I'd use static inline, but static would work just as well.
extern and extern inline are out because you'd get multiple external definitions if the header is included in more than one translation unit, so you need to consider static, static inline and inline specification.
Heptic correctly states in his answer that most compilers consider functions for inlining regardless of whether inline is specified or not, ie the main impact of inline is its effect on linkage.
However, static definitions have internal linkage, so there's not much difference between static and static inline; I prefer static inline for function definitions in header files for purely stylistic reasons (rule of thumb: header files should only contain extern declarations, static const variable definitions and static inline function definitions).
inline without static or extern results in an inline definition, which the standard states (C99 6.7.4, §6)
provides an alternative to an external definition, which a translator may use to implement
  any call to the function in the same translation unit. It is unspecified whether a call to the
  function uses the inline definition or the external definition.
ie inline definitions should always be accompanied by external definitions, which is not what you're looking for.
Some more information about the subtleties of C99 inline semantics can be found in this answer, on the Clang homepage and the C99 Rationale (PDF).
Keep in mind that GCC will only use C99 semantics if -std=c99 or -std=gnu99 is present...