Questions on proving, manipulating and applying inequalities.

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0
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1answer
14 views

Two ways to define an increasing map: help me find this less strange!

Let $A$ and $B$ be totally ordered sets. I just realized that: The following are equivalent: $x \leq y$ implies $f(x) \leq f(y)$. $f(x) < f(y)$ implies $x < y$. Proof: ...
3
votes
0answers
20 views

Prove that $\sum_{i,j} \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0$

Let $v_1 \dots v_n, w_1 \dots w_n \in H$ an inner product space. I am trying (unsuccesfully) to show that $$ \sum_{i,j=1}^n \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0 .$$ Any hints?
1
vote
0answers
12 views

Solving a system of inequalities in modulo N

I have a problem that boils down to two unknowns, $X_1$ and $X_2$, where: $X_1 \cdot M + A\bmod N = X_2$ And: $X_1 \lt L_1\bmod N$ $X_2 \lt L_2\bmod N$ I can try every possible $X_1 \lt L_1$ ...
2
votes
1answer
18 views

If $X \geq X_t$ why is $\frac{X}{(1+|X|)} \geq \frac{X_t}{(1+|X_t|)}$? So a monotone, 1-1 transformation doesn't affect the inequality?

I am wondering why if $X=\sup_t \{ X_t \} $ for $t \in T$ which is some index set, we have that $\frac{X}{1+|X|} \geq \frac{X_t}{1+|X_t|}$. Clearly, $ \ast X \geq X_t \forall t$. My beginning is to ...
0
votes
1answer
44 views

If $N$ is a $4$ digit number $x_1x_2x_3x_4$, then prove that $\frac{N}{x_1+x_2+x_3+x_4}\le1000$

So $N=1000x_4+100x_3+10x_2+x_4$ $0<x_4\le 9$ $0\le x_3\le 9$ $0\le x_2\le 9$ $0\le x_1\le 9$ $0<{x_1+x_2+x_3+x_4}\le 36$ What should be my approach?
0
votes
1answer
19 views

Figuring out variable pairs in an inequality

Let $x$ and $y$ be positive integers such that $45 < 8x + 5y < 60$ How many $(x,y)$ pairs can be found? (Ans: 16) Of course, there is a way to write it one by one. On the other hand ...
3
votes
1answer
49 views

Prove $x^n < n^n 2^x$

Given that $$x < 2^x$$ is always true, use it to prove that $$x^n < n^n2^x$$ Here are the steps that I've taken so far: Reduce $$x < 2^x$$ to $$\log(x) < x$$ Then $$x^n < n^n2^x$$ ...
1
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0answers
20 views

Expressing a set of discrete inequalities as a continuous differential equation

I'm trying to work out the solution to a problem of sequential inequalities. I believe the solution collapses to a set of differential equations, but I'm having trouble organizing things and I think I ...
0
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1answer
34 views

Show that $|x-y|\leq |x|+|y|$.

Show that $|x-y|\leq |x|+|y|$. I know this probably applies the triangle inequality in some way, but I can't figure it out.
0
votes
2answers
53 views

Proof if $0\leq a,b<1$ then $a+b<1+ab$

I need to prove that is $0\leq a,b<1$ then $a+b<1+ab$. What I did is see this is equivalent to $a+b-ab<1$. We have: $a+b-ab=a+(1-a)b<a+(1-a)1=1$ as desired. Is there a straightforward ...
1
vote
1answer
16 views

Binary Linear Codes of Minimum Distance 3

Let $B_n$ denote the maximum size of a binary linear code (a binary code that is closed under addition) whose codewords have length $n$ and whose minimum distance is $3$. I have been searching for the ...
2
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2answers
36 views

Proving that $\tan^n\angle A + \tan^n\angle B + \tan^n\angle C \ge 3 + \frac{3n}{2}$

Given a acute $\triangle ABC$. Prove that $$\tan^n\angle A + \tan^n\angle B + \tan^n\angle C \ge 3 + \dfrac{3n}{2}$$ I have tried by using a inductive proof. In case $n=0$, the equality holds. ...
0
votes
2answers
29 views

