Questions on proving and manipulating inequalities.

learn more… | top users | synonyms (1)

0
votes
1answer
44 views

How prove that $ 3(a^3+b^3)+1-3c\ge \frac{a^2+b^2-c^2+1-4c}{a+b}$?

Let $ a,b,c>0$ be such that $ a+b+c=1$. How prove that $ 3(a^3+b^3)+1-3c\ge \frac{a^2+b^2-c^2+1-4c}{a+b}$?
4
votes
0answers
47 views

Higher interior regularity

From PDE by Evans, 2nd edition, pages 332-333. My question and work shown are at the bottom of this post. THEOREM 2 (Higher interior regularity). Let $m$ be a nonnegative integer, and assume ...
1
vote
2answers
34 views

a proof of constants are null from a given inequality

Problem: given constants $a,b\text{ and }c$, and a variable $x$, assume that for all $x\in\mathbb{R}$ holds that $|ax^2+bx+c|\le|x|^3$, then proof that $a=b=c=0$ My try: substitute $x=0$ into the ...
0
votes
1answer
15 views

Nesbitt inequality symmetric proof

I was trying to proof Nesbitt inequality using symmetry. So i asummed that $$a+b=x , b+c =y ,c+a=z$$ and then I rewrite inequality as:$$\sum \limits_{cyc} \frac {\frac {x+z-y}{2}}{y}\geq ...
2
votes
1answer
33 views

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$?

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$ for $x \in (0,\pi)$?
5
votes
0answers
150 views

A weird inequality maybe solvable by a power series

$a,b,c,d>0$ satisfying $a^3+b^3+c^3+d^3=1$. Prove $$\frac{1}{1-bcd}+\frac{1}{1-cda}+\frac{1}{1-dab}+\frac{1}{1-abc}\le \frac{16}{3}$$. I tried to go the normal way, by Cauchy-Schwartz, but that ...
3
votes
1answer
77 views

Prove $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$ [duplicate]

if $a,b,c$ are positive real numbers,Prove:$$a^3+b^3+c^3\geq a^2b+b^2c+c^2a$$ Things I have done so far: I know the fact that $$a^3+b^3+c^3\geq\frac{1}{2}[ab(a+b)+bc(b+c)+ca(c+a)]$$ However i ...
13
votes
2answers
333 views

Comparing sums of surds without any aids

Without using a calculator, how would you determine if terms of the form $\sum b_i\sqrt{a_i} $ are positive? (You may assume that $a_i, b_i$ are integers, though that need not be the case) When there ...
2
votes
2answers
69 views

Proving that a number is non-negative?

The numbers $a$,$b$ and $c$ are real. Prove that at least one of the three numbers $$(a+b+c)^2 -9bc \hspace{1cm} (a+b+c)^2 -9ca \hspace{1cm} (a+b+c)^2-9ab$$ is non-negative. Any hints would be ...
1
vote
0answers
52 views

Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
1
vote
0answers
91 views

How to prove this inequality $b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$

let $a,b,c>0$,and $$abc=1$$ show that $$b(a-1)(c-1)+c(b-1)(a-1)+a(c-1)(b-1)\le 0$$ since $$b(a-1)(c-1)=b(ac-a-c+1)=abc-ab-bc+b=1-ab-bc+b$$ so we only prove $$3-2(ab+bc+ac)+a+b+c\le 0 $$ oh,this ...
8
votes
6answers
536 views

How to solve the inequality $x^2>10$ using square roots?

Solve the inequality: $$x^2>10$$ How am I supposed to do this? It doesn't make sense when I take into account that if $x^2=10$ then $x=+\sqrt{10}$ and $x=-\sqrt{10}$ But how am I supposed to ...
1
vote
2answers
53 views

Prove Schwarz inequality in $R^2$

Can someone please show me how you would prove the following in $R^2$ $\int f(x)* g(x) dx \leqslant \int f(x)^2 dx * \int g(x)^2 dx $ starting from $\int [\lambda*f(x) - g(x)]^2 dx \geqslant ...
0
votes
1answer
24 views

Applying Markov's inequality to a sequence of random variables

Does the Markov inequality also work for infinite $a$ or only for constant $a$? More precisely: If $X(n)$ is a sequence of random variables and $f(n)$ is some sequence of numbers,is it allowed to ...
8
votes
1answer
169 views

$a;b;c\in \mathbb{R}^+$. Prove : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$

$a;b;c\in \mathbb{R}^+$. Prove : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$ Thanks :) I have proved that : $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq ...
0
votes
1answer
24 views

Continued fraction inequality: $q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$

In an article it is used the fact that $$q_n\left|q_n\alpha-p_n\right|(a_{n+1}+1)>1$$ where $\alpha=[a_0;a_1,\ldots]$ is an irrational number and $q_i$ is the series of the best approximation ...
6
votes
2answers
98 views

