Bounded starlike functions of complex order

Abstract

LetF(b, M) (b ≠ 0 complex,M>1/2) denote the class of functionsf(z) =z + Σ ∞ a n z n analytic in U={z:|z|<1} which satisfy for fixedM, f(z)/z ≠ 0 inU and $$\left| {\frac{{b - 1 + \left[ {zf'{{\left( z \right)} \mathord{\left/ {\vphantom {{\left( z \right)} {f\left( z \right)}}} \right. \kern-\nulldelimiterspace} {f\left( z \right)}}} \right]}}{b} - M} \right|< M, z \in U$$ . In this note we obtain various representations for functions inF(b, M). We maximize |a3=μa 2 2 | over the classF(b, M). Also sharp coefficient bounds are established for functions inF(b, M). We also obtain the sharp radius of starlikeness of the classF(b, M).