Questions on proving and manipulating inequalities.

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9
votes
2answers
136 views

when index is irrational number with inequality

Let $x>0$, show that $$x^{\sqrt{3}}+x^{\frac{\sqrt{3}}{2}}+1\ge 3\left(\dfrac{1+x}{2}\right)^{\sqrt{3}}$$ we consider $$f(x)=2^{\sqrt{3}}(x^{\sqrt{3}}+x^{\dfrac{\sqrt{3}}{2}}+1)- ...
1
vote
3answers
34 views

an inequality with noninteger order

I want to Show that the following inequality is true or not: For $0<q<1$ and $a\geq b\geq0$, $$a^q-b^q\leq 2(a-b)^q.$$ Could you please help me in showing this inequality is true or not? ...
1
vote
3answers
71 views

Positivity of power function.

Prove that $6^a-7^a+2\cdot 4^a-3^a-5^a\ge0$ for $-\frac{1}{2}\le a\le0$. I tried to do it by first derivative test but derivative become more complicated (same with 2nd derivative for ...
1
vote
1answer
127 views

Are these inequalities true?

The following is a question regarding an inequality direction. With $a,b,c $ being real and non-negative. Can I say that following is ALWAYS true. $$1+(a+b+c)^2 \le 2(a+b+c)^2$$ What about the ...
0
votes
1answer
31 views

Question regarding an inequality relationship [closed]

If $a,b,c >0$, is true that the following is always satisfied? $$(a+b+c)^2 > {(a+\sqrt{b^2+c^2})^2} $$
5
votes
1answer
69 views

Proving $\frac {2n}{(a+b)^n} \le \frac {1}{a^n} + \frac {1}{b^n}$ for $a,b>0, n\in\mathbb{N}$ by induction

prove using induction: $$ \frac {2n}{(a+b)^n} \le \frac {1}{a^n} + \frac {1}{b^n} $$ $$a,b \gt 0 , n \in N$$ my attempt: base $n=1$: $$ \frac {2}{(a+b)} \le \frac {1}{a} + \frac {1}{b}$$ ...
1
vote
1answer
35 views

Precalculus equation solving with inequalities

For which a $\in \mathbb{R}$ does the equation $$|x-1| + 2|x-2| = a$$ have two solutions? I divided this into three cases: Case 1 $x \geq 2$: $$x-1 +2(x-2) = a$$ $$x-1+2x-4 = a$$ $$3x-5 = a$$ ...
0
votes
1answer
25 views

How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality?

Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u ...
0
votes
1answer
36 views

How to bound this difference between two logarithmic expression

I want to bound the difference between two logarithmic expression shown below with a constant number i.e not function of $x,y,z$ where $x,y,z \in \mathbb{C}$. The difference is $$ ...
4
votes
1answer
55 views

Let $l$ be a natural number. Prove that $n\lt\sqrt{n ^ 2 + l}\lt n+1$ for almost every $n$.

In my assignment I have to prove the following statement: Let $l$ be a natural number. Prove that for almost every $n$ the following inequality is true: $$n\lt\sqrt{n ^ 2 + l}\lt n+1$$ I chose ...
0
votes
2answers
22 views

Inequality Solution Set

Find the solution set for the inequality: $|x-2| > |x+6|$ I'm just not sure how to work this out? I've found the answer is $(-\infty,-2)$ using wolframAlpha, however there isn't a step by step. ...
2
votes
0answers
43 views

On the second part of solution of a question due to Erdos

Problem. Let $a_1<a_2<\dotsb<a_n\le 2n$ be a sequence of positive integers. Then $$ \min [a_i,a_j]\le 6\left(\Big[\frac n2\Big]+1\right), $$ where $[a_i,a_j]$ denotes the least ...
3
votes
3answers
126 views

Prove that $a^2+b^2+c^2\geq [2(a-b)^2(b-c)^2(a-c)^2]^{1/3}$

Mathematica seems to know that this statement is true, yet I am struggling to prove it. Possible useful inequalities are Minkowski and the geometric mean. Using the geometric mean inequality I can ...
5
votes
4answers
308 views

Compare three numbers, expressed as powers: $4^{68}$, $5^{51}$ and $12^{23}$

So I have these numbers: $4^{68}, 5^{51}, 12^{23}$ They need to be ordered from the smallest to greatest. I have no idea how to solve this. Typically, one should break down the exponents somehow to ...
0
votes
4answers
93 views

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$.

Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 1$. Find the maximum value of $2x - 5y$. I do know how to solve this problem using trigonometry, however I need to solve it by using vectors. ...
2
votes
1answer
72 views

The implication $\sqrt{(x-2)^2+(y-1)^2+(z-1)^2+(w-3)^2}<b\implies |xyzw-6|<a$

Given a real number $a>0$ find a $b>0$ such that $\sqrt{(x-2)^2+(y-1)^2+(z-1)^2+(ω-3)^2}<b\Longrightarrow |xyzω-6|<a$ I tried the procedure followed in another one of my questions, but ...
0
votes
0answers
22 views

“Transference” Argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
2
votes
1answer
41 views

“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 $$ ...
0
votes
1answer
34 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
vote
1answer
48 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
votes
4answers
189 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 ...
1
vote
1answer
67 views

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

Let $a, b, c >0$ and $a^2+b^2+c^2 =1$ How prove $a + b +c + \frac{1}{abc} \geq 4\sqrt{3}$?
2
votes
1answer
41 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
votes
3answers
31 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 $|x-2| = p,$ ...
0
votes
2answers
70 views

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

Given that $a,b$ are real numbers. How can one show that $(a^2+1)(b^2+1)>a(b^2+1)+b(a^2+1)$ ?!!
2
votes
2answers
80 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?
0
votes
2answers
50 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 ...
0
votes
1answer
14 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 ...
1
vote
1answer
38 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
votes
1answer
71 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 ...
1
vote
2answers
62 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!
-5
votes
1answer
206 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 ...
1
vote
0answers
47 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 ...
0
votes
1answer
21 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 ...
1
vote
0answers
59 views

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 ...
0
votes
0answers
29 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 $t_n = nh, n=0,\ldots,N$ ...
0
votes
2answers
28 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
9
votes
7answers
937 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
votes
9answers
120 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.
2
votes
0answers
41 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 $P=\frac{2}{3+xy+yz+zx}+(\frac{xyz}{(x+1)(y+1)(z+1)})^\frac{1}{3}$
0
votes
0answers
13 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
votes
1answer
72 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. ...
1
vote
2answers
66 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: ...
2
votes
6answers
105 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 ...
0
votes
2answers
35 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 ...
-2
votes
2answers
43 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.)
3
votes
1answer
55 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 $A$ ...
8
votes
1answer
75 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 $$ ...
1
vote
2answers
28 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 ...
1
vote
1answer
60 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 ...