Questions on proving and manipulating inequalities.

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2answers
39 views

Easy question on integrals

I have some problems understanding this inequality: $$\int_{x-\varepsilon x}^x \vartheta\left(t\right)dt \leq \vartheta\left(x\right)x\varepsilon$$ where $\vartheta\left(x\right)$ is the Čebyšëv (or ...
0
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1answer
42 views

I'm new to proving inequalities. How does one prove this?

If a, b, and c are non-negative real numbers and $a + b + c = 2 $, prove that $ 2 \ge a^2 b^2 + b^2 c^2 + c^2 a^2 $
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0answers
12 views

Regarding the $\sigma (n)$ function.

This question relates to Robin's Inequality. Is $\sigma{(n^2)}$ < (2 n) $\sigma{(n)}$ ? For what integer values of n is this satisfied?
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1answer
52 views

Why does $2x_1x_2y_1y_2 \leq x_1^2y_2^2+x_2^2y_1^2$?

When I tried to prove the triangle inequality $|z_1+z_2| \leq |z_1| + |z_2|$ algebraically for complex variables $z_1$ and $z_2$, I came across this inequality and found that this is always true no ...
3
votes
2answers
102 views

Proving the bound $\left ( 1+\frac{x}{n} \right)^n \leqslant 3^x$, $\forall x \in \mathbb{R^+}$

I'm trying to directly prove the above bound. I have tried expanding it $$\left ( 1+\frac{x}{n} \right)^n = \sum_{k\geqslant 0} \binom{n}{k}\left ( \frac{x}{n}\right)^k$$ $$= \sum_{k=0\dots n ...
1
vote
1answer
28 views

if $x,y,z\ge0$ Prove $x^3+y^3+z^3+3xyz\ge xy(x+y)+yz(y+z)+zx(z+x)$

if $x,y,z\ge0$ Prove $$x^3+y^3+z^3+3xyz\ge xy(x+y)+yz(y+z)+zx(z+x)$$ Things I have done: At first i thought this can be solved by AM-GM but I was wrong. I tried to move $RHS$ to $LHS$ and result ...
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2answers
27 views

Relation of length of a projection of a point to a line

In the given figure, can it be said that $x \leq a + b - d$?
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1answer
32 views

if $a$ is a real number that $a\neq1$ and $a^5-a^3+a=2$,Prove $3<a^6<4$

if $a$ is a real number that $a\neq1$ and $a^5-a^3+a=2$,Prove $3<a^6<4$ Things I have done: using AM-GM for $a\ge0$ $$a^5+a\ge2a^3$$ The equality occurs only in $a=1$ which is not true as ...
1
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1answer
20 views

When can I expect the derivative of an inequality to always hold true?

Let $f,g$ be real-valued functions. Suppose I have an inequality $$f(x)>g(x)$$ for $x\in D$, where $D\subseteq\mathbb{R}$ is some domain. After some "tinkering" I see that we can not always expect ...
2
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2answers
52 views

show $\frac{1}{15}< \frac{1}{2}\times\frac{3}{4}\times\cdots\times\frac{99}{100}<\frac{1}{10}$ is true

Prove $\frac{1}{15}< \frac{1}{2}\times\frac{3}{4}\times\cdots\times\frac{99}{100}<\frac{1}{10}$ Things I have done: after trying many ways and failing, I reached the fact ...
1
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0answers
22 views

Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
1
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4answers
57 views

Proving $4^n > n^4$ holds for $n\geq 5$ via induction.

I know that it holds for $n=5$, so the first step is done. For the second step, my IH is: $4^n > n^4$, and I must show that $4^{n+1} > (n+1)^4$. I did as follows: $4^{n+1} = 4*4^n > 4n^4$, ...
0
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0answers
27 views

About questions related to the Riemann's Hypothesis.

