Questions on proving and manipulating inequalities.

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1
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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?
4
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5answers
142 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 ...
1
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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 $\frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4})$?
1
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1answer
22 views

$S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
6
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2answers
134 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, ...
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} + ...
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!!
2
votes
3answers
47 views

if $ax-2by+cz=0$ and $ac-b^2>0$ , Prove $zx-y^2\leq0$

for real numbers like $a,b,c,x,y,z$ that $ax-2by+cz=0$ and $ac-b^2>0$ Prove:$$zx-y^2\leq0$$ Additional info: The Proof should be by contradiction.we can use Cauchy , AM-GM and other simple ...
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 $x \in (0,\pi)$?
0
votes
0answers
25 views

Proof of inequality related to complex function

Let $f(z)$ be a nonsingular complex function whose domain is $D=\{z\in C\ ;\ |z|<R\}$. $f(z)$ satisfies $f(0)=1$, and Re$f(z)$ is positive everywhere on $D$. Let $g(z)$ be $$g(z) = ...
0
votes
1answer
39 views

Prove $\sum\limits_{cyc}\frac{(b+c-a)^4}{a(a+b-c)}\geq ab+bc+ca$

if $a,b,c$ are positive real numbers,Prove:$$\sum\limits_{cyc}\frac{(b+c-a)^4}{a(a+b-c)}\geq ab+bc+ca$$ Additional info: Problem should be solved with AM-GM inequality only. Things i have tried ...
8
votes
1answer
103 views

Prove $(1-a)(1-b)(1-c)(1-d)\geq abcd$ if $a^2+b^2+c^2+d^2=1$

Let $a,b,c,d\geq0$, $a^2+b^2+c^2+d^2=1$ Prove $\displaystyle (1-a)(1-b)(1-c)(1-d)\geq abcd$ I mutiplied both with $\displaystyle (1+a)(1+b)(1+c)(1+d)$ to use $1-a^2=b^2+c^2+d^2$ and try using the ...
5
votes
1answer
61 views

Prove $\frac{(xy)^7}{x^8+(xy)^7+y^8}+\frac{(yz)^7}{y^8+(yz)^7+z^8}+\frac{(zx)^7}{z^8+(zx)^7+x^8}\leq1$

If $x,y,z$ are positive real numbers that $xyz=1$ , Prove a) $\frac{xy}{x^8+xy+y^8}+\frac{yz}{y^8+yz+z^8}+\frac{zx}{z^8+zx+x^8}\leq1$ ...
0
votes
0answers
24 views

Minkowski inequality for $0<p<1$

I'm trying to prove this, $$\left ( \sum_{i=1}^{n}(x_i+y_i)^p \right) \geq \left ( \sum_{i=1}^{n}(x_i)^p \right)^\frac{1}{p} + \left ( \sum_{i=1}^{n}(y_i)^p \right)^\frac{1}{p} $$ for $0<p<1$. ...
3
votes
5answers
59 views

Determine variables that fit this criterion…

There is a unique triplet of positive integers $(a, b, c)$ such that $a ≤ b ≤ c$. $$ \frac{25}{84} = \frac{1}{a} + \frac{1}{ab} + \frac{1}{abc} $$ Just having trouble with this Canadian Math ...
6
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1answer
64 views

How to show $ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$ [duplicate]

When $p\geq 2$ and $0\leq x\leq1$, how does one show the inequalities $$ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$$ and $$ 2(1+x^p)\leq (1+x)^p + (1-x)^p \ ?$$ ...
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2answers
42 views

Upper bound on the rational function of $z$ in terms of $|z|$

Show that: $$\frac{|2z^2-5|}{|z^2+1||z^2+4|} \le \frac{2|z|^2+5}{(|z|^2-1)(|z|^2-4)}$$ I started by considering that for the above to hold $|2z^2-5|\le (2|z|^2+5)$, and $|z^2+1|\ge (|z|^2-1)$, ...
2
votes
1answer
145 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 ...
6
votes
2answers
97 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
2answers
45 views

Given $(x+3)(y−4)=0$, what is the relationship between $xy$ and $-12$?

