For functionspanalytic in the open unit discU={z:|z|<1}with the normalizationp(0)=1, we consider the families𝒫[A,−1],−1<A≤1, consisting ofpsuch thatp(z)is subordinate to(1+Az)/(1−z)inUand𝒫(1,b),b>0, consisting ofp, which have the disc formulation|p−1|<binU. We then introduce subordination criteria for the choice ofp(z)=zf′(z)/f(z), wherefis analytic inUand normalized byf(0)=f′(0)−1=0. We also obtain starlikeness and convexity conditions for such functionsfand consequently extend and, in some

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... es, improve the corresponding previously known results.