Let n be an integer , if n satisfies Robin's Inequality ($\omega(n)$ /n < $e^{\gamma}$ lnlnn) say n is 'regular'. If n doesn't satisfy Robin's Inequality say n is a 'counter' or a counter-example. ...
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0answers
22 views

A question regarding ODEs that must satisfy certain conditions

Let us say that I have two quantities: $a$ and $b$, where both are from the set of reals. Always, $a \geq 0$ and $b \geq 0$. In addition, $a + b = C$, where $C$ is a real number such that $C > 0$. ...
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0answers
14 views

Finding the range of $x$ in ${c_1}^x+\sqrt{\frac{\log(c_2)x}{c_3}}+\frac{\log(c_2)x}{c_4}\le c_5,$

Is there any way to find the range of $x$ that satisfies the following inequality: $${c_1}^x+\sqrt{\frac{\log(c_2)x}{c_3}}+\frac{\log(c_2)x}{c_4}\le c_5,$$ for $x>1$, $0<c_1<1$, ...
1
vote
1answer
21 views

Inequalities with variables that are integers

If $a, b, c, d$ are all positive integers, is it true that if $a \gt b$, and $c \gt d$, then we can say that $ac \gt bd$ ?
2
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1answer
49 views

Find the minimum value of $x+y+z.$

Let $x,y,z$ are nonegative such that $(x - y)(y - z)(z - x) \geq 1.$ Find the minimum value of $x+y+z.$
2
votes
4answers
41 views

Inequality for a rational function of three variables

$x,y,z$ are positive real numbers such that $$x^2+y^2+z^2=1$$ Prove that $\dfrac{x^2}{1+2yz}+\dfrac{y^2}{1+2xz}+\dfrac{z^2}{1+2xy} \geqslant \dfrac{3}{5}$.Again, I try with Engel form of Cauchy ...
1
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1answer
22 views

Prove sum of distance from triangle vertices to a point inside triangle is more than semiperimeter and less than perimeter

If $O$ is a point inside $\triangle ABC$,Prove: $$\frac{\overline{AB}+\overline{BC}+\overline{CA}}{2}<\overline{AO}+\overline{BO}+\overline{CO}<\overline{AB}+\overline{BC}+\overline{CA}$$ ...
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1answer
25 views

binominal inequality - checking

I have to show that $\displaystyle \binom{n}{k}<\binom{n}{l}$ for $\displaystyle 0 \le k <l \le\frac{n}{2}$ where $n,k,l$ are an integers. I think I solved it but I'm not sure if my approach is ...
8
votes
3answers
99 views

Inequality $\int^{1}_{0}(u(x))^2\,\mathrm{d}x \leq \frac{1}{6}\int^{1}_{0} (u'(x))^2\,\mathrm{d}x+\left(\int_{0}^{1} u(x)\,\mathrm{d}x \right)^2$

I've been scratching my brain on this one for about a week now, and still don't really have a clue how to approach it. Show that for $u \in C^1[0, 1]$ the following inequality is valid: ...
-2
votes
4answers
92 views

Proof of inequality $\frac{2-a}{2+a}<e^{-a}$

How can I prove that $$\frac{2-a}{2+a}<e^{-a}$$ for all $a \geq 0$ ? For $a \geq 2$ it is clear, but how can it be shown for $0<a<2$ ?
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4answers
53 views

Solve the inequality: $|x^2 − 4| < 2$

This is a question on a calculus assignment our class received, I am a little confused on a few parts to the solution, can someone clear a few things up with it? Since $x^2-4 = 0$ that means $x = 2$ ...
1
vote
2answers
17 views

Is the following inequality valid or not?

