Abstract
In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping.
Highlights
Let D1 and D2 be domains in the complex plane C
By Hol (D1, D2) we denote the set of holomorphic mappings of D1 into D2
If D is a domain in C, the set Hol (D) := Hol (D, D) forms a semigroup with composition being the semigroup operation
Summary
Let D1 and D2 be domains in the complex plane C. Assuming that Φ is holomorphic in the open unit disk ∆, the numbers ρ and r are often called the Bloch radii [6, 7]. Φ(z) z on some subsets of ∆ or on the whole unit disk (see, for example, [1, 13]) In this connection, we observe that if a function Φ ∈ Hol (∆, C), Φ(0) = 0, satisfies the condition. Re [Φ(z)z] ≥ |z|2, z ∈ ∆, by the argument principle, Φ covers the open unit disk ∆ and for every z ∈ ∆ there is a unique solution w of the equation Φ(w) = z This observation links to a notion, which comes from operator theory and nonlinear analysis. The inverse function Φ−1 preserves the disk ∆R invariant, where (0), if Re Φ (0) ≥ 2, Re Φ (0) − 1, if Re Φ (0) < 2
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