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Let A,B,D,E∈[−1,1] and let p(z) be an analytic function defined on the open unit disk, p(0)=1. Conditions on A, B, D, and E are determined so that 1+βzp'(z) being subordinated to (1+Dz)/(1+Ez) implies that p(z) is subordinated to (1+Az)/(1+Bz). Similar results are obtained by considering the expressions 1+β(zp'(z)/p(z)) and 1+β(zp'(z)/p2(z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike.