Questions on proving and manipulating inequalities.

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2
votes
2answers
63 views

How find the maximum of the value $x^2_{1}+x^2_{2}+\cdots+x^2_{2014}$

Question: let $x_{i}\in[-11,5],i=1,2,\cdots,2014$,and such $$x_{1}+x_{2}+\cdots+x_{2014}=0$$ find the maximum of the value $$x^2_{1}+x^2_{2}+\cdots+x^2_{2014}$$ since ...
5
votes
3answers
110 views

Inequality with $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}$

Inspired by this recent question, I suggest this. Let $n=2,3,4, \ldots .$ Then $$ \frac{7}{12} < \cfrac 1 {1 + \cfrac {1^2} {1 + \cfrac {2^2} {\ddots + \cfrac \vdots { 1 + \, {n^2} \,}}}} \leq ...
1
vote
1answer
31 views

Computational complexity and the big $\mathcal{O}$

I have a question about this Big $\mathcal{O}$ problem. I have the question down $90\%$, but the other $10\%$ isn't getting to me. I will write out the entire question and I'll point out the step, ...
5
votes
4answers
329 views

Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
5
votes
3answers
211 views

Is symmetry a valid option in inequalities?

Consider two questions: Q1. $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ $$a,b,c,d,e\in\mathbb{I^+_0}$$ Find maximum value of 'e'? My answer: Since when e is maximum when all other variables are equal ...
4
votes
8answers
159 views

Show that the inequality holds $\frac{1}{n}+\frac{1}{n+1}+…+\frac{1}{2n}\ge\frac{7}{12}$

We have to show that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{7}{12}$ To be honest I don't have idea how to deal with it. I only suspect there will be need to consider two ...
-2
votes
3answers
69 views

$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=1$ prove that $a+b+bc+ac-abc \leq 2$

$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4$ prove that $ab+bc+ac-abc \leq 2$ can any one help me with this problem,I believe Dirichlet's theorem is the key for this sorry for making mistake over and over ...
1
vote
2answers
61 views

Induction inequality on sum of reciprocals

I have to prove that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{1}{2}$ for natural $n$ Checking for $n=1$ we have $\displaystyle 1+\frac{1}{2}=\frac{3}{2}\ge \frac{1}{2}$ ...
2
votes
1answer
71 views

Prove $\sum \limits_{cyc} \frac{1+b^2+c^4}{a^+b^2+c^3}\geq 3$

If $a,b,c$ are positive real numbers, prove $$\sum \limits_{cyc} \frac{1+b^2+c^4}{a+b^2+c^3}\geq 3$$ Additional info: We should only use Cauchy (preferred to used at least once and more than ...
1
vote
1answer
43 views

To control first derivative with the function itself.

Let $f$ be a compactly supported positive $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2< C f(x) $ by showing that for every point ...
1
vote
3answers
24 views

Inequality involving floor function and fractions

I have little to no experience working with floor inequalities so I am kind of stuck on this one. It seems pretty intuitive though. So basically I want to show that ...
4
votes
2answers
85 views

prove $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq 3(a^2+b^2+c^2)$

If $a,b,c$ are positive real numbers and $a+b+c=1$,Prove: $$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq 3(a^2+b^2+c^2)$$ Additional info:We can use AM-GM and Cauchy inequalities mostly.We are ...
4
votes
2answers
44 views

prove $\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$

If $a,b,c$ are positive real numbers,prove:prove $$\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}$$ Additional info:We can use AM-GM and Cauchy ...
0
votes
0answers
29 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
0
votes
1answer
43 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
-1
votes
1answer
45 views

Finding two sided bounds on $(x+y)/(xy)$ given inequalities for $x$ and $y$

Given $\dfrac{1}{6} < x < \dfrac{1}{2}$ and $\dfrac{1}{7} < y < \dfrac{1}{3}$, can we determine bounds for $\dfrac{x+y}{xy}$?
1
vote
0answers
40 views

