Questions on proving, manipulating and applying inequalities.

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0
votes
2answers
41 views

Does this inequality hold for all n>=1?

$$\ln ( \ln ( ( n+1 ) ^ {1/2} ) ) - \ln ( \ln ( n ^ {1/2} ) ) < \frac1 { (n / \ln (n ) ) ^ 2 + 1}$$ It seems that this is true for all $n\geq1$. I tried proving that by induction but I ...
2
votes
1answer
31 views

What is the best time complexity of checking the inequality $a_1x_1 + \cdots + a_mx_m \le K$ to have a non-negative integer solution?

We know that all the coefficients $a_1, a_2, \ldots , a_m$ are integer. Also, $K$ is an integer number. I only need to know if the inequality has a integer solution or not. It means that there is no ...
1
vote
0answers
56 views

Where does this inequality in a paper come from?

It's probably simple but I'm not sure why I'm not seeing it. The inequality is from a paper: $$\begin{align*} \sum_{i=1}^4 \rho_i (x_i-1)(1-\sum_{j=1}^4 \alpha_{ij}x_j) &\leq\begin{split} ...
1
vote
3answers
64 views

Help with proof of $(n+1)^n > n! 2^n$

I have already managed to prove it using induction and Bernoulli's inequality but I wonder if there is another way. My proof goes like this: (This is my first time using MathJax, so I apologize for ...
0
votes
2answers
53 views

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$ options $a.)\ 1<x<6\\ b.)\ 1 \leq x<6\\ c.)\ 1<x\leq 6\\ \color{green}{d.)\ 1\leq x \leq 6}$ I ...
2
votes
2answers
59 views

$x_1,x_2,…,x_n$ are positive real numbers, $\sum_{i=1}^n x_i^2 = 1$, to find the minimum value of:

$$\sum_{i=1}^n \frac{x_i^5}{s - x_i}$$ with $$s = \sum_{i=1}^{n}x_i$$ I used Cauchy Schwarz inequality: $$(\sum_{i=1}^n \frac{x_i^5}{s - x_i})(\sum_{i=1}^{n}\frac{s-x_i}{x_i}) \geq 1$$ ...
4
votes
3answers
47 views

$a_1,a_2,…,a_n$ are positive real numbers, their product is equal to $1$, show: $\sum_{i=1}^n a_i^{\frac 1 i} \geq \frac{n+1}2$

it says to use the weighted AM-GM to solve it, because the inequality is not homogenous I've tried to use $$\lambda _ i = \frac{a_i^{\frac1i -1}}{\sum_{k=1}^n a_k^{\frac1k -1}}$$ this $\lambda$ is ...
2
votes
1answer
30 views

$a_1,a_2,a_3,b_1,b_2,b_3$ are positive real numbers, show: $\sqrt[3]{(a_1+b_1)(a_2+b_2)(a_3+b_3)} \geq \sqrt[3]{a_1a_2a_3} + \sqrt[3]{b_1b_2b_3}$

The question says one only needs the AM-GM inequality, I've been stuck here for more than one hour. $$(a_i + b_i) \gt a_i$$ and $$a_i + b_i \gt b_i$$ therefore, $$ ...
4
votes
3answers
112 views

How to show $ \Big\vert \frac{\sin(x)}{x} \Big\vert $ is bounded by $1$?

This may be a silly question, but I cannot figure it out. I want to prove that $ \Big\vert \frac{\sin(x)}{x} \Big\vert \leq 1 $ for $x\in[-1,0)\cup(0,1]$, but I don't even know where to start.
0
votes
0answers
33 views

Solving differential equation and the inequality

I have stuck in a small step were I need to solve for t: $$ e^{t(\lambda_3-\lambda_1)} \geq 1 \Rightarrow t \geq\frac{1}{\lambda_3 - \lambda_1}$$ I don´t understand how the solution is required. ...
12
votes
3answers
220 views

This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$

let $0<a\le b\le c\le d$, and such $abcd=1$,show that $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$ it seems harder than This inequality $a+b^2+c^3\ge ...
2
votes
5answers
36 views

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$.

