Questions on proving and manipulating inequalities.

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3
votes
1answer
76 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 ...
1
vote
0answers
54 views

AM-GM Inequality Practice?

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 ...
11
votes
1answer
238 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 ...
2
votes
5answers
151 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 ...
-2
votes
0answers
10 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 $\|A\|_{1} \leq \upsilon(A) \leq n \|A\|_1$.
1
vote
1answer
28 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:$$\sum ^n_{i=1} \frac{x_i}{\sqrt{1-x_i}}\geq \frac{1}{\sqrt{n-1}}\sum ^n_{i=1} x_i$$ ...
2
votes
2answers
35 views

Calculate the inequality

$a,b,c$ are the sides of a triangle.Then show that $$(a+b+c)^3 > 27(a+b-c)(b+c-a)(c+a-b)$$ Also give the case where equality holds i.e. $$(a+b+c)^3=27(a+b-c)(b+c-a)(c+a-b)$$ I tried triangle ...
0
votes
3answers
396 views

Solution to cubic inequality

How do I find the solutions to this equation? $$(3x^3)-(14x^2)-(5x) \leq 0$$
0
votes
4answers
41 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.
5
votes
2answers
107 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 ...
1
vote
1answer
42 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 ...
0
votes
0answers
18 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 ...
3
votes
3answers
105 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
162 views

Proving that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$ for all positive $n$

Hello I am having difficulty proving the following inequality: $$ \sqrt[n]{n} < 1 + \sqrt{\frac{2}{n}} \quad \text{for all positive integers}\,\,\, n. $$ I am trying to use mathematical induction ...
0
votes
0answers
18 views

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
65 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
3answers
123 views

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac n2 \rceil$ or $ \lfloor \frac n2\rfloor $?

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac{n}{2} \rceil $ or $ \lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...
1
vote
3answers
36 views

Greatest value of the binomial coefficient. [duplicate]

How should I prove the greatest value of the binomial coefficient $C(n,r)$ occurs for $r=\left[\cfrac{(n+1)}{2}\right]$ ?
0
votes
0answers
28 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 ...
3
votes
0answers
44 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 ...
1
vote
1answer
36 views

Is the following sequence monotonic?

Suppose $a_i>0$ for all $i$ 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 this? Otherwise, can it be shown that ...
0
votes
0answers
13 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 ...
0
votes
2answers
56 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
4answers
107 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? ...
-2
votes
1answer
37 views

Lower bound for $\ln x$ using Lagrange's mean value theorem or Rolle's theorem

I have to prove the inequality $$ \ln x>\frac{2(x-1)}{x+1} $$ for $x>1$ by using either Lagrange or Rolle. Can you give me a little hint?
8
votes
3answers
870 views

Proofs of the Cauchy-Schwarz Inequality?

How many proofs of the Cauchy-Schwarz inequality are there? Is there some kind of reference that lists all of these proofs?
1
vote
0answers
84 views
+50

How prove $\sum_{cyc}\sqrt{PA+PB}\ge 2\sqrt{\sum_{cyc}h_{a}}$

Question: let $\Delta ABC$,and the altitude is $h_{a},h_{b},h_{c}$,where $AB=c,BC=a,AC=b$ and for any $P$ show that $$\sqrt{PA+PB}+\sqrt{PB+PC}+\sqrt{PA+PC}\ge 2\sqrt{h_{a}+h_{b}+h_{c}}$$ ...
0
votes
2answers
54 views

Proof of Schwarz inequality

Schwarz inequality can be proved by using vector addition property and pythagoras theorem. Is there any other way of proving it?
4
votes
2answers
101 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 ...
8
votes
2answers
47 views

How to prove an inequality

$a$, $b$, $c$, $d$ are rational numbers and all $> 0$. $\max \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\} \geq \dfrac{a+c}{b+d}\geq \min \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\}$ Hope someone ...
1
vote
2answers
26 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 ...
3
votes
1answer
274 views

Use induction to prove that $ 1 + \frac {1}{\sqrt{2}} + \frac {1}{\sqrt{3}} … + \frac {1}{\sqrt{n}} < 2\sqrt{n}$

