Inequalities, Modulus & Logarithms

Axiomatic Overview

Inequalities form the backbone of analytical bounds and extremal determinations. In competitive mathematics, algebraic inequalities often provide instantaneous shortcuts to optimization problems without using multivariable calculus.

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1. AM $\ge$ GM $\ge$ HM Inequality

📊Visual Mathematical Intuition

Semicircle Geometric Proof of AM ≥ GM ≥ HM

Arithmetic Mean:$AM = \frac{a+b}{2} \text{ (Semicircle Radius)}$

Geometric Mean:$GM = \sqrt{ab} \text{ (Altitude to Hypotenuse)}$

Harmonic Mean:$HM = \frac{2ab}{a+b} \implies AM \ge GM \ge HM$

Classical Greek semicircle proof of $AM \ge GM \ge HM$: On a semicircle of diameter $a+b$, the radius is $AM = \frac{a+b}{2}$ and the vertical altitude is $GM = \sqrt{ab}$. The hypotenuse always exceeds the altitude.

Arithmetic, Geometric & Harmonic Mean Inequality

For any set of $n$ positive real numbers $a_1,a_2,\dots ,a_n>0$:

$$\frac{\sum a_i}{n}\ge \sqrt[n]{\prod a_i}\ge \frac{n}{\sum \frac{1}{a_i}}$$$$\text{AM}\ge \text{GM}\ge \text{HM}$$

Equality holds if and only if $a_1=a_2=\cdots =a_n$.

Weighted AM-GM Inequality

For positive weights $w_1,w_2,\dots ,w_n>0$ with $W=\sum w_i$:

$$\frac{\sum w_i{a}_i}{W}\ge {(\prod a_i^{w_i})}^{1/W}$$

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2. Cauchy-Schwarz Inequality

Cauchy-Bunyakovsky-Schwarz Inequality

For real numbers $(a_1,\dots ,a_n)$ and $(b_1,\dots ,b_n)$:

$${\left(\sum _{i=1}^n{a}_i{b}_i\right)}^2\le \left(\sum _{i=1}^n{a}_i^2\right)\left(\sum _{i=1}^n{b}_i^2\right)$$

Equality holds if and only if vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are linearly dependent ($\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}$).

Bergström / Engel Form (Titu's Lemma)

For real $x_i$ and positive $y_i>0$:

$$\sum _{i=1}^n\frac{x_i^2}{y_i}\ge \frac{(\sum x_i{)}^2}{\sum y_i}$$

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3. Jensen's Inequality for Convex Functions

Jensen's Convexity Inequality

If $f^{″}(x)\ge 0$ (convex function on interval $I$) and $x_1,\dots ,x_n\in I$:

$$f\left(\frac{\sum x_i}{n}\right)\le \frac{\sum f(x_i)}{n}$$

If $f^{″}(x)\le 0$ (concave function), the inequality reverses.

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4. Logarithmic Identities & Inequations

For base $a>0,a\ne 1$:

1. ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$
2. ${\log }_a(x/y)={\log }_{a}x-{\log }_{a}y$
3. ${\log }_{b^k}(x^m)=\frac{m}{k}{\log }_{b}x$
4. Base Change Identity: ${\log }_{a}b=\frac{\ln b}{\ln a}=\frac{1}{{\log }_{b}a}$
5. $a^{{\log }_{b}c}=c^{{\log }_{b}a}$

Logarithmic Inequality Behavior

• If $a>1$: ${\log }_{a}x>{\log }_{a}y⟺x>y>0$ (Monotonically Increasing).
• If $0<a<1$: ${\log }_{a}x>{\log }_{a}y⟺0<x<y$ (Monotonically Decreasing, inequality flips).