Subordination criteria for starlikeness and convexity

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 + Az)/(1 − z) in U and 𝒫(1, b), 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) = zf′(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.