Algebra theory weighted means examples

Ex1: Prove that

$$\left( \frac{a + b}{2} \right)^{a + b} \geq a^b \cdot b^a$$

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$$\frac{a + b}{2} > (a^b b^a)^{\frac{1}{a + b}}$$

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WAM ≥ WGM

$$\text{WAM} \geq \text{WGM} \quad \left\{ \text{a}^b, \text{b}^a \right\}$$

$a, a, a, \ldots$ — $b$ times
$b, b, b, \ldots$ — $a$ times

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$$\text{WAM} = \frac{a_1 w_1 + a_2 w_2 + \ldots + a_n w_n}{w_1 + w_2 + \ldots + w_n} = \frac{a b + b a}{a + b}$$

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$$\text{WGM} = (a_1^{w_1} a_2^{w_2} \ldots a_n^{w_n})^{\frac{1}{w_1 + w_2 + \ldots + w_n}} = (a^b b^a)^{\frac{1}{a + b}}$$

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$$\text{WAM} \geq \text{WGM}$$ $$\frac{a b + b a}{a + b} \geq (a^b b^a)^{\frac{1}{a + b}}$$ $$\frac{2 a b}{a + b} \geq (a^b b^a)^{\frac{1}{a + b}} \tag{1}$$

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AM ≥ HM { a, b }

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

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$$\frac{a + b}{2} \geq \frac{2 a b}{a + b} \Rightarrow \left( a^b b^a \right)^{\frac{1}{a + b}}$$

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$$\frac{a + b}{2} \geq (a^b b^a)^{\frac{1}{a + b}}$$ $$\left( \frac{a + b}{2} \right)^{a + b} \geq a^b b^a$$

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Ex2: Prove that

$$\left( \frac{1 + n a}{1 + n} \right)^{n + 1} > a^n$$

WAM ≥ WGM

$$\begin{cases} a_1 = 1,\ \omega_1 = 1 \\ a_2 = a,\ \omega_2 = n \end{cases}$$ $$\frac{a_1 \omega_1 + a_2 \omega_2}{\omega_1 + \omega_2} \geq \left( a_1^{\omega_1} a_2^{\omega_2} \right)^{\frac{1}{\omega_1 + \omega_2}}$$ $$\frac{1(1) + a n}{1 + n} \geq \left( 1^1 \cdot a^n \right)^{\frac{1}{n + 1}}$$ $$\frac{1 + a n}{1 + n} \geq a^{\frac{n}{n + 1}}$$ $$\left( \frac{1 + a n}{1 + n} \right)^{n + 1} \geq a^n$$

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Ex3:

 Prove that

$$\frac{a^3 b}{(a + b)^4} \leq \frac{27}{256}$$

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$$\frac{a^3 b}{(a + b)^4} \leq \frac{27}{256}$$ $$\frac{(a + b)^4}{256} \geq \frac{a^3 b}{27}$$ $$\frac{(a + b)^4}{2^8} \geq \left( \frac{a}{3} \right)^3 b$$ $$\left( \frac{a + b}{4} \right)^4 \geq \left( \frac{a}{3} \right)^3 b$$ $$\frac{a + b}{4} \geq \left( \left( \frac{a}{3} \right)^3 b \right)^{\frac{1}{4}}$$ $$\frac{3 \left( \frac{a}{3} \right) + b}{4} \geq \left( \left( \frac{a}{3} \right)^3 b \right)^{\frac{1}{4}} \quad \checkmark$$

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WAM ≥ WGM on

$$\begin{cases} a_1 = \frac{a}{3},\ \omega_1 = 3 \\ a_2 = b,\ \omega_2 = 1 \end{cases}$$ $$\frac{a_1 \omega_1 + a_2 \omega_2}{\omega_1 + \omega_2} \geq \left( a_1^{\omega_1} \cdot a_2^{\omega_2} \right)^{\frac{1}{\omega_1 + \omega_2}}$$ $$\frac{\left( \frac{a}{3} \right)(3) + b(1)}{3 + 1} \geq \left( \left( \frac{a}{3} \right)^3 \cdot b \right)^{\frac{1}{3 + 1}}$$ $$\frac{a + b}{4} \geq \left( \frac{a^3 b}{27} \right)^{\frac{1}{4}}$$

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Ex4:

Find min of

$$\frac{9}{p} + \frac{16}{q} \quad \text{if} \quad p + q = 7$$

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$$\frac{p}{3} + \frac{p}{3} + \frac{p}{3} + \frac{q}{4} + \frac{q}{4} + \frac{q}{4} + \frac{q}{4} = 7$$ $$\left( \frac{p}{3} \right)3 + \left( \frac{q}{4} \right)4 = 7$$ $$\begin{cases} a_1 = \frac{p}{3}, \quad \omega_1 = 3 \\ a_2 = \frac{q}{4}, \quad \omega_2 = 4 \end{cases}$$

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WAM ≥ WHM

$$\frac{a_1 \omega_1 + a_2 \omega_2}{\omega_1 + \omega_2} \geq \frac{\omega_1 + \omega_2}{\frac{\omega_1}{a_1} + \frac{\omega_2}{a_2}}$$

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$$\frac{\left( \frac{p}{3} \right)3 + \left( \frac{q}{4} \right)4}{3 + 4} \geq \frac{3 + 4}{\frac{3}{p/3} + \frac{4}{q/4}}$$ $$\frac{p + q}{7} \geq \frac{7}{\frac{9}{p} + \frac{16}{q}}$$ $$1 \geq \frac{7}{\frac{9}{p} + \frac{16}{q}}$$ $$\frac{9}{p} + \frac{16}{q} \geq 7$$

Answer = 7