Sufficient conditions for starlikeness

Abstract

We obtain the conditions on β so that 1+ βzp′(z) ≺ 1+4z/3+ 2z2/3 implies p(z) ≺ (2+z)/(2−z), 1+(1-α)z, (1+(1-2 α)z)/(1-z), (0 ≤ α < 1), exp(z) or √1 + z. Similar results are obtained by considering the expressions 1+βzp′(z)/p(z), 1+βzp′(z)/p2(z) and p(z)+ βzp′(z)/p(z). These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition | log(zf'(z)/f(z))| < 1 or |(zf'(z)/f(z))2 -1| < 1 or zf′(z)/f(z) lying in the region bounded by the cardioid (9x2 + 9y2 − 18x + 5)2 − 16(9x2 + 9y2 − 6x + 1) = 0.