Let be a real analytic function with . Let and be a compact subset in . Then the (classical) \L ojasiewicz inequality asserts that:
- There exist such that
\noindent Let be a real analytic function with and . The \L ojasiewicz gradient inequality asserts that:
As a consequence, in (1), the order of zero of an analytic function is finite, and if is close to then is close to the zero set of . However, if is not compact, the latter is not always true and the inequality (1) does not always hold. The inequality (2) is similar to (1), it is not true in the case of is non-compact.
Two inequalities (1) and (2) have some special cases. For example, in the inequality (1), if has only isolated zero, i.e. , this implies . Hence, we have
On the other hand, being different from (2), we have another inequality:
There are some relations between and in complex case and real cases…