Algebra theory rearrangement inequality

General form or Rearrangement inequality:
if x1 >= x2 >= ... xn
also, y1 >= y2 > = .. yn
then
x1.y1 + x2.y2 + ... xn.yn >= any other xy pairing(for e.g. x1y2 + x2y1 + ... xn.yn)

 Rearrangement Inequality

Given:

$$a \leq b \leq c$$ $$\begin{bmatrix} a & b & c \\ a & b & c \end{bmatrix} \geq \begin{bmatrix} a & b & c \\ c & b & a \end{bmatrix}$$ $$a^2 + b^2 + c^2 \geq ac + b^2 + ac$$ $$a^2 + b^2 + c^2 \geq b^2 + 2ac$$

---

Example with Boxes and Values
(₹10) B₁ — 2
(₹50) B₂ — 3
(₹100) B₃ — 5

$$\begin{array}{ccc} 2 & 3 & 5 \Rightarrow \text{Max} \\ 5 & 3 & 2 \Rightarrow \text{Min} \end{array}$$

---

Conclusion:
If $a \leq b \leq c$ are 3 positive numbers,
then $a^2 + b^2 + c^2$ will always be greater
than any other rearrangement of $a, b, c$.

---

(1) Prove that

$$a^2 + b^2 + c^2 \geq ab + bc + ca$$

Given:

$$a \leq b \leq c \quad \text{(any other arrangement)}$$ $$\begin{bmatrix} a & b & c \\ a & b & c \end{bmatrix} \geq \begin{bmatrix} a & b & c \\ b & c & a \end{bmatrix}$$ $$a^2 + b^2 + c^2 \geq ab + bc + ca$$

---

(2) Prove that $a, b, c \geq 0$

$$a^3 + b^3 + c^3 \geq a^2 b + b^2 c + c^2 a$$

Given:

$$a \leq b \leq c \Rightarrow a^2 \leq b^2 \leq c^2$$ $$\begin{bmatrix} a & b & c \\ a^2 & b^2 & c^2 \end{bmatrix} \geq \begin{bmatrix} a & b & c \\ c^2 & a^2 & b^2 \end{bmatrix}$$ $$a^3 + b^3 + c^3 \geq a c^2 + b a^2 + c b^2$$

---

(3)

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

---

Let $a \leq b \leq c$

Then:

$$\frac{1}{b + c} \leq \frac{1}{a + c} \leq \frac{1}{a + b}$$

---

$$\begin{bmatrix} a & b & c \end{bmatrix} \geq \begin{bmatrix} a & b & c \end{bmatrix} \quad \text{paired with} \quad \begin{bmatrix} \frac{1}{b + c} & \frac{1}{a + c} & \frac{1}{a + b} \end{bmatrix}$$

Using Rearrangement Inequality:

$$\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} \geq \frac{a}{a + c} + \frac{b}{a + b} + \frac{c}{b + c} \tag{1}$$ $$\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} \geq \frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{a + c} \tag{2}$$

Adding (1) and (2):

$$2 \left( \frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} \right) \geq 3$$ $$\frac{a}{b + c} + \frac{b}{c + a} + \frac{c}{a + b} \geq \frac{3}{2}$$

(4):

Prove that

$$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b + c$$

given that $abc = 1$

$$$$Solution:
Assume
a/b <= b/c <= c/a

Then:

$$\left( \frac{a}{b} \right)^{1/3} \leq \left( \frac{b}{c} \right)^{1/3} \leq \left( \frac{c}{a} \right)^{1/3} \quad \left( \frac{a}{b} \right)^{2/3} \leq \left( \frac{b}{c} \right)^{2/3} \leq \left( \frac{c}{a} \right)^{2/3}$$ $$\left[ \begin{array}{ccc} \left( \frac{a}{b} \right)^{1/3} & \left( \frac{b}{c} \right)^{1/3} & \left( \frac{c}{a} \right)^{1/3} \\ \left( \frac{a}{b} \right)^{2/3} & \left( \frac{b}{c} \right)^{2/3} & \left( \frac{c}{a} \right)^{2/3} \end{array} \right] \geq \left[ \begin{array}{ccc} \left( \frac{a}{b} \right)^{1/3} & \left( \frac{b}{c} \right)^{1/3} & \left( \frac{c}{a} \right)^{1/3} \\ \left( \frac{c}{a} \right)^{1/3} & \left( \frac{a}{b} \right)^{1/3} & \left( \frac{b}{c} \right)^{1/3} \end{array} \right]$$ $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq \left( \frac{a}{b} \right)^{1/3} \left( \frac{c}{a} \right)^{2/3} + \left( \frac{b}{c} \cdot \frac{a^2}{b^2} \right)^{1/3} + \left( \frac{c}{a} \cdot \frac{b^2}{c^2} \right)^{1/3}$$ $$\geq \left( \frac{a c^2}{b a^2} \right)^{1/3} + \left( \frac{b a^2}{c b^2} \right)^{1/3} + \left( \frac{c b^2}{a c^2} \right)^{1/3}$$ $$\geq \left( \frac{c^2}{ab} \right)^{1/3} + \left( \frac{a^2}{bc} \right)^{1/3} + \left( \frac{b^2}{ac} \right)^{1/3}$$ $$\geq \left( c^3 \right)^{1/3} + \left( a^3 \right)^{1/3} + \left( b^3 \right)^{1/3}$$ $$\geq c + a + b$$

---

(5) Find min of

$$\frac{\sin^3(x)}{\cos(x)} + \frac{\cos^3(x)}{\sin(x)}, \quad x \in \left(0, \frac{\pi}{2} \right)$$

Using identity:

$$\sin^2 \theta + \cos^2 \theta = 1$$ $$\sin x \leq \cos x \Rightarrow \sin^3 x \leq \cos^3 x \quad \text{and} \quad \frac{1}{\sin x} \geq \frac{1}{\cos x}$$ $$\left[ \begin{array}{cc} \frac{\sin^3 x}{\cos x} & \frac{\cos^3 x}{\sin x} \end{array} \right] \geq \left[ \begin{array}{cc} \frac{\sin^3 x}{\sin x} & \frac{\cos^3 x}{\cos x} \end{array} \right]$$ $$\frac{\sin^3 x}{\cos x} + \frac{\cos^3 x}{\sin x} \geq \sin^2 x + \cos^2 x$$ $$\frac{{\sin }^{3}x}{\cos x}+\frac{{\cos }^{3}x}{\sin x}\ge 1$$

Answer = 1