$ 1 > 1 - \dfrac{1}{x}$ for every $x \in \mathbb{N} $. Obviously the limit of $1-\dfrac{1}{x}$ as x approaches infinity is $1$, but is this inequality 100% valid? Does it need a proof, and if so, ...
1
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3answers
44 views

prove inequality

I have to show: $\displaystyle\frac{1}{1^4}+\frac{1}{2^4}+...+\frac{1}{n^4} \le 2-\frac{1}{\sqrt{n}}$ for natural $n$ I tried to show it by induction (but I think it could be possible to show it ...
2
votes
3answers
45 views

Showing that $x+ cos x - 1 > 0$ for all $x > 0$

I got this problem: Show that for all $0<x$, $0<x+cos x - 1$ I tried to show it several times but none worked. I showed that $lim_{x\to\infty} (x+cos x - 1) = \infty$ by using the Squeeze ...
3
votes
1answer
110 views

symetric inequality for a rational function of three variables

$a,b,c$ are positive real numbers such that $a+b+c=1$. Prove that: $$\dfrac{a^3}{a^2+b^2}+\dfrac{b^3}{b^2+c^2}+\dfrac{c^3}{c^2+a^2} \geqslant \dfrac{1}{2} $$ I have tried with Cauchy-Schwarz ...
1
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0answers
23 views

Find condition on $A$ and $B$ so that $|q_o|+|q_1|<1$.

I have to find condition on $A$ and $B$ so that $$|q_o|+|q_1|<1$$ Where $$q_0=\frac{-1-4A+C+(2B+1)\sqrt{8A+(B+C-1)^2}-B(2B+2C+1)}{8A-4(B+C)}$$ and ...
1
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1answer
29 views

Inequality regarding exponential function

For every positive $x$ and for every $n$ show that $(1+\sqrt{\frac{x}{n}}+\sqrt[3]{\frac{x}{n}})e^{n\arctan{\frac{x}{n}}}>e^x$. I plotted it, it seems that the inequality holds. Any ideas how to ...
6
votes
3answers
129 views

Inequality $\binom{2n}{n}\leq 4^n$

I would like to prove the following inequality, for $n=0,1,2,...$, $$ \binom{2n}{n}\leq 4^n.$$ I already proved it by induction, and I'm looking for another proof.
0
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3answers
42 views

Using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$?

So using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$, and we're given $\int_1^\infty\frac{1}{4x^2}dx$. So I have been trying to set up an inequality to use, but I can't seem ...
0
votes
1answer
39 views

Verification of proof that $x^p+y^p<(x+y)^p$

I came across the inequality $x^p+y^p<(x+y)^p$, $x,y \in \mathbb{R}^+$, $p>1$ during working on an assignment. While I think I have a proof down, I'm not 100% confident about the steps taken. ...
1
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1answer
34 views

If$ x^2 + y^2 + Ax + By + C = 0 .$ Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.

Also find the center and radius of the circle Here's my solution, I'm not sure if it's correct or not (specifically the conditions on $A$, $B$ and $C$. I feel that my conditioning is invalid and that ...
4
votes
3answers
56 views

Show that the solution of an initial value problem is always less than a given constant

My try is that $$\frac{dy}{dt} =(y-3)e^{\cos ty}$$ $$\frac{dy}{y-3}= e^{\cos ty}dt$$ $$\ln (y-3)=-\frac{e^{\cos ty}}{\sin ty} +c$$ my steps is correct or I made mistakes ? please help to solve ...
0
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2answers
55 views

How prove this integral inequality $1\le \int_{0}^{2}f(x)\,dx\le 3$

Let $f(x)$ be differentiable on $[0,2]$,and such $f(0)=f(2)=1,|f'(x)|\le 1,x\in[0,2]$. Show that $$1\le \int_{0}^{2}f(x)\,dx\le 3.$$ I want use Taylor theorem ...
1
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1answer
52 views

Symmetric inequality for a rational function of three variables

If $x,y,z$ are positive real numbers such that $xyz \geqslant 1$ prove: $$\dfrac{x^3+y^3}{x^2+xy+y^2}+\dfrac{y^3+z^3}{y^2+yz+z^2}+\dfrac{x^3+z^3}{x^2+xz+z^2} \geqslant 2$$ I have tried with ...
2
votes
1answer
24 views

