Products of Composition and Differentiation between the Fractional Cauchy Spaces and the Bloch-Type Spaces

The operators D C Φ and C Φ D are defined by D C Φ f = f ∘ Φ ′ and C Φ D f = f ′ ∘ Φ where Φ is an analytic self-map of the unit disc and f is analytic in the disc. A characterization is provided for boundedness and compactness of the products of composition and differentiation from the spaces of fractional Cauchy transforms F α to the Bloch-type spaces B β , where α > 0 and β > 0 . In the case β < 2 , the operator D C Φ : F α ⟶ B β is compact ⇔ D C Φ : F α ⟶ B β is bounded ⇔ Φ ′ ∈ B β , Φ Φ ′ ∈ B β and Φ ∞ < 1 . For β < 1 , C Φ D : F α ⟶ B β is compact ⇔ C Φ D : F α ⟶ B β is bounded ⇔ Φ ∈ B β and Φ ∞ < 1 .

STARLIKENESS AND CONVEXITY OF CAUCHY TRANSFORMS ON REGULAR POLYGONS

For $n\geq 3$ , let $Q_n\subset \mathbb {C}$ be an arbitrary regular n-sided polygon. We prove that the Cauchy transform $F_{Q_n}$ of the normalised two-dimensional Lebesgue measure on $Q_n$ is univalent and starlike but not convex in $\widehat {\mathbb {C}}\setminus Q_n$ .

Finite term relations for the exponential orthogonal polynomials

The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function g supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthogonal polynomials. On function theory side, we offer an effective way based on the Cauchy transforms of g,z̅g,…,z̅dg, to decide whether a (d + 2)-term relation among the exponential orthogonal polynomials exists; in that case we indicate how the shade function g can be reconstructed from a resulting polynomial of degree d and the Cauchy transform of g. A discussion of the relevance of the main concepts in Hele-Shaw dynamics completes the article.