Given $(x+3)(y−4)=0 $ Quantity $A = xy $ Quantity $B = -12 $ A Quantity $A$ is greater. B Quantity $B$ is greater. C The two quantities are equal. D The relationship cannot be determined from ...
3
votes
2answers
59 views

Prove that $\frac{a^3}{x} + \frac{b^3}{y} + \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$ a,b,c,x,y,z are positive real numbers.

I stumbled upon it on some olympiad papers. Tried to AM>GM but didn't get any idea to move forward.
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
3answers
76 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
4
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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; $$\{[x_{1}],[x_{2}],[x_{3}],\cdots,[x_{n}],\}=\{1,2,3,\cdots,n\},n\ge 2 $$ ...
2
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0answers
71 views

Diophantine inequality that comes up after Vieta Jumping Hurwitz technique

I am blaming this on Prove the equality EDITTTTT: allowing $x_1 \geq x_2$ and $x_2 \geq x_n,$ I would rather not explain what that was about and the only changes are in $n=3,4,$ already settled. ...
1
vote
3answers
50 views

$\exists x \in N, \forall y \in N, x \ge y$

$\exists x \in N, \forall y \in N, x \ge y$ Why is this a false statement? Intuitively, it seems that if you let x always equal y, then the statement always holds true.
0
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2answers
41 views

How to solve the system $ax>y+z$, $by>x+z$, $cz>x+y$ in positive numbers?

Let $a>b>c>1$. How to find solutions in positive numbers of the following system? \begin{cases} ax>y+z \\ by>x+z \\ cz>x+y \end{cases}
3
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3answers
213 views
+50

How to prove this inequality $\sum_{i=1}^{n}\left(x^k_{i}\ln{x_{i}}\ln{\frac{x_{i}}{n}}\right)\le 0$

Let $x_{i}\ge 0$ for $i\in\{1,2,\cdots,n\}$ and $x_{1}+x_{2}+\cdots+x_{n}=n$ for $n\ge 3$ Show that for all strictly positive integers $k\ge2$ the following inequality holds : ...
0
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0answers
29 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
0
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2answers
50 views

Mathematical Induction - Inequality

Does anyone have any idea on how to complete the inductive step? Thm: For all $n >= 0~~~~ 6^n + 4 > n^3$ Pf: by Induction     Let $P(n)$ be proposition that $~6^n + 4 ...
0
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0answers
30 views

$a,b,c \geq0$ and $a+b+c=1$ find Maximum value of : $M=\frac{1+a^2}{1+b^2}+\frac{1+b^2}{1+c^2}+\frac{1+c^2}{1+a^2}$ [closed]

$a,b,c \geq0$ and $a+b+c=1$ find Maximum value of : $M=\frac{1+a^2}{1+b^2}+\frac{1+b^2}{1+c^2}+\frac{1+c^2}{1+a^2}$ i've posted this problem in another post,howerver i took the advise to seperate it ...
0
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1answer
30 views

How prove this inequality?

Question: let sequence $x_{i}\neq 0,i=1,2,\cdots,n$ are real numbers, show that $$\left|x_{l}-\dfrac{\displaystyle\max_{1\le k\le n}|x_{k}|\min_{1\le k\le n}|x_{k}|}{x_{l}}\right| \le \max_{1\le ...
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0answers
23 views

$a,b,c \geq 0$ such that $a+b+c=1$ prove that $a^2b+b^2c+c^2a+abc \le \frac{4}{27}$ [closed]

$a,b,c \geq 0$ such that $a+b+c=1$ prove that $a^2b+b^2c+c^2a+abc \le \frac{4}{27}$ i've posted this problem in another post,howerver i took the advise to seperate it and double check the problem.
0
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3answers
72 views

$a,b,c > 0$ such that $\sqrt{a^2+b^2+c^2}=\sqrt[3]{ab+bc+ca} $ prove that $a^2b+b^2c+c^2a+abc \le \frac{4}{27}$

I have a series of problems in inequalities that I can not solve,please help me if you can. problem 1 :$a,b,c \geq 0$ such that $\sqrt{a^2+b^2+c^2}=\sqrt[3]{ab+bc+ca} $ prove that ...
2
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1answer
55 views

How prove $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?