Prove $\frac{a^2+b^2+c^2}{ab+bc+ca} + 8\frac{abc}{(a+b)(b+c)(c+a)} \ge 2$

Let $a,b,c>0$, prove that $$\frac{a^2+b^2+c^2}{ab+bc+ca}+\frac{8abc}{(a+b)(b+c)(c+a)}\ge 2.$$ I tried using the equality $(a+b)(b+c)(c+a)=(a+b+c)(ab+bc+ca)-abc$ and the Schur inequality but it's ...
0
votes
1answer
28 views

Alternative solutions of an inequality problem

Let $x, y, z$ be distinct real numbers. Prove $ \sqrt[3]{x - y} + \sqrt[3]{y - z} + \sqrt[3]{z - x} \neq 0$ I'm curious about different ways to solve this inequality. My solution:
4
votes
2answers
56 views

Find the smallest constant K satisfying the inequality

Find the smallest constant $K$satisfying the inequality $$x^{1\over 3}+y^{1\over 3} \le K(x+y)^{1\over 3}$$ The official proof makes the substitution $a=x^{1\over 3}$ and $b=y^{1\over 3}$, which does ...
1
vote
2answers
42 views

How to show that $\frac {q + \frac {1}{2}}{p - \frac {1}{2}} > \sum_{i = p}^q\frac {1}{i}$ if $q\ge p > 0?$

How to show that : $$\frac{2q+1}{2p-1}>\sum_{i=p}^q\frac{1}{i}$$ if $q\ge p>0$
2
votes
1answer
35 views

How prove that $q \geq b+d$ for $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$?

Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$. How prove that $q \geq b+d$?
6
votes
5answers
198 views

Alternate Proof for $e^x \ge x+1$

This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it: Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) ...
0
votes
1answer
37 views

Is there any easy way to approximate the answer? [closed]

I have recently faced a long question with a lot of confusing words :( May be I am too weak in English, I don't know. But, can anybody here help me to find out the question with analysis of the ...
4
votes
5answers
143 views

How to solve this inequality? From MSU entrance exam '66

$\frac{\log _{10}\left(2\right)}{\log _{10}\left(\sin \left(x\right)\right)}\le \frac{\log _{10}\left(4\sin ^2\left(x\right)\right)}{\log _{10}\left(\sin \left(x\right)\right)}$ From the title. Not ...
3
votes
2answers
52 views

Proving an convexity-looking inequality

If $0 \le \alpha \le 1$ and $0 \le \lambda \le 1$, then $$\lambda^\alpha x^\alpha +(1-\lambda^\alpha) y^\alpha \ge (\lambda x + (1-\lambda)y)^\alpha$$ whenever $0 \le y \le x$. This looks ...
4
votes
1answer
77 views

Showing that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$.

I know from intuition that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$. The way I would prove it is to use the triangle inequality: $|x-y| = |x+(-y)| \leq |x| +|-y| = |x|+|y|$ for $x.y \in ...
9
votes
1answer
74 views

If a polynomial has only real zeros then $a_{0}+a_{1}+\cdots+a_{n}\le\frac{(n+1)^n}{\binom{n}{s}(n-s)^{n-s}(s+1)^s}\cdot\max_{k}a_{k}$

Question: For all real polynomials $P(x)=a_{0}+a_{1}x+\cdots+a_{n}x^n$ of degree $n$, with only real zeros,we have ...
5
votes
1answer
59 views

Prove $\sin^{2m}\alpha\cdot\cos^{2n}\alpha\leq\frac{m^m n^n}{(m+n)^{(m+n)}}$

If $n$ and $m$ are natural numbers, Prove: $$\sin^{2m}\alpha\cos^{2n}\alpha\leq\frac{m^mn^n}{(m+n)^{(m+n)}}$$ Additional info:We should only use AM-GM inequality.We can use Trigonometry ...
2
votes
4answers
57 views

Prove the inequality $2\sqrt{x}\ge3-\frac1x $

Given that $x\gt0$ prove the following inequality: $2\sqrt{x}\ge3-\frac1x $ I have done it using calculus but how can I do it using elementary methods?Thanks!!
3
votes
3answers
287 views

Sufficient conditions for bound

Let $m\leq n$ be nonnegative integers and $x > 0$. I would like to find sufficient conditions on $m,n,x$ (as tight as possible) s.t. $$\frac{ \binom{n}{m} \sum_{j=0}^m j\binom{n}{m-j}x^j }{ x ...
0
votes
1answer
46 views

Show That $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ And $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ Are Orthogonal Trajectories

Show that the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ are orthogonal trajectories if $A^2< a^2$ and $a^2-b^2 = A^2+B^2$. What I've ...
1
vote
1answer
46 views

Regarding 'non-square- free ' numbers.