Arithmetic Mean and Geometric Mean Question, Guidance Needed

I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff ...
10
votes
1answer
112 views

prove that : $\frac{a^2+b^2}{a+b} + \frac{b^2+c^2}{b+c}+ \frac{c^2+a^2}{c+a} \geq 3$

For $a^2+b^2+c^2 =3$, with $a,b,c$, positive real numbers, prove that $$\frac{a^2+b^2}{a+b} + \frac{b^2+c^2}{b+c}+ \frac{c^2+a^2}{c+a} \geq 3.$$ can any one help me with this problem.
1
vote
1answer
27 views

Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
2
votes
1answer
39 views

prove $\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$

If $x$,$y$,$z$ are positive real numbers,Prove:$$\sum \limits_{cyc} \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq 1$$ Using this two inequality: $\sum ^n_{i=1} \sqrt{a_ib_i}\leq\sqrt {ab} $ (we call it ...
2
votes
2answers
57 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
2
votes
2answers
121 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
2
votes
3answers
86 views

How to prove $ \frac{e^{x}+e^{-x}}{2} \le e^{\frac{x^2}{2}} $?

Let $x\in \mathbb{R}$, how to prove $$ \frac{e^{\large x}+e^{\large-x}}{2} \le e^{\large\frac{x^2}{2}} $$
3
votes
3answers
40 views

prove $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

$a,b,c$ are positive real numbers.prove:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$ Additional info:we cant use induction.we should mostly use Cauchy inequality.other ...
2
votes
0answers
45 views

$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $ [on hold]

Show that for every arbitrary complex number a,b and c we have $$|a+b|+|b+c|+|c+a| \leq |a|+|b|+|c|+|a+b+c| \ $$ Thanks.
1
vote
0answers
75 views

Practicing the arithmetic-geometric means inequality

I am struggling with learning the AM-GM Inequality that is considered vital to know for math olympiads, contests, etc. I just don't really know when to use it, when it is necessary to use, the purpose ...
2
votes
6answers
213 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
6
votes
2answers
127 views

prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$

If $a_i$ and $b_i$ are positive, and $b= \sum ^n_{i=1} b_i$,$a= \sum ^n_{i=1} a_i$ prove $$\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$$ Additional: we should just use Cauchy ...
-3
votes
0answers
12 views

Maximum absolute column sum of the matrix norm inequality [on hold]

$A=(a_{ij})\in \mathbb{C}^{n\times n}$, $\upsilon(A)=n \max_{ij} |a_{ij}|$ is matrix norm. Prove that $\|A\|_{1} \leq \upsilon(A) \leq n \|A\|_1$.
2
votes
1answer
33 views

Prove $\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$

If $\space x_1+x_2+\cdots+x_n=1$ and all $x_1,x_2,\cdots,x_n$ are positive and real numbers, prove:$$\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$$ ...
0
votes
4answers
43 views

Prove that $r(n+1−r) \ge n$ for any positive integer $n$ and $1 \le r \le n$. [on hold]

Prove that $r(n+1−r) \ge n$ for any positive integer $n$ and $1 \le r \le n$. Nothing I tried worked, because the coefficient of $r$ is negative so I only get maximums and not minumums.
0
votes
0answers
19 views

Deriving inequalities featuring bounded variables

I have a model which fits certain thermodynamic data, of the form $$y = \frac{x}{ 1 + (a - 1)x} + bx(1 - x) \quad a,b \in \mathbb{R} \quad 0 \leq x \leq 1$$ Thermodynamics dictate that ...
1
vote
1answer
44 views

Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
3
votes
3answers
111 views

Prove that $n^n \le (n!)^2$.

