Questions on proving, manipulating and applying inequalities.

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2
votes
2answers
34 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
22 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
votes
4answers
112 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
21 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
93 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
26 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
20 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 ...
10
votes
3answers
2k 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
42 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
48 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
118 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
30 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
votes
1answer
30 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
39 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
80 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
10 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
43 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
60 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
16 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
104 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
48 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
65 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
49 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
29 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
35 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- ...
0
votes
1answer
29 views

Why is the integral of a square always larger than the square of an integral?

I learned in physics that $\langle x^2 \rangle - \langle x \rangle ^2 = \sigma_x^2 \ge 0$ and thus $\langle x^2 \rangle \ge \langle x \rangle ^2$. In the case of continuous distribution, it becomes ...
0
votes
2answers
37 views

Is it true that $|f(x)|\leq |f^2(x)|$?

Is the following true for all $x\in\mathbb{R}$ and for all real functions f? $$\left| f(x)\right| \leq \left| f^2(x)\right|$$ Also, is it true that $|f(x)|\leq |f^3(x)|$?
3
votes
3answers
80 views

Proving an inequality between $\frac 1{n+1}$ and $\frac 1n$ and a definite integral

For all natural numbers $n$, prove that $$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$$ I have tried working with $\frac 1{t+1} < \frac 1t < \frac 1{t-1}$ but this doesn't ...
-3
votes
0answers
39 views

$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$ [on hold]

For all natural numbers $n$, prove that $$\frac 1{n+1} < \int_n^{n+1} \frac 1t \, dt < \frac 1n$$ (Do not use induction.) Please help me on the first step. :)
4
votes
0answers
58 views

Prove $\cos(\sin x)>\sin(\cos x)$ [duplicate]

Prove that $\cos( \sin x)>\sin(\cos x), \forall x\in\mathbb{R}$. I have thought that we should consider their difference and show it is positive for all x, so: Let $$A=\cos\sin x-\sin\cos ...
4
votes
2answers
124 views

A singular Gronwall inequality

Let $f : [0,T] \to R^+$ be a continuous function such that $f(0)=0 $ and : $$ f(t)\le C\int_0^t s^{-1}f(s) ds,\; \forall t\in [0,T] $$ for some constant $C>0.$ Is it true that $f(t)=0,\; \forall ...
0
votes
2answers
41 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
1
vote
2answers
146 views
+100

Finding a function satisfying a certain inequality

This is a continuation of this post where I tried to find a function $f(n)$ that would satisfy the induction step of an inductive argument and it was shown that such function does not exist. Trying ...
3
votes
4answers
83 views

Show $\frac{\sin(x)}{x}>\cos(x)$ for $0<x<\pi$ using the Mean Value Theorem

I'm trying to show the inequality $$\frac{\sin(x)}{x}>\cos(x)$$ by for $0<x<\pi$ using the Mean Value Theorem, but I don't know how to start. I can show that $\sin(x)<x$, but I can't see ...
0
votes
0answers
21 views

helping inequality for cyclic three variable inequality

Let $a\ge b \ge c\ge 0$ be reals and $a+b+c=3$ .Then prove $$c(24a^2b+25)(b^2+ac)+50b(a^2+c^2)+5bc^2\le 200+3b^2c^4$$ this one has a proof replacing $b=3-a-c$ and then using calculus but uggly ...
8
votes
0answers
104 views
+100

cyclic three variable inequality

Let $a,b,c$ be nonnegative real numbers and $a+b+c=3$. Prove the inequality $$ \sqrt{24a^2b+25}+\sqrt{24b^2c+25}+\sqrt{24c^2a+25}\le 21 $$ I have tried to find the solution using classical ...