  Miller and Mocanu proved in their 1997 paper a greatly useful result which is
now known as the Open Door Lemma. It provides a sufficient condition for an
analytic function on the unit disk to have positive real part. Kuroki and Owa
modified the lemma when the initial point is non-real. In the present note, by
extending their methods, we give a sufficient condition for an analytic
function on the unit disk to take its values in a given sector.