Prove that $n^n \le (n!)^2$. There is an elementary solution, which I haven't been able to find. So far I tried manipulating and pairing terms but nothing worked. I would appreciate any help!
2
votes
1answer
75 views
+50

Entropy of the product of two random variables

Consider a random matrix $X$ and a random vector $Y$. Let the Shannon entropies $H(X) = H(Y) = n$. Is there a simple upper bound for entropy $H(XY)$? I believe $H(XY) \leq 2n$ as that is a simple ...
4
votes
2answers
66 views

How to prove that $ 1- \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1-\frac{x}{n})^n$

How would I prove this inequality (assuming its true, its from a textbook) $$1 - \frac{x^2}{n} \leq (1+\frac{x}{n})^n\cdotp(1+\frac{-x}{n})^n$$ if $n > |x|$, $x\in R$ and $n\in N$ I first ...
3
votes
0answers
45 views

How prove $\max\lbrace \cot{A}(\cot{Y}+\cot{Z}),\cot{B}(\cot{Z}+\cot{X}),\cot{C}(\cot{X}+\cot{Y})\rbrace\ge\frac{2}{3}$

let $\Delta ABC,\Delta XYZ$ are acute triangle show that $$\max\lbrace\cot{A}(\cot{Y}+\cot{Z}),\cot{B}(\cot{Z}+\cot{X}),\cot{C}(\cot{X}+\cot{Y})\rbrace\ge\dfrac{2}{3}$$ My idea: since ...
0
votes
0answers
20 views

Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
1
vote
1answer
49 views

Does the following sequence converge?

Suppose $a_i>0$ for all $i$, $\frac{\sum_{i=1}^n a_i}{n}\to \infty$ and p>1. Let $$y_n = \frac{(\sum_{i=1}^n a_i)^p}{n^{p-1}\sum_{i=1}^n(a_i^p)}.$$ Is $y_n$ monotonic? How can you prove or disprove ...
5
votes
3answers
147 views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that ...
0
votes
2answers
58 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
1
vote
2answers
27 views

Where did I go wrong with this inequality involving absolute value function?

Question: Find all $x \in \mathbb R$ such that the inequality $4<|x+2|+ |x-1|<5$ is satisfied. This is my attempt at solving the problem: Case (i): If $x+2 \geq 0 $ and $ x-1\geq0$, then ...
1
vote
4answers
109 views

Is the minimum of the product of two functions equal to the product of their minima?

I have stuck with following equality, For all $x$, assume function $a(x)$, $b(x)$ have nonzero, and non negative values. (i.e $a(x)>0$, $b(x)>0$, Is the following equality true? ...
13
votes
1answer
273 views
+50

How prove this $x_{1}+x_{2}+\cdots+x_{n}<\frac{5}{3}$

Question: Let $x_{1},x_{2},\cdots,x_{n}\ge 0$ with $$x_{i}x_{j}\le 4^{-|i-j|}$$ for all $i, j = 1, \dots, n.$ Show that $$x_{1}+x_{2}+\cdots+x_{n}<\dfrac{5}{3}.$$ This problem is from ...
10
votes
3answers
57 views

Solve inequality: $-5 < \frac{1}{x} < 0$

Solve inequality: $-5 < \frac{1}{x} < 0$ I thought about how I can solve this. If I multiply all sides by $x$ I'm afraid I'm removing the answer, cause $\frac{x}{x}=1$. And when $x$ 'leaves' ...
5
votes
2answers
104 views

How prove $x^3+y^3+z^3-3xyz\ge C|(x-y)(y-z)(z-x)|$

let $x,y,z\ge 0$,and such $$x^3+y^3+z^3-3xyz\ge C|(x-y)(y-z)(z-x)|$$ Find the maximum of the $C$ witout loss of we assume that $$x+y+z=1$$ I think ...
0
votes
0answers
37 views

A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
0
votes
0answers
39 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
1
vote
1answer
64 views

Is the following Eigenvalue inequality holds or not?

Can anyone help me with the following problem? Suppose $u=(u_1,u_2,...u_n)^T$, $e=(1,1,...1)^T$, and we have $u\geq e$. Now for any symmetric matrix $A\in S^n$ with $diag(A)=0$, can we claim the ...
0
votes
0answers
25 views

How to solve the inequality: $\prod_{k=1}^N\left(x^k-k^2\right)\gt0$

Given the inequality: $$\displaystyle\prod_{k=1}^N\left(x^k-k^2\right)\gt0$$ how can I solve it? I suppose there is a difference if $N=2n$ or $N=2n+1$ with $n\in\mathbb{N}$, but I'm unable to find a ...