Big O notation family

This is a family of notations related to the big O notation. A good mnemonic summary of all notations would be:

Big O notation ( $O (n)$ )

Module bound above, possibly multiplied by a constant:

f (x) = O (g (x))

is defined as:

\exists M > 0 \exists x_{0} \forall x > x_{0} : ∣ f (x) ∣ \leq M g (x)

E.g.:

$\forall c \in R x + c = O (x)$ . For $c < 0$ , $M = 1$ is enough. Otherwise, any $M > 1$ will do, the bottom line will always catch up to the top one eventually.

Stronger version of the big O notation, basically means that ratio goes to zero. In big O notation, the ratio does not need to go to zero.

So in informal terms, big O notation means

\leq

<

E.g.:

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