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 R α d ( Ω ) of) generalized prestarlike functions of order α ⩽ 1 and ask for Hadamard multipliers g analytic at z = 0 for which f ∈ R α d ( Ω ) implies g ∗ f ∈ R α d ( Ω ) . We prove that such a multiplier necessarily has to be analytic in Ω ∗ : = { u v : u ∈ Ω , v ∈ C ∖ Ω } . In many cases (we prove this for all proper disks containing the origin) we actually find that R α d ( Ω ∗ ) is the precise description of the set of all such multipliers. For these disks, Ω γ say, the domains Ω γ ∗ turn out to be bounded by the outer loops of certain Limaçons of Pascal. The parameter γ is related to the characteristic q ( Ω γ ) = ( 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 R α d ( Ω ∗ ) are being explicitly determined. For the case γ = 0 , i.e. Ω γ = Ω γ ∗ = 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 R α d ( Ω ) . The previously determined class of ‘universally prestarlike functions’, defined in the slit-domain C ∖ [ 1 , ∞ ] , is identified as the class of ‘universal multipliers’ for R α d ( Ω ) in any disk-like domain Ω.