The Radii of Starlikeness and Convexity of Certain Nevanlinna Analytic Functions

Abstract

Abstract. We determine the radius of starlikeness and the radius of convexity of analytic functions having the forms / L i dpHt)/{z -t) and / L i zdpJ(j)/(l -tz) where p(t) is a probability measure.

Typically real harmonic functions

“…Consequently, any function f ∈ T is univalent on the preimage of the unit disk under the function ψ given by (3.2), which is the lens domain L. In 1936 Robertson observed that an analytic function F with real coefficients is univalent and convex in the vertical direction if and only if the function z → zF ′ (z) is typically real (see [8], p. 206). Hence the functions given by (3.3) are convex in the direction of the imaginary axis (see also [13], [12]). Therefore the sets f (L), f ∈ T, are convex in the vertical direction.…”

“…Now we observe that a function f ∈ T O H is univalent on the given region in (b) if and only if function f • ψ, where ψ is given by (3.2) is univalent on the disk D(0; √ 2 − 1). The last follows from the fact that an analytic function F given by (3.3) maps the disk D(0; √ 2 − 1) onto a convex domain (see [13], p. 292] ) and from the shearing theorem of Clunie and Sheil-Small.…”