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Bounding techniques

Also see functional analysis ref.

Effectively bound one function with another.

General ideas

Use convexity of functions! See proof of AM/ GM inequalities, for example.

Bound f over [a, b] with a constant c

Show that $f$ is monotone over $[a, b]$, perhaps by taking derivative; take $c$ to be value at $a$ or $b$.

Asymptotics

Asymptotic notation: o, O, $\Theta, \Omega, \omega$.

Hierarchy of growth of functions

$$c, \lg^{*} n, \log n, n^{k}, n^{\log n}, \ n^{n^{1/3}} = e^{n^{1/3}\ln n}, e^{n}, n! = \theta(\sqrt{n}(\frac{n}{e})^{n}), n^{n}$$.

Iterative logarithm

$\lg^{} n$: 0 if $n\leq 1, 1+\lg^{}\lg n $ otherwise. Very slow growing.