Call an integer 'n' that is not a square or a prime power or a square-free a 'square-in'.Let n be square-in. Then between n and (2 n) is there another square-in? This is a kind of 'variation' on ...
2
votes
2answers
55 views

Trigonometric inequation $\sin x \ne \sin y$

How can I solve the following trigonometric inequation? $$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$ Why I'm asking this question... I was doing my calculus homework, ...
6
votes
2answers
135 views

Prove two of $\frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq 6,\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq 6,\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq 6$ are True

if $a,b,c$ are positive real numbers that $a+b+c\geq abc$, Prove that at least $2$ of following inequalities are true. $\frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq 6, ...
2
votes
1answer
147 views

How prove this ineuality$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}\le\sqrt{xy+yz+zx+9}$

let $$x,y,z\in(-1,1), x+y+z=-xyz$$ show that $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}\le\sqrt{xy+yz+zx+9}$$ This problem is my frends ask me,I remenber this is old inequality,But Now I can't it ...
2
votes
2answers
50 views

Writing solutions of inequalities: $3<x$ versus $x>3$

My son wrote a solution to a number line graph as 3 < x instead of what his teacher said was the correct answer of x > 3. When he brought his paper back in to bring it up he was told that the ...
22
votes
27answers
4k views

How can I prove that $xy\leq x^2+y^2$?

How can I prove that $xy\leq x^2+y^2$?
1
vote
5answers
59 views

Inequality involving a finite sum

this is my first post here so pardon me if I make any mistakes. I am required to prove the following, through mathematical induction or otherwise: $$\frac{1}{\sqrt1} + \frac{1}{\sqrt2} + ...
2
votes
1answer
56 views

Using logical OR to combine inequalities.

I have a physical system that must satisfy one of two inequalities: $x\leq y$ OR $p\leq q$ But not necessarily both simultaneously. Is there a way to combine this into a single inequality? ...
4
votes
1answer
97 views

How find this maximum and minimum of the value $\sum_{i=1}^{n-1}[x_{i+1}-x_{i}]$

Question: let $x_{1},x_{2},\cdots,x_{n}\in \mathbb{R}$,and Assume that the following two sets are equivalent; $$\{[x_{1}],[x_{2}],[x_{3}],\cdots,[x_{n}],\}=\{1,2,3,\cdots,n\},n\ge 2 $$ ...
1
vote
2answers
29 views

Regarding square-free numbers and their doubles.

Is it true that between any non-prime square-free number and it's double is another non-prime square-free number?
1
vote
2answers
37 views

Radical Inequality

$\sqrt{2x-1}$ + $\sqrt{3x-2}$ > $\sqrt{4x-3}$ + $\sqrt{5x-4}$ I have attempted to solve this by squaring each side, resulting in $5x + 2\sqrt{2x-1}\sqrt{3x-2} - 3 > 9x + 2\sqrt{(4x-3)(5x-4)} - 7 ...
1
vote
1answer
33 views

How prove $ \frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4}) $?

For any $x,y\in[0,\frac{\pi}{2}]$ , how prove the inequality $\frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4})$?
2
votes
3answers
315 views

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
2
votes
1answer
66 views

How prove $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$

How prove that if $x, y \in (0,\sqrt{\frac{\pi}{2}})$ and $x \neq y$, then $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$?
3
votes
1answer
225 views

Inequalities involving arithmetic, geometric and harmonic means

Let $A$, $G$ and $H$ denote the arithmetic, geometric and harmonic means of a set of $n$ values. It is well-known that $A$, $G$, and $H$ satisfy $$ A \ge G \ge H$$ regardless of the value $n$. ...
5
votes
1answer
51 views

How prove that $ x+y+z>4$ for $ a+b+c=4$ and $ ax+by+cz=xyz$?

Given positive reals $ a,b,c,x,y,z$ such that $ a+b+c=4$ and $ ax+by+cz=xyz$. How prove that $ x+y+z>4$?
-1
votes
3answers
135 views

An inequality with absolute value and a parameter: $|x-4|>a$

Solve : $|x-4|>a$. Case 1: $a>0$; Case 2: $a<0$ Progress I am getting answers which look similar in both cases: Let $a>0$ so $x>4+a$ or $x<4-a$ , Let $a<0$ so ...
4
votes
2answers
109 views

Prove that $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}} \le 2 \sqrt{n} - 1$ is true for $n \in \mathbb{N}^{\ge 1}$

I'm trying to solve these induction exercises proposed by the department of mathematics of Oxford University. I don't know how to give a valid proof for the third one which says the following: ...
3
votes
2answers
74 views

Prove the inequality $n!\lt n^{n+\frac12} e^{-n+1}$ [closed]

Prove the following inequality: $$n!\lt n^{n+\frac12} e^{-n+1}.$$ Try to avoid induction if possible. Thanks!!