Inequality for the expected value of the sum of Bernoulli random variables

I'm stuck with this seemingly simple inequality. Suppose that $X_1,X_2,\ldots$ are Bernoulli random variables and denote $S_n=X_1+\ldots+X_n$. Let $n_k=\inf\{n:\operatorname ES_n\ge k^2\}$ for ...
0
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4answers
114 views

How to prove that $3^\pi > \pi^3$ [duplicate]

I need to prove this inequality that $3^\pi > \pi^3$ How can i start to answer this problem. What concept should I apply?
0
votes
1answer
26 views

An inequality used in elliptic PDE

$$\sum a^{ij}\xi_i\eta_j\leq\epsilon \sum a^{ij}\eta_i\eta_j+\frac{1}{4\epsilon}\sum a^{ij}\xi_i\xi_j$$ The summations are $1\leq i,j \leq n$, all the variables are positive. Can anybody prove this? ...
2
votes
3answers
114 views

Inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$

I am trying to find a proof of the following inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$ and ...
1
vote
1answer
29 views

Appling Jensen's inequality

I have to prove that for every $a,b,c \in \mathbb{R}$ $$1+\sqrt[3]{e^{2a}}\sqrt[5]{e^b}\sqrt[15]{e^{2c}}\le \sqrt[3]{(1+e^a)^2}\sqrt[5]{(1+e^b)}\sqrt[15]{(1+e^c)^2}.$$ We can prove that ...
0
votes
1answer
29 views

Inequtions problem - how to calculate total of sales for a determined ROI?

A company has determined that the cost of production of X cellphones is according to this formula: $$ C = 150x + x^2 + 25$$ If each cellphone is sell at 220, how many of them must be produced and ...
14
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4answers
3k views

How to find out which number is larger without a calculator?

So I have a question which is: Which is larger? $$2.2^{3.3} \text{ or } 3.3^{2.2} $$ Now I need to find out with using a calculator but the answer is $3.3^{2.2}$. The only thing I could think of ...
1
vote
2answers
43 views

Show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$

Show that in a $\Delta ABC$, $\sin\frac{A}{2}\leq\frac{a}{b+c}$ Hence or otherwise show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ for all $n\geq1$. ...
6
votes
3answers
49 views

Prove that $3x-x^3<\frac2{\sin2x}$

Prove that $$3x-x^3<\frac2{\sin2x},\forall x\in\left(0,\frac\pi2\right)$$ I have tried by proving that $$3x-x^3<\frac9{5\pi}x+\frac32<\frac2{\sin2x},\forall ...
3
votes
4answers
124 views

How to show that $2\times 10^{18}<20!<3 \times 10^{18}$ without calculator? [on hold]

I want to find the first digit of $20!$ By calculator $20! = 2.43290200817664 \times 10^{18}$. So I want to show that $2\times 10^{18}<20!<3 \times 10^{18}$ Thank you.
0
votes
0answers
33 views

Maximum and minimum of $f(x)=x\sqrt{1+\sqrt{1-x^2}}+\sqrt{(1-x^2)(1-x)}$ when $-1\leq x\leq 1$

I am trying to find the maximum and minimum of $f(x)=x\sqrt{1+\sqrt{1-x^2}}+\sqrt{(1-x^2)(1-x)}$ when $-1\leq x\leq 1$. It seems that the minimum is $-1$, but I could not prove it. Anyway, does any ...
-1
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1answer
33 views

How can I find the minimum of these two functions? [on hold]

Here are these two functions: $$P=x^{3}+y^{3}+3(xy-1)(x+y-2)$$, where $$x^{2}+y^{2}-8(x+y)+2xy\leq 0$$ and $$Q=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$, where $$x^{2}+y^{2}+z^{2}=1$$ I've no idea how ...
0
votes
1answer
34 views

An inequality involving the AM-GM inequality: $| x + \frac1x | \ge 2 $ (for $x<0$).

