On Refinement of the Coefficient Inequalities for a Subclass of Quasi-Convex Mappings in Several Complex Variables
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
ON REFINEMENT OF THE COEFFICIENT INEQUALITIES FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES∗
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Yuanping LAI (
Qinghua XU (
School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China E-mail : [email protected]; [email protected] Abstract Let K be the familiar class of normalized convex functions in the unit disk. In ∞ P [14], Keogh and Merkes proved that for a function f (z) = z + ak z k in the class K, k=2
|a3 − λa22 | ≤ max
1 , |λ − 1| , λ ∈ C. 3
The above estimate is sharp for each λ. In this article, we establish the corresponding inequality for a normalized convex function f on U such that z = 0 is a zero of order k + 1 of f (z) − z, and then we extend this result to higher dimensions. These results generalize some known results. Key words
Fekete-Szeg¨ o problem; subclass of quasi-convex mappings; sharp coefficient bound
2010 MR Subject Classification
1
32H30; 32H02; 30C45
Introduction Let A be the class of functions of the form ∞ X f (z) = z + an z n ,
(1.1)
n=2
which are analytic in the open unit disk
U = {z ∈ C : |z| < 1}. We denote by S the subclass of A consisting of all functions which are also univalent in U. Let K denote the subclass of S consisting of convex functions on U. Let X be a complex Banach space with norm k · k, X ∗ be the dual space of X, E be the unit ball in X, N be the set of all positive integers. Furthermore, let ∂Un denote the boundary of Un , and let (∂U)n be the distinguished boundary of Un . ∗ Received November 2, 2019; revised September 11, 2020. Supported by National Natural Science Foundation of China (11971165, 11561030).
1654
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
For each x ∈ X \ {0}, we define T (x) = {Tx ∈ X ∗ : kTxk = 1, Tx(x) = kxk}. According to the Hahn-Banach theorem, T (x) is nonempty. In [3], Fekete and Szeg¨o obtained the following classical result: Let the function f (z) be defined by (1.1). If f ∈ S, then 2λ
|a3 − λa22 | ≤ 1 + 2e− 1−λ for λ ∈ [0, 1]. The above inequality is known as the Fekete and Szeg¨o inequality. After the appearance of this, there were many papers considering the corresponding problems for various subclasses of the class S, and many interesting results were obtained. For example, in [14], Keogh and Merkes obtained the following result for K: Theorem A ([14]) Let the function f (z) be defined by (1.1). If f ∈ K, then 1 |a3 − λa22 | ≤ max , |λ − 1| , λ ∈ C. 3 z 1+z The above estimation is sharp for the function f (z) = 1−z if |λ − 1| ≥ 31 , and for f (z) = 12 ln 1−z 1 if |λ − 1| ≤ 3 . Although the Fekete and Szeg¨ o inequalities for various subclasses of the class S have been established, only a few results are known for the inequalities of homogeneous expansions for subclasses of biholomorphic mappings in several complex variables. Some best-possible results concerning the coefficient estimates for subclasses of holomorphic mappings in several variables were
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