Universally prestarlike functions as convolution multipliers

Abstract

Universally prestarlike functions (of order α ≤ 1) in the slit domain := C\[1, ∞] have recently been introduced in Ruscheweyh et al. (Israel J Math, to appear). This notation generalizes the corresponding one for functions in the unit disk D (and other circular domains in C). In this paper we study the behaviour of universally prestarlike functions under the Hadamard product. In particular it is shown that these function classes (with α fixed), are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.

An extension of Suffridge's convolution theorem

“…Ruscheweyh and Salinas (and Sugawa) [8,6,7] have recently found amazing extensions of Theorem E to functions which are pre-starlike on so-called disk-like domains, i.e. domains which are the union of disks or halfplanes, all containing the origin.…”

Suffridge's convolution theorem for polynomials and entire functions having only real zeros

“…It has found numerous applications in complex analysis and other fields. For instance, it forms the basis of the geometric convolution theory which was developed by Ruscheweyh, Suffridge, and Sheil-Small (see [23,24,28,29,30,32] and, more recently, [25,26,27]) and it can be used to classify all linear operators which preserve the set of polynomials whose zeros lie in a given circular domain (cf. [24, Thm.…”

“…We believe that Suffridge's work [32], the recent work of Ruscheweyh and Salinas [25,26,27], and the results of this paper and [13] (the methods of proof presented here and in [13] also seem to have some kind of resemblance to the methods used in [10]), strongly hint at a very deep lying extension of Grace's theorem which will lead to a much better understanding of the relation between the zeros and the coefficients of complex polynomials.…”

Suffridge's convolution theorem for polynomials with zeros in the unit disk

“…Subsequently, more extensions of Suffridge's theorem and other special cases of Grace's theorem were found by Ruscheweyh, Salinas, Sheil-Small, and the author (cf. [6,7,10,11,12,13,14,16,17]). The extensions of Grace's theorem found in these papers show strong similarities; it thus seems very likely that there should be a generalization of Grace's theorem which unifies all partial extensions that have been discovered until now.…”