Nunokawa, Owa, and others have proved a number of theorems involving criteria for functions to be multivalently starlike. In this paper, the author obtains some improvements on these earlier results. For example, a corollary of the main theorem of the paper shows that if
is analytic in the unit disk with
there and satisfies Re
(and this result is sharp). The main theorem of the paper has a number of other corollaries and is of some independent interest: Let
be given. Suppose
is analytic and non-zero in the unit disk. If Re
, then Re
in the disk. There are several other similar results.