Abstract. We obtain the conditions on β so that 1+βzpThese 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.
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