Sufficient Conditions For Janowski Starlike Functions With Fixed Second Coefficient

Abstract

Let A,B,D,E belong to [-1, 1] and let p(z) be an analytic function with fixed initial coefficient defined in the open unit disk. Conditions on A,B,D,and E are determined so that 1+{\alpha}zp'(z) being subordinated to (1+Dz)/(1+Ez) implies that p(z) is subordinated to (1+Az)/(1+Bz) and other similar implications involving 1+{\alpha}zp'(z)/p(z), {\alpha}p2(z)+{\lambda}zp'(z),{\alpha}p(z)+(1-{\alpha})p2(z)+{\lambda}zp'(z),and (1-{\alpha})p(z)+{alpha}(1+zp'(z)/p(z)). Also, sufficient conditions for Janowski starlikeness with fixed second coefficient are obtained.