SUFFICIENT CONDITIONS FOR STARLIKENESS

We obtain the conditions onso that 1+�zp ' (z) ≺ 1+4z/3+ 2z 2 /3 implies p(z) ≺ (2+z)/(2−z), 1+(1−�)z, (1+(1−2�)z)/(1−z), (0 ≤ � < 1), exp(z) or √ 1 + z. Similar results are obtained by considering the expressions 1+�zp ' (z)/p(z), 1+�zp ' (z)/p 2 (z) and p(z)+�zp ' (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 (9x 2 + 9y 2 − 18x + 5) 2 − 16(9x 2 + 9y 2 − 6x + 1) = 0.