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$. options $a.)\ -101<x<25\\ b.)\ [-\infty,3]\\ c.)\ x\leq 3\\ \color{green}{d.)\ x<3}\\ $ I tried , Case $1$ ,for $ ...
1
vote
0answers
33 views

Inequality in 3 variables (conjecture)

Let $a, b, c$ be nonnegative real numbers such that $a+b+c=3$. If $0<k\leq 3+2\sqrt{3}$, then $$\frac{a}{b^2+k}+\frac{b}{c^2+k}+\frac{c}{a^2+k}\geq \frac{3}{1+k}$$ If $k=3+2\sqrt{3}$, then equality ...
2
votes
0answers
18 views

An inequality involving an exponential rate of sum

I'm having trouble understanding the conclusion in the proof of Cramér's Theorem in $\mathbb{R}^d$ in the book by Dembo/Zeitouni: We have the following: $\delta>0$ is fixed, $B_{y,\delta}$ is the ...
2
votes
0answers
22 views

Optimization by Symmetry?

Let $$f(x,y,a,b) := \frac{xa+yb}{\sqrt{xa^2+yb^2}},$$ where $x,y,a,b$ are all positive. Define $$g(a,b) = \min_{x+y=1,\,x,y\ge 0}f(x,y,a,b).$$ How would one solve for $g(a,b)$? I have solved this by ...
4
votes
4answers
233 views

This inequality $a+b^2+c^3\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}$

Let $0\le a\le b\le c,abc=1$, then show that $$a+b^2+c^3\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}$$ Things I have tried so far: ...
2
votes
1answer
43 views

Seeking Better (Symmetry Exploiting) Solution and Generalization of An Inequality

Given positive variable $(x,y,a,b)$ where $x+y=1$, how does one "slickly" prove the following inequality? $$f(x,y) := \frac{xa+yb}{\sqrt{xa^2+yb^2}}\ge \frac{2\sqrt{ab}}{a+b}.$$ or simply $$f(x,y) := ...
2
votes
3answers
58 views

Proving $n \lt 2^n$ for $n\geq 1$ using induction

Very close to understanding this, hopefully. Via induction, I'm following a proof but can't understand one of the last steps. Claim: $n < 2^n$ for natural numbers $n = 1, 2, 3,\ldots$ For step ...
0
votes
0answers
65 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle ...
1
vote
1answer
29 views

The AM-GM inequality with rational weights

This is Problem 2.2 of Steele's "The Cauchy-Schwarz Master Class": Suppose that $p_1, \ldots, p_n$ are nonnegative rationals whose sum is $1$. Show that for any real numbers $a_1, \ldots, a_n$, one ...
0
votes
3answers
62 views

Induction Inequality with Summation [closed]

I can't seem to figure out this problem. Do you have any ideas? For an integer $n > 1$, show that $$ \sum_{k=1}^n {1\over \sqrt{{n^2}+k}} > {{\sqrt{1+{1\over n}}}\over 2} $$
2
votes
1answer
58 views

Elementary inequality? [duplicate]

Let $x,y>0$, it seems (with numerical simulations) that $x^x+y^y \geq x^y +y^x$. If this is true, it has to be well known by some people. Does this inequality have a name? several proofs?
1
vote
3answers
51 views

Show $|\exp(-x/2) - \exp(-y/2)| \leq |x-y|/4$ for $x,y\geq 0$. [closed]

I am trying to show this inequality: $$ \left|e^{-x/2} - e^{-y/2}\right| \leq \frac{|x-y|}{4} $$ for $x,y\geq 0$. I've gotten stuck and could use some kind assistance. Many thanks in advanced!
-1
votes
3answers
63 views

Prove the inequality by induction [duplicate]

Prove the inequality by induction: $3^n > n^3\ $ for $\ n \geq 4$ Edit: 1) Base case: $n=4$, $3^4>4^3, 81>64$ 2) Assume true for n=k: so $3^k>k^3$ 3) Consider $(k+1)^3$, $(k+1)^3 = ...
0
votes
1answer
30 views

