Open-Door Lemma for Functions with Fixed Second Coefficients.

We extend the well-known open-door lemma by using the theory of differential subordination for functions with fixed initial coefficient and apply it to study some integral operators. Further, using an extension of Nunokawa lemma, we determine some sufficient conditions for the radii and order of starlike functions with fixed second coefficient. Our results improve and generalize some previously known results. [ABSTRACT FROM AUTHOR]