Modified Roper-Suffridge operator for some holomorphic mappings

Abstract

In this paper, we will study the operator given by $F(z) = (f(z_1 ) + f'(z_1 )P(z_0 ),(f'(z_1 ))^{1/k} z_0 ^T )^T ,$ where z = (z ₁, z ₀ ^T)^T belongs to the unit ball B ⁿ in ℂ ⁿ, z ₁ ∈ U = B ¹, z ₀ = (z ₂, …, z _n)^T ∈ ℂ ⁿ⁻¹, and P: ℂ ⁿ⁻¹ → ℂ is a homogeneous polynomial of degree k (k ⩾ 2), the holomorphic branch is chosen such that (f′(0))^1/k = 1. We will give different conditions for P such that the modified operator preserves the properties of almost spirallikeness of type β and order α, spirallikeness of type β and order α, and strongly spirallikeness of type β and order α, respectively.

Keywords

Roper-Suffridge operator / almost spirallike mappings of type β and order α / spirallike mappings of type β and order α / strongly spirallikeness of type β and order α

References

[1] Duren P. L. Univalent Functions, 1983, New York: Springer-Verlag.

[2] Feng S. X., Liu T. S. The generalized Roper-Suffridge extension operator. Acta Math Sci, 2008, 28B 1 63-80

[3] Feng S. X., Liu T. S., Ren G. B. The growth and covering theorems for several mappings on the unit ball in complex Banach spaces. Chin Ann Math, 2007, 28A 2 215-230

[4] Graham I. Growth and covering theorems associated with Roper-Suffridge extension operator. Proc Amer Math Soc, 1999, 127, 3215-3220

[5] Graham I., Hamada H., Kohr G. Extension operators for locally univalent mappings. Michigan Math J, 2002, 50, 37-55

[6] Graham I., Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. J Anal Math, 2000, 81, 331-342

[7] Graham I., Kohr G. An extension theorem and subclass of univalent mappings in Banach space. Complex Variables, 2002, 47, 59-72

[8] Graham I., Kohr G., Kohr M. Loewner chains and Roper-Suffridge extension operator. J Math Anal Appl, 2000, 247, 448-465

[9] Kohr G. Loewner chains and a modification of the Roper-Suffridge extension operator. Mathematica, 2006, 71, 41-48

[10] Muir J. R. A modification of the Roper-Suffridge extension operator. Comput Methods Funct Theory, 2005, 5, 237-251

[11] Muir J. R. A class of Loewner chain preserving extension operators. J Math Anal Appl, 2008, 337, 862-879

[12] Nehari Z. Conformal Mapping, 1982, New York: Dover Publications.

[13] Roper K. A., Suffridge T. J. Convex mappings on the unit ball of ℂⁿ. J Anal Math, 1995, 65, 333-347

[14] Wang J. F., Liu T. S. A modification of the Roper-Suffridge extension operator for some holomorphic mappings. Chin Ann Math, 2010, 31A 4 487-496