Solution of linear inequality

I have the following system of linear inequality on $x_1, x_2, \dots, x_n$, $x_i \in \mathbb{R} \; \forall i$ $x_i - 2x_j < b \; \forall i, j$ The right hand side of the inequality ($b \in ...
2
votes
0answers
22 views

Schwarz Inequality of function from upper half plane to disc

So I've been working on this problem and I have everything nailed down (I think) except for the very end. In particular I get a bound, but I can't seem to reduce it down to the one the question is ...
9
votes
2answers
140 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$

Edit: This problem 1 is a 2014 Sydney mathematics competition problem (8th grade). It seems difficult to solve. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such ...
1
vote
4answers
63 views

Prove that $\sum_{i=1}^{i=n} \frac{1}{i(n+1-i)} \le1$

$$f(n)=\sum_{i=1}^{i=n} \dfrac{1}{i(n+1-i)} \le 1$$ For example, we have $f(3)=\dfrac{1}{1\cdot3}+\dfrac{1}{2\cdot2}+\dfrac{1}{3\cdot1}=\dfrac{11}{12}\lt 1$ If true, it can be used to prove: ...
0
votes
1answer
28 views

Inequality about Frobenius norms for matrices [closed]

For any square matrix A, but not necessarily symmetric, what are some ways to prove the inequality $$ \|A^2\|_F^2\leq\|A^TA\|_F^2, $$ where $\|B\|_F^2=tr(B^TB)$ is the Frobenius norm of matrix $B$ ?
1
vote
4answers
153 views

Solve: $-\frac{1}{\sqrt{2}} < \sin \theta + \cos \theta < \frac{1}{\sqrt{2}}$

The question is: Solve $$-\frac{1}{\sqrt{2}} \lt \sin\theta + \cos\theta < \frac{1}{\sqrt{2}}$$ for values of $\theta$ between $0^\circ$ and $180^\circ$. I realized that: $$\begin{align} ...
7
votes
0answers
79 views

A matrix with a dense submatrix - application of Chernoff’s Inequality

I am trying to solve an exercise from this book, which I will post here for convenience. I have a bit of a problem understanding how the hint of using Chernoff's bound implies the claim. Specifically ...
5
votes
2answers
68 views

Proving $x \ln^2 x - (x-1)^2<0$ for all $x\in(0,1)$

I want to prove that for all $x\in(0,1)$,$$f(x):=x \ln^2 x - (x-1)^2<0$$ Using the derivative ($f'(x)=-2x+\ln^2 x+2\ln x +2$), I tried to prove that $f$ is monotonically increasing in $(0,1)$, and ...
2
votes
1answer
38 views

Integral inequality similar to Hardy's

I am trying to solve following puzzle: We are given functions $f$, where $f(x) > 0$ and $F := \int_0^x f(t) dt$ and some real $p>1$. Does $\int_0^\infty f(x)^p e^{-x}dx < \infty$ imply ...
0
votes
1answer
55 views

Area of the figure within the circle and outside a polygon

For which values of the parameter $c \in \mathbb{R}$, the area $S$ of the figure $F$, consisting of the points $(x,y)$ such that $$\begin{gathered} \max \{ \left| x \right|,y\} \geqslant 2c \hfill ...
14
votes
1answer
255 views
+50

On the inequality $\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
1
vote
1answer
23 views

Prove that $ m_{a} \geq m_{g} \geq m_{h} $ using strict inequalities unless $ a = b $.

