Indicator Diagram of an Entire Function of One Variable with Nonnegative Indicator

Abstract

The idea to construct a plane indicator diagram of an entire function f of order ρ > 0, p ≠ 1, of normal type, and with nonnegative indicator h is as follows. By Theorem 1.9.25, the cyclic group of rotations about the origin (the point 0*⁾ in ℂ) with the generator e ^i2πρ operates on the locally convex curve Γ _h (see Def. 1.9.25) associated with h. If h(φ) ≥ 0, φ ∈ ℝ, then the support straight line l _φ to Γ _h (see 1.9.16) rotates about 0 as φ runs through the interval (−∞, ∞). If ρ is an integer, then Γ _h is a closed curve in ℂ, and in one circuit along it the straight line l _φ makes ρ complete turns about 0 in ‒. If, moreover, 0 < h(φ) < ∞, then Γ _h is the image under the mapping α = ζ^ρ of a closed Jordan curve in ℂ that bounds a so-called ρ-convex compact set. This compact set may be viewed as the indicator diagram of f. Properties of ρ-convex sets are considered in detail in 3.1. The geometry of ρ-convex sets is in many ways similar to the geometry of convex sets. For instance, since ρ-convex sets are star-like, it is possible to give their analytic discription by an analog of the Minkowski functional.