Let $x\ge 1$. How prove that $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?
0
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2answers
22 views

$\|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $ holds?

I want to find the relation of $p$ and $k$ such that the inequality $$ \|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $$ holds when r.h.s $<\infty$. Here $f$ ...
7
votes
1answer
96 views

How to solve $\log _{x^{2}-3}(x^{2}+6x)<\log _{x}(x+2)$?

How to solve the following inequality $$\log _{x^{2}-3}\left(x^{2}+6x\right)<\log _{x}(x+2)\ ?$$
0
votes
1answer
33 views

Inequality with little-o notation

I'm having trouble justifying the following: For large $n$, \begin{align*} -\log f(n) & < \log n + o(\log n)\\ \implies f(n) &> n^{-1} \log^3(n) \log(10) \end{align*} I think basically ...
1
vote
3answers
45 views

Bounding a modified Bessel function of the first kind.

Let $I_0$ be the zeroth-order modified Bessel function of the first kind. We know that, asymptotically as $x\to \infty$, $I_0(x) \sim e^x/\sqrt{2\pi x}$. Does anybody have a reference for the maximum ...
4
votes
0answers
46 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 ...
2
votes
1answer
43 views

verifying extrema found by Lagrange multipliers

This question was inspired by reading this problem: Prove the inequality $\frac 1a + \frac 1b +\frac 1c \ge \frac{a^3+b^3+c^3}{3} +\frac 74$ Suppose I have a function $f(x,y,z)$ with continuous ...
0
votes
1answer
26 views

Equivalence of inequalities

I'm having trouble showing the following. If $a>b\ge 0$, then $$\frac{a+b}{2}-\sqrt{ab}<1 \iff \begin{cases}a+b<2, \text{ or,} \\ a+b \ge 2 \text{ and } (a-b)^2 < ...
0
votes
0answers
27 views

Interior $H^2$ regularity - using a textbook's identity to show an important estimate

This is yet another continuation of my previous question, and concerns the same long textbook proof which I asked in those two questions as well. This is now from page 331 of PDE by Evans, 2nd ...
0
votes
2answers
27 views

Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
7
votes
2answers
91 views

Inequality of integrals $\int_0^1(f(x))^2 dx \geq 4$ if $\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$

If $$\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$$ prove that $$\int_0^1(f(x))^2 dx \geq 4$$ EDIT My attempt is as follows - I can use only the $\int xf(x)$dx part to get a bound $\int f^2(x) dx \geq ...
2
votes
1answer
28 views

Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
5
votes
0answers
148 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 ...
2
votes
1answer
18 views

How to prove the second inequality

This might be very trivial to show. But I still cannot figure it out. Let $a \in [-1, 1]$ and $b_i, c_i \in \mathbb R$ with $i \in \mathbb N$. Show that $$\sum_i ab_ic_i \leq |\sum_i ab_ic_i| \leq ...
2
votes
3answers
32 views

Minimum of $\max(1-2x+y,1+2x+y,x^2-y^2)$ for $x,y(y\ge 0)$?

We define $$S(x,y)=\max(1-2x+y, \, 1+2x+y, \, x^2-y^2)$$ on $\mathbb{R}\times\mathbb{R}_{\geq 0}$. How do we find the minimum value of $S(x,y)$?
1
vote
3answers
85 views

Conditional extreme value of a function

Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.