Use induction to prove that $ 1 + \frac {1}{\sqrt{2}} + \frac {1}{\sqrt{3}} ... + \frac {1}{\sqrt{n}} < 2\sqrt{n} $ My attempt was as follows: Lets assume the inequality is true for n = k $S_k ...
4
votes
1answer
74 views

how to solve this elementary induction proof: $\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\le\ 2-\frac{1}{n}$

this is a seemingly simple induction question that has me confused about perhaps my understanding of how to apply induction the question; $$\frac{1}{1^2}+ \cdots+\frac{1}{n^2}\ \le\ 2-\frac{1}{n},\ ...
1
vote
1answer
63 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 ...
10
votes
3answers
54 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' ...
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 $ ...
2
votes
1answer
304 views

Proof of Frechet-Hoeffding Copula bounds

How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from $C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0$ by setting $(v_1,v_2)=(1,1)$. I'm struggling to ...
1
vote
0answers
51 views

Almost Jensen's Inequality

Let $a,b$ and $c$ three positive reals numbers such that $abc=1$. Define the function $f$ by $f(x)=\frac{^1}{1+(n-1)x^n}$ where $n$ is a positive integer. Prove that ...
16
votes
4answers
284 views

An inequality on sequences with each term dividing sum of two neighbouring terms

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is $$\frac{x_{i-1}+x_{i+1}}{x_i} = k_i $$ is an integer for ...
2
votes
1answer
77 views

An inequality involving vectors

Let $n$ be a positive integer number. If $S$ is a finite set of vectors in the plane, let $N(S)$ denote the number of two-element subsets $\{\mathbf{v}, \mathbf{v'}\}$ of $S$ such that ...
1
vote
1answer
107 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
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 ...
1
vote
1answer
37 views

Rank of a Matrix Sum

I have matrices of $3\times3$ dimension such that S=A+B. I know there is one inequality connecting rank of the matrices A,B and its sum S? Could you write down that here. It will be a great help for ...
2
votes
2answers
37 views

minimum value of $y= \frac {x^n+a}{x^m}$

Question if $n>m$, $\frac {a}{x^m} > 0$ and $x^{n-m} > 0$,prove $y= \frac {x^n+a}{x^m}$ is minimum when $x= \sqrt[n]{\frac {am}{n-m}}$ and value of minimum is equal to $y= ...
3
votes
2answers
57 views

prove $\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$

Show that if $a,b,c,d \geq 0$ and $ab+bc+cd+da=1$ :$$\sum\limits_{cyc} \frac {a^3} {b+c+d} \geq \frac {1} {3}$$ yet again it should be solved with Cauchy inequality. thing i have done so far: ...
0
votes
1answer
63 views

Prove that: $\sum \frac{a^2+2bc}{(b+c)^2}\geq \sum \frac{3}{2}\frac{a}{b+c}$

Let $a, b, c > 0$.Prove that: $\sum \frac{a^2+2bc}{(b+c)^2}\geq \sum \frac{3}{2}\frac{a}{b+c}$ p/s: I tried to solve the problem by $S.O.S$. But I cannot solve it !! I have: The inequatily ...
0
votes
3answers
36 views

How do I prove $x^n < x^m$ when $m > n$ and $x > 1$

Title I made an attempt at it here: $x^n < x^m$ when $m > n$ and $x > 1$, $m$ and $n$ are naturals so divide both sides by $x^n$ so $1 < x^{m-n}$ but here i am stuck. Please help!
-3
votes
1answer
63 views

Prove that: $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac{1}{3}$ [on hold]

Let $a,b,c$ be positive real numbers satisfying $a+b+c+d=4$. Prove that: $\frac{1}{11+a^2}+\frac{1}{11+b^2}+\frac{1}{11+c^2}+\frac{1}{11+d^2} \leq \frac{1}{3}$ p/s: I have no idea about the problem ...
2
votes
0answers
51 views

How prove this inequality $\frac{3(x^2+y^2+z^2)}{(x+y+z)^2+2(yz+xz+xy)}\ge\sum_{cyc}\frac{x^2}{x^2+(y+z)^2}$

Question: let $x,y,z\ge 0$.prove or disprove $$\dfrac{3(x^2+y^2+z^2)}{(x+y+z)^2+2(yz+xz+xy)}\ge\sum_{cyc}\dfrac{x^2}{x^2+(y+z)^2}$$ My idea: let $x+y+z=1$, then we can only ...
0
votes
2answers
32 views

Solving inequalities with modulus in addition

How would you solve an inequality with modulus in addition? Question is: $$|2x-1| + |x-3| \geq 10$$ How to start here? What I tried: Well you can obviously solve the equation for each possibility ...