$ m_{a} = \frac{1}{2} (a + b) $ $ m_{g} = \sqrt{ab} $ $ \frac{1}{m_{h}} = \frac{1}{2}(\frac{1}{a} + \frac{1}{b})$ Attempted Solution: I believe I have shown the first step, which is $ m_{a} ...
1
vote
4answers
54 views

Prove that If $0<x<\ln 2$ then $x^2\geq e^x-x-1$

If $0<x<$ln $2$ then $x^2\geq e^x-x-1$ I got this problem while reading a proof. Tried to prove it but failed. $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...$. So $e^x\geq 1+x$ for all $x$ but ...
1
vote
6answers
72 views

Given $a\leq b \leq c \leq d $ .How to prove $ c - b \leq d-a$

Given $a\leq b \leq c \leq d $ .How to prove $ c - b \leq d-a$. I am facing this inequality again and again in Riemann integration ,but i couldnot figure it out Thanks
0
votes
1answer
63 views

An inequality: $(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)$

Question: Find the smallest possible integer value of $n$, for which the following inequality holds true:- $$(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)$$ Where $x,y,z \in R $. I used the Cauchy inequality ...
0
votes
4answers
50 views

Find the minimum value

If $a,b,c,d$ are positive real numbers and $abcd=1$ then Find the minimum value of $(4+a)(4+b)(4+c)(4+d)$. Find the condition when minimium value holds. I've used AM-GM Inequality $4+a \ge 2 ...
1
vote
1answer
89 views

What is the non-inductive proof of this inequality? [duplicate]

$$\dfrac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} < \dfrac{1}{\sqrt{3n+1}}.$$ However I've non-inductive proof of $\dfrac{1 \cdot 3 \cdot 5 \cdots(2n-1)}{2 \cdot 4 \cdot 6 ...
3
votes
1answer
77 views

Inclusion - exclusion-like inequality

Let $(\Omega,\mathcal{F},\mathbb{P})$ be some probabilistic space and $A_1,\ldots,A_n\in \mathcal{F}$. Is it true that: $$\sum\limits_{i=1}^{n} \mathbb{P}(A_i)^2 - \sum\limits_{1\le i<j\le ...
0
votes
0answers
38 views

Behaviour of the sum of divisors function via logarithmic means versus an elementary problem equivalent to the Riemann Hypothesis due to Lagarias

It is known the following (see [1], here is an open access in his homepage www.math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf): Theorem (Lagarias, 2002). Let $\sigma(n)$ denote the sum of the ...
1
vote
3answers
79 views

Prove that $n^a < a^n$ for $a>1$ and $n$ big enough

How can I solve this? I'm trying to prove using limits but it's not working.. Thanks
5
votes
4answers
126 views

$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{2}{3}$

This is from the book Problems in Mathematical Analysis I by Kaczor and Nowak: Show that, for $n\in \mathbb{N}$, $$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{2}{3}$$ The solution in the ...
1
vote
3answers
37 views

Inequalities Theorem Proof

While studying, I read that the solutions of $f(x)/g(x)<0$ is equal to the solutions of $f(x)*g(x)<0$. I don't get why this is so. Could someone explain the reason the solutions to these two ...
1
vote
4answers
32 views

how to solve inequality using logarithm

I was given the following expression :$$(0.87)^n\leq 0.1$$ And the next step was: $$n\geq \frac{log(0.1)}{log(0.87)}$$ What was the steps betweens?
2
votes
3answers
120 views

How can I find the minimum value for $F(x,y,z,w)=x^2+y^2+z^2+w^2+xy+zw-xz-yw-yz$

Let $x,y,z,w$ be integer numbers,and $xw=yz+1$ Find this minimum of the value $$x^2+y^2+z^2+w^2+xy+zw-xz-yw-yz$$ This is how did it and I would like to know if I made a mistake Let ...
9
votes
2answers
108 views

If $x+y+z=3$, then $\sum_{\text{cyc}}\frac{x^2}{2y^2-y+3}\ge\frac{3}{4}$

Let $x,y,z>0$, be such that $x+y+z=3$. Show that $$\dfrac{x^2}{2y^2-y+3}+\dfrac{y^2}{2z^2-z+3}+\dfrac{z^2}{2x^2-x+3}\ge\dfrac{3}{4}.$$ I've tried many things but all have failed. ...
0
votes
1answer
32 views

Bound for non-integer power of sum

Let $x > 1$, $y \in (0,1)$ and $z \in (0,1)$. I need to bound $$(x+y)^z - x^z \leq B_z(x)$$ where I guess something like $B_z(x) \approx x^{z-1}$. Is there anything known on these non-integer ...