Suppose $x \neq 0 $, then $| x + \frac{1}{x} | \geq 2 $. I have shown this using the am gm inequality $(a+b)/2 \geq \sqrt{ab} $. In fact, with $a = x^2 $ and $b=1$ works. So, for $x > 0 $ we have ...
2
votes
4answers
45 views

Prove $\log(x) < n(x)^{1/n}$, for all positive integer values of $n$, and $x > 0$

Given that $$lg(u) < u$$ is always true, how do we use that to prove that $$lg(x) < n(x)^\frac 1n$$ These are the steps that I have taken so far: $$1: lg(x) < n(x)^\frac 1n$$ $$2: \frac ...
0
votes
1answer
20 views

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$.

Let $A\subset R$ and $sup\{f(x)^2:x\in A\}=M(f^2)$, $inf\{f(x)^2:x\in A\}=m(f^2)$. Show that $M(f^2)=(M(|f|))^2$ and $m(f^2)=(m(|f|))^2$. I'm having difficulty showing the above equalities. I ...
5
votes
2answers
81 views

Given $a,b,c\ge1;abc\ge8$. Proving that $\sqrt{a^2-1}+\sqrt{b^2-1}+\sqrt{c^2-1}\ge 3\sqrt3$

Given $a,b,c\ge1;abc\ge8$. Proving that $$\sqrt{a^2-1}+\sqrt{b^2-1}+\sqrt{c^2-1}\ge 3\sqrt3$$ I have tried by using Jensen's inequality: We consider the inequality: ...
0
votes
0answers
12 views

An elementary inequality in the context of Strictly Convex Function

Suppose: whenever $\epsilon \gt 0$ , define: $\zeta (y, \epsilon) = \frac{\eta (y+\epsilon) - \eta (y)}{\epsilon} - \eta'_{+}(y)$ ; where: $\eta$ is STRICTLY CONVEX CONTINUOUS FUNCTION & ...
2
votes
3answers
44 views

Prove that $\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$

Prove that $$\tan x < \frac{4}{\pi}x,\forall x\in \left( 0;\frac{\pi}{4} \right)$$ I have known the solution that uses convex function. But I'd like another solution don't use it. :D
4
votes
2answers
61 views

Why $\tan x>\sin x$ in this question?

The question asks me to prove the identity $\tan ^2x-\sin ^2x=\tan^2 x \sin^2 x$ and use this result to explain why $\tan x>\sin x$ for $0<x<90$ I've proved the identity and I can't explain ...
0
votes
1answer
58 views

Conditions on $c$ such that the inequality dont hold.

I want to find conditions on $c$ such that the inequality don't hold. $$1-ac(a-2)(a-1)^2 < 0 \ \ \ \ \ \ \text{for } a>2, c>0$$ If $\phi(a) = ac(a-2)(a-1)^2 \Rightarrow \phi'(a) = ...
2
votes
2answers
42 views

Error in proving inequality $1 - x \leq e^{-x}$

Fact states as following, $$1 - x \leq e^{-x}$$ This is how I try to prove it: \begin{align*} \ln (1 - x) &\leq \ln (e^{-x})\\ \ln 1/ \ln x &\leq -x\\ \ln 1 &\leq -x \times \ln x ...
0
votes
0answers
20 views

Rank of product of two rectangular matrices

Given $A_{m \times n}$ matrix with rank $m$, and $B_{n \times p}$ matrix with rank $p$, where $n > p \geq m$. We know that $$ \operatorname{rank}(AB) \leq ...
2
votes
4answers
108 views

Geometric proof of $QM \ge AM$

Prove by geometric reasoning that: $$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$ The proof should be different than one well known from Wikipedia: DISCLAIMER: I think I devised such proof ...
1
vote
1answer
27 views

$f(x) = x^{p}(1-x)^{q}$ for all $x\in \left[0,1\right]\;,$ Where $p,q\in \mathbb{Z^{+}}$, Then Max. of $f(x)$ at $x=$

