## 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

### MSC:

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