On analytic prolongation of a family of operators. (English) Zbl 0753.30005

Let a family of operators \(\{{\mathcal L}(a)^ \lambda\}_{\lambda\geqq 0}\) depending on a parameter \(a>0\) be defined by \({\mathcal L}(a)^ \lambda f(z)=a^ \lambda/\Gamma(\lambda)\cdot\int^ 1_ 0f(zt)t^{a- 2}\log^{\lambda-1}(1/t)dt\) . Its domain of definition is the class of analytic functions \(f\) which are holomorphic in the unit disk and normalized by \(f(0)=f'(0)-1=0\). The family possesses the additivity property \({\mathcal L}^ \lambda{\mathcal L}^ \mu={\mathcal L}^{\lambda+\mu}\).
The main purpose of the present paper is to show the possibility of analytic prolongation of the family \(\{{\mathcal L}(a)^ \lambda\}_{\lambda\geq 0}\) with respect to its parameters \(\lambda\) and \(a\) within single-valuedness into the whole complex plane cut along the negative real axis on the \(a\)-plane. Based on the fact that \({\mathcal L}(a)\) is inverse to the differential operator \(\Theta(a)={1\over a}\left({d\over d\log z}+a-1\right)\), a way is obtained to define the differentiation of any complex order.
Reviewer: Y.Komatu


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
47B38 Linear operators on function spaces (general)