Hull subordination and extremal problems for starlike and spirallike mappings

Abstract

Let F \mathfrak {F} be a compact subset of the family A \mathcal {A} of functions analytic in Δ = { z : | z | > 1 } \Delta = \{ z:\;|z| > 1\} , and let L \mathcal {L} be a continuous linear operator of order zero on A \mathcal {A} . We show that if the extreme points of the closed convex hull of F \mathcal {F} is the set { f 0 ( x z ) } ( | x | = 1 ) \{ {f_0}(xz)\} (|x| = 1) , then L ( f ) \mathcal {L}(f) is hull subordinate to L ( f 0 ) \mathcal {L}({f_0}) in Δ \Delta . This generalizes results of R. M. Robinson corresponding to families F \mathcal {F} of functions that are subordinate to ( 1 + z ) / ( 1 − z ) (1 + z)/(1 - z) or to 1 / ( 1 − z ) 2 1/{(1 - z)^2} . Families F \mathcal {F} to which this theorem applies are discussed and we identify each such operator L \mathcal {L} with a suitable sequence of complex numbers. Suppose that Φ \Phi is a nonconstant entire function and that 0 > | z 0 | > 1 0 > |{z_0}| > 1 . We show that the maximum of Re ⁡ { Φ [ log ⁡ ( f ( z 0 ) / z 0 ) ] } \operatorname {Re} \{ \Phi [\log (f({z_0})/{z_0})]\} over the class of starlike functions of order a is attained only by the functions f ( z ) = z / ( 1 − x z ) 2 − 2 α , | x | = 1 f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;|x| = 1 . A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.