigma_i-\sigma_{i-1}\right) w_i}{\sum_{i=0}^{n} \sigma_i^2}$$ I need to come up with a lower bound to ...
3
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1answer
63 views

Prove that $\Big|\frac{f(z)-f(w)}{f(z)-\overline{f(w)}}\Big|\le \Big|\frac{z-w}{z-\overline w}\Big|$

Let $\mathbb{H}$ denote the upper half plane of $\mathbb{C}$, i.e. \begin{equation*} \mathbb{H}=\{z \in \mathbb{C}: Im(z)> 0\} \end{equation*} Suppose $f:\mathbb{H}\to\mathbb{H}$ is analytic. ...
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2answers
59 views

Given $r>0$, find $k>0$ such that $\sqrt{(x-2)^2+(y-1)^2}<k$ implies $|xy-2|<r $

Using the axioms, theorem, definitions of high school algebra concerning the real numbers, then prove the following: Given $r>0$, find a $k>0$ such that: $$\text{for all }x, y: ...
3
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0answers
99 views

Prove this inequality $\frac{a}{b+cd}+\frac{b}{c+da}+\frac{c}{d+ab}+\frac{d}{a+bc}≥\frac{16}{5}$ [closed]

Show that if $a, b, c, d \in (0, \infty)$ and $a + b + c + d = 1$, then $$\frac{a}{b+cd}+\frac{b}{c+da}+\frac{c}{d+ab}+\frac{d}{a+bc}≥\frac{16}{5}$$
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6answers
103 views

Prove this inequality $25ab+25a+10b\le38$

let $a,b>0$,and such $a^2+b^2=1$,show that $$25ab+25a+10b\le38$$ Now I have found this inequality $"="$,if and only if $a=\dfrac{4}{5},b=\dfrac{3}{5}$ then How to prove this inequality by AM-GM ...
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2answers
30 views

Solution set of inequality

This is the question: $$\frac{1-2x-3x^2}{3x-x^2-5} \gt 0$$ What I did : I got the answer as $$\left(x-3\right)\left(x+1\right) \gt 0$$ giving me the solution set : $x \in (-\infty,-1 ...
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2answers
41 views

Inequality of numbers.

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a\le0$. (May be Jensen's inequality help but need help how to apply.)
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1answer
48 views

How find this minimum

Help me! Let $x,y,z\ge0$ such that: $xy+yz+zx=1$. Find the minimum value of: $A=\dfrac{1}{x^2+y^2}+\dfrac{1}{y^2+z^2}+\dfrac{1}{z^2+x^2}+\dfrac{5}{2}(x+1)(y+1)(z+1)$ I found minimum value of $A$ ...
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1answer
71 views

If $u\in L^1(0,1)$ is nonnegative and $E_n = \int_0^1 x^n u(x) \, dx$, prove $E_{n-k} E_k \leq E_0 E_n$.

$\textbf{Question:}$ Let $ u \in L^1(0,1)$ be a nonnegative function. Define $$E_n := \int_0^1 x^n u(x) dx$$ Prove the following inequality, $\forall n \ge 0$, and $\forall k \in [0,n]$, we have $$ ...
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2answers
21 views

Cardinality of the union of two sets

I am having trouble attempting to prove the inequality $|X\cup Y| \le |X|+|Y|$. Here is my intuitive argument when we take the union of $X\cup Y$ if there are repeated elements then they are not ...
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0answers
45 views

Floor inequality with prime

If $a$ and $b$ are positive integers and $a\ge b$ and $b$ is an odd prime, show that: $$\left\lfloor \frac{6a-1}{b}\right\rfloor+\left\lfloor\frac{a}{b}\right\rfloor\ge \left\lfloor ...
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1answer
26 views

Floor inequality: $\lfloor \frac{6a-1}{b}\rfloor+\lfloor\frac{a}{b}\rfloor\ge \lfloor \frac{2a}{b}\rfloor+\lfloor \frac{3a-1}{b}\rfloor+\cdots$

If $a$ and $b$ are positive integers and $a\ge b$, show that: $$\left\lfloor \frac{6a-1}{b}\right\rfloor+\left\lfloor\frac{a}{b}\right\rfloor\ge \left\lfloor \frac{2a}{b}\right\rfloor+\left\lfloor ...
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1answer
37 views

Floor inequality: $\lfloor x+y\rfloor\ge \lfloor x\rfloor+\lfloor y\rfloor$

I remember seeing the inequality $\lfloor x+y\rfloor\ge \lfloor x\rfloor+\lfloor y\rfloor$ somewhere which is true for all reals. So I was wondering what's wrong with this proof? For all reals $a,b$ ...
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0answers
13 views

Difficulty understanding the source of an inequality solution.

