New Pólya–Schoenberg type theorems

In this paper the theory of Hadamard product multipliers is extended from the unit disk in the complex plane to arbitrary so-called disk-like domains, i.e. such domains which are the union of disks or half-planes, all containing the origin. In such a domain, say Ω  , we define (the class Rαd(Ω) of) generalized prestarlike functions of order α⩽1α⩽1 and ask for Hadamard multipliers g   analytic at z=0z=0 for which f∈Rαd(Ω) implies g∗f∈Rαd(Ω). We prove that such a multiplier necessarily has to be analytic inΩ∗:={uv:u∈Ω,v∈C∖Ω}. In many cases (we prove this for all proper disks containing the origin) we actually find that Rαd(Ω∗) is the precise description of the set of all such multipliers. For these disks, ΩγΩγ say, the domains Ωγ∗ turn out to be bounded by the outer loops of certain Limaçons of Pascal. The parameter γ   is related to the characteristic q(Ωγ)=(1−γ)/(1+γ):=r/sq(Ωγ)=(1−γ)/(1+γ):=r/s of the disk, where r is the shortest distance of the origin to the boundary of that disk, and s   the largest. Large subclasses of Rαd(Ω∗) are being explicitly determined. For the case γ=0γ=0, i.e. Ωγ=Ωγ∗=D, this result coincides with an old one by Ruscheweyh and Sheil-Small, previously conjectured by G. Pólya and I.J. Schoenberg. The notion of the characteristic of a disk (containing the origin) is then extended to general disk-like domains, and some multipliers are identified for those general classes Rαd(Ω). The previously determined class of ‘universally prestarlike functions’, defined in the slit-domain C∖[1,∞]C∖[1,∞], is identified as the class of ‘universal multipliers’ for Rαd(Ω) in any disk-like domain Ω.