For functions p analytic in the open unit disc U = { z : | z | < 1 } with the normalization p ( 0 ) = 1 , we consider the families 𝒫 [ A , − 1 ] , − 1 < A ≤ 1 , consisting of p such that p ( z ) is subordinate to ( 1 + A z ) / ( 1 − z ) in U and 𝒫 ( 1 , b ) , 0$"> b > 0 , consisting of p , which have the disc formulation | p − 1 | < b in U . We then introduce subordination criteria for the choice of p ( z ) = z f ′ ( z ) / f ( z ) , where f is analytic in U and normalized by f ( 0 ) = f ′ ( 0 ) − 1 = 0 . We also obtain starlikeness and convexity conditions for such functions f and consequently extend and, in some cases, improve the corresponding previously known results.