On the coefficients of Bazilevič functions

Source abstract

Let B ( α ) B(\alpha ) denote the class of normalized ( f ( 0 ) = 0 , f ′ ( 0 ) = 1 ) (f(0) = 0,f’(0) = 1) , Bazilevič functions of type α \alpha defined in Δ : | z | > 1 , i.e. f ( z ) = [ α ∫ 0 z P ( ζ ) g ( ζ ) α ζ − 1 d ζ ] 1 / α \Delta :|z| > 1,{\text {i.e.}}f(z) = {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }} where g ( ζ ) g(\zeta ) is starlike in Δ , P ( ζ ) \Delta ,P(\zeta ) is regular with ℜ P ( ζ ) > 0 \Re \,P(\zeta ) > 0 in Δ \Delta and α > 0 \alpha > 0 . Let B m ( α ) {B_m}(\alpha ) denote the subclass of B ( α ) B(\alpha ) which is m -fold symmetric ( f ( e 2 π i / m z ) = e 2 π i / m f ( z ) , m = 1 , 2 , ⋯ ) (f({e^{2\pi i/m}}z) = {e^{2\pi i/m}}f(z),m = 1,2, \cdots ) . Functions in B ( α ) B(\alpha ) have been shown to be univalent. The authors obtain sharp coefficient inequalities for functions in B m ( 1 / N ) {B_m}(1/N) where N is a positive integer. In addition an example of a Bazilevič function which is not close-to-convex is given.