Supremum of sum of two sequences: $\sup (x_n+y_n) \le \sup x_n + \sup y_n$

Prove that $\sup\{x_n+y_n\}\leq \sup\{x_n\}+\sup\{y_n\}$, if both sups are finite. Furthermore, prove that $\limsup\{x_n+y_n\}\leq \limsup\{x_n\}+\limsup\{y_n\}$ if both limsups are finite.
4
votes
4answers
72 views

How to prove this inequality with $a+b+c=1$

Question: For any $a,b,c\in \mathbb{R}$ such that $a+b+c=1$ and $abc>0$, show that $$ab+bc+ac<\dfrac{\sqrt{abc}}{2}+\dfrac{1}{4}.$$ My idea: let $$a+b+c=p=1, \quad ab+bc+ac=q,\quad ...
-1
votes
1answer
29 views

Inequality of expectation of random variables [on hold]

Let $X$ and $Y$ be two continuous random variables in $\mathbb{R}$. $X$ and $Y$ may not have any kind of stochastic dominance properties. Let $f$ be a strictly monotonically increasing concave ...
0
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1answer
59 views

$x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$, for $xyz=1$ [on hold]

$x,y,z > 0$ such that $xyz=1$ can you prove that $x^2+y^2+z^2 +3 \geq 2(xy+xz+yz)$ without lagrange multiplier Thanks
1
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1answer
23 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
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votes
2answers
56 views

Find the smallest $a>1$ such that $\frac{a+\sin x}{a+\sin y} \leq e^{(y-x)}$ for all $x \leq y$

Can anyone please help me with the following question: Find the smallest $a>1$ such that $$\frac{a+\sin x}{a+\sin y} \leq e^{(y-x)}$$ for all $x \leq y$ My attempt: I think we should rearrange ...
1
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2answers
52 views

How to write a formal proof of the statement: if $x<3$ then $10-2x>4$?

Prove: For all real numbers $x$, if $x<3$ then $10-2x>4$. Proof: Let $x \in \mathbb{R}$, such that $x<3$. We have the following sequence of implications: $10-2x>4 \Rightarrow -2x>4 ...
-1
votes
1answer
26 views

Induction Proof of Inequality Involving Summation [closed]

I really need help with the following exercise! Show with induction that $\sum _{k=1}^n\left(k^2\right)<n^3$, for $n>1$ .
1
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3answers
35 views

Limit-related inequalities with absolute values

Recently I decided to learn calculus on my own and I stumbled across something which I cannot figure why is correct. Let $f$ be some function for which you know only that if $0<|x-3|<1$, then ...
1
vote
2answers
50 views

minimal value of $x^2+2y^2+5z^2$ with constraint.

$x,y,z>0$, and $xy+yz+zx=1$. I need to find the minimum value of $x^2+2y^2+5z^2$ In general what can we say about the minimal value of $\frac{ax^2+by^2+cz^2}{xy+xz+yz}$, over all positive numbers ...
0
votes
0answers
26 views

Characterize the set that solves the Diophantine inequality

Given that $a, b, c$ are integers and $2\max(|b|, |3a + b|) \le \min(|c|, |3a+2b+c|)$ What characterizes the solution set in $\mathbb{Z}^3$? Obviously this is equivalent to the system of linear ...
0
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0answers
18 views

question on inequalities

Consider the following inequality: b + g(1-2b) > 0 which of the following two is correct? b > -g(1-2b) => b > g(2b-1) => g < b/(2b-1) or g(1-2b) > -b => g > -b/(1-2b) => g > b/(2b-1) ...
0
votes
1answer
57 views

prove this inequality related to probability and information theory

How do I prove this? I'm thinking I should use Jensen's inequality somehow. $$\sum_K p_k(1-p_k) \le -\sum_K p_k\log p_k$$ The assumption that $\sum_K p_k=1$ holds.
1
vote
4answers
86 views

How to show that $a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$

How do you show that $$a+b> \sqrt{a^2+b^2-ab}, \qquad a, b >0$$ I could write $\sqrt{a^2+b^2-ab}=\sqrt{(a+b)^2-3ab}$, but this seems to lead nowhere.