I used Wolfram-Alpha to verify an answer that I got for simplifying an inequality, and it turns out I was only partially correct. Original equation: ...
1
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2answers
47 views

this sum inequality $\sum_{i=1}^{n}\frac{1}{4i(i+1)-1}<\frac{2}{7}$

show that $$\sum_{i=1}^{n}\dfrac{1}{4i(i+1)-1}<\dfrac{2}{7}\tag{1}$$ we have $$4i(i+1)-1>4i^2$$ But $$\sum_{i=1}^{n}\dfrac{1}{i^2}<\dfrac{8}{7}$$ it is clear not hold, because ...
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3answers
18 views

A store is offering a 20% discount on a certain item. The store’s sale of the item is subject to a 6% sales tax.

Quantity A : The item’s purchase price, if the discount is applied to the after-tax price Quantity B : The item’s purchase price, if the tax is applied to the discounted price Which of the ...
1
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0answers
27 views

How to show that $0<b'<b$?

If $a, b, q$ are natural numbers with $\frac{1}{q + 1} < \frac ba < \frac1q$, show that $$ \frac{b}{a} - \frac{1}{q+1} = \frac{b'}{a(q+1)} $$ where $0<b'<b$. I did the following: ...
2
votes
1answer
44 views

Proving an Inequality (terms won't cancel out)

Problem: If $x$ and $y$ are real numbers such that $y \geq 0$ and $y(y+1) \leq (x+1)^2$, prove that $y(y-1) \leq x^2$. This is what I tried: \begin{align} y(y+1) \leq (x+1)^2 &\implies y^2 + y ...
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2answers
20 views

How to simplify the inequality $5c * 3^{n-2} + 3 \le c3^n$

I'm stuck on trying to simplify the inequality: $5c * 3^{n-2} + 3 \le c3^n$ I'm looking for an expression such as $c \ge x$ without the n to show when this inequality holds true Edit: This ...
0
votes
3answers
56 views

Proof By Induction $2^n \ge n^2$ for $n\ge4$

I am trying to prove the following, and here is what I have done: Can somebody help to complete this? $2^n \ge n^2$ for $n\ge 4$ $n=4$, LHS: $2^4 = 16$, RHS: $4^2=16$, $16=16$ Therefore TRUE Assume ...
2
votes
2answers
34 views

Proof By Induction $n^2 > 3n$ where $n\ge 4$

I am trying to prove the following example, however I seem to be getting a little stuck: For $n\in\mathbb N$, $n\ge 4, n^2>3n$ What I have Done: Base Case:$ n=4$, LHS: $4^2 = 16$, RHS: $3\cdot 4 ...
1
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1answer
74 views

Is $a \sin x + b \sin y \leq \sin(ax + by)$ true?

Studying math essay exam, I saw the following strange formula $$ a \sin x + b \sin y \leq \sin(ax + by), $$ where $x, y$ are arbitrary angles and $a + b = 1.$ Is the above inequality true, and can it ...
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1answer
22 views

Inequality regarding modulus

I am trying to prove this limit to be true: $$\lim_{x\to a}(x^2)=(a^2)$$ using the Epsilon Delta Limit Definition. So far I can understand how it works but I got stumped on this inequality ...
2
votes
2answers
83 views

Inequality with condition $x+y+z=xy+yz+zx$

I'm trying to prove the following inequality: For $x,y,z\in\mathbb{R}$ with $x+y+z=xy+yz+zx$, prove that $$ \frac{x}{x^2+1}+\frac{y}{y^2+1}+\frac{z}{z^2+1}\ge-\frac{1}{2} $$ My approach: After ...
1
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0answers
46 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...