The function $f(x) = x^{p}(1-x)^{q}$ for all $x\in \left[0,1\right]\;,$ Where $p,q$ are positive integers, has maximum value for $x=$ $\bf{Using\; Derivative}$ Let $$f(x) = ...
-1
votes
1answer
50 views

The choice of scalar factors in the proof of the Schwarz inequality

In this proof for the Schwarz Inequality, they seemingly arbitrarily choose $r = w\cdot w$ and $s =-(v\cdot w)$. Why did they make these selections? I don't understand where these values for $r$ and ...
1
vote
4answers
66 views

Prove that $1^2 + 2^2 + … + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + … + n^2$

I'm having trouble on starting this induction problem. The question simply reads : prove the following using induction: $$1^{2} + 2^{2} + ...... + (n-1)^{2} < \frac{n^3}{3} < 1^{2} + 2^{2} + ...
2
votes
2answers
52 views

If $a,b,c,d,e,f$ are non negative real numbers such that $a+b+c+d+e+f=1$, then find maximum value of $ab+bc+cd+de+ef$

$(a+b+c+d+e+f)^2=$ sum of square of each number (X)+ $2($ sum of product of two numbers (Y) $)$ $ab+bc+cd+de+ef \le Y$ since all are positive. Therefore $1\ge X+(ab+bc+cd+de+ef)$ Edit: From AM GM ...
0
votes
2answers
51 views

generalized Cauchy-Schwarz inequality

How to prove $A'B(B'B)^{-1}B'A \leq A'A$, where $A$,$B$ are $n\times k$ matrices and $B'B$ is assumed to be positive definite? I don't see why it is a Cauchy-Schwarz inequality.
0
votes
0answers
22 views

Upper bound for incomlete Gamma function

It is well-known, that for real arguments $a \geq 0$ and $x \geq 0$ the upper incomplete Gamma function $$\Gamma(a,x) = \int_x^\infty e^{-t} t^{a-1} \, \mathrm{d} t$$ behaves for sufficiently large ...
0
votes
0answers
25 views

Is there any smart way to check triangle inequality for a matrix?

Here is the description of the problem: We have a matrix with: all (i,i) cells are 0; Some cells are filled with certain number while others are left blank. Now, we want to fill the blanks with ...
0
votes
1answer
31 views

What exactly does this inequality do?

I this paper which is titled "KSVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation", in the section about "kmeans algorithm for vector quantization", there is the ...
0
votes
1answer
13 views

An inequality for power of positive functions

Let $f,g,h$ be positive real vlaued functions on a finite set $\mathbb{X}$. Let $p >1$. I am wondering whether the following should be true? $$\sum_{x\in ...
0
votes
0answers
31 views

What is (if there is) the generic term for equalities and inequalities

I'm writing a text about a particular linear programming (LP)I optimization problem, that is described using a mixture of inequalities (, ...
3
votes
1answer
76 views

Difficulty to prove this inequality in Binomial Coefficient.

This inequality is found in a book titled as Randomized Algorithms, by Rajeev Motwani and Prabhakar Raghavan, in Chapter 3, during explaining Occupancy Problems, to see the book click here PP. 43-44 ...
0
votes
0answers
37 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
1
vote
3answers
36 views

Prove that from the equalities, $\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$ follows $x^yy^x=y^zz^y=z^xx^z$.

Problem : Prove that from the equalities, $$\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$$ follows $$x^yy^x=y^zz^y=z^xx^z$$. My approach : $$\frac{x(y+z-x)}{\log ...
3
votes
0answers
36 views

Matrix product bound

Consider the following inequality \begin{align*} AB^{-1}A^\top \preceq cI \end{align*} where $A\in\mathbb{R}^{n\times m}$, $B\in\mathbb{R}^{m\times m}$, $c\in\mathbb{R}$ (given), and $I$ is the ...
6
votes
4answers
101 views

Showing that $\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x$

Show that $$\left (\frac{\sin x}{x} \right )^3\geq \cos^{2}x,\forall x\in \left ( 0;\frac\pi2 \right )$$ Firstly, I had use the differentiation of $f(x)=\left (\frac{\sin x}{x} \right )^3- ...