Some Geometric Properties of Multivalent Convex Function for Operator on Hilbert Space

By making use of the operator on Hilbert space, we introduce and study some properties of geometric of a subclass of multivalent convex functions with negative coefficients. Also we obtain some geometric properties, such as, coefficient inequality, growth and distortion theorem, extreme points, convex set, closure theorem, radius of close-to-convexity, weighted mean and inclusive properties. 1. Introducti0n In this paper, the aim is to mention the basic facts about geometric function theory, which this is obtained from mixing of geometry and analysis. Its origin started from the 19th century, but it continued and continually applicable till now. Geometric Function Theory is an important branch of complex analysis; It deals with the geometric properties of the analytic functions. In particular , we will concentrate on the important ideas in this theory. The fundamentals of this theory are explained in most text books on this subject. Also, we review and consider the basic ideas, principles, definitions and the general principles of complex analysis, which underline the geometric function theory of a complex variable rather than the basic lemmas which are needed in the proofs of our results. A full discussion of these principles can be found in standard text books, [1], [2]. The study of multivalent functions is one of the main branches of geometric function theory and plays a central role in complex analysis. Let be the class of functi0ns of the f0rm: ∑ { } which are analytic and valent in the open unit disk { | | }. Let denote the subclass of consisting of functions of the f0rm: ∑ { } Definition (1.1): A function is said to be in the class if it satisfies | | Journal of University of Babylon for Pure and Applied Sciences (JUBAS) by University of Babylon is licened under a Creative Commons Attribution 4.0 International License. 2018.


Introducti0n
In this paper, the aim is to mention the basic facts about geometric function theory, which this is obtained from mixing of geometry and analysis.Its origin started from the 19th century, but it continued and continually applicable till now.Geometric Function Theory is an important branch of complex analysis; It deals with the geometric properties of the analytic functions.In particular , we will concentrate on the important ideas in this theory.The fundamentals of this theory are explained in most text books on this subject.Also, we review and consider the basic ideas, principles, definitions and the general principles of complex analysis, which underline the geometric function theory of a complex variable rather than the basic lemmas which are needed in the proofs of our results.A full discussion of these principles can be found in standard text books, [1], [2].
The study of multivalent functions is one of the main branches of geometric function theory and plays a central role in complex analysis.Let be the class of functi0ns of the f0rm: where .
Let H be a Hilbert space on the complex field.Let be a linear operator on H.For a complex analytic function on the unit disk , we denoted , the operator on H defined by the usual Riesz-Dunford integral [3] ∫ , where is the identity operator on H, is a positively oriented simple closed rectifiable contour lying in Ủ and containing the spectrum of in its interi0r domain [4].Also can be defined by the series ∑ , which converges in the norm topology [5].

Definition (1.2):
Let H be a Hilbert space and be an operator on such that  and ‖ ‖ .Let be real numbers such that .
An analytic functi0n on the unit disk belong to the class if it satisfy the inequality The operator on Hilbert space were consider recently by [6]- [12].

Main results:
The first theorem gives C0efficient inequality for a function to be in the class .

Theorem (2.1):
Let be defined by (2).Then for all  if and only if where .
The result is sharp for the function given by .
Proof: Suppose that the inequality (3) holds.Then, we have Setting in the above inequality, we get Up0n clearing denominator in (5) and letting , we obtain ∑ ∑ .
Thus ∑ , which completes the proof.
Next, we obtain the growth and distortion for a function to be in the class .

Theorem (2.2): If
, and ‖ ‖ ∅, then The result is sharp for the function given by .
Proof: According to the Theorem (2.1), we get Therefore the proof is complete.
In the following, we prove extreme points of the class . The0rem Then if and only if it can be expressed in the form where and ∑ .
Proof: Assume that can be expressed by (6).Then, we have and so .
Conversely, Suppose that given by ( 2 Next, we obtain the convex set of the class .

Theorem (2.4):
The class is a convex set.
Proof: Let and be the arbitrary element of .Then for every , we show that .Thus, we have ∑ . Hence,
We will consider the function defined, for every by We want to find the value such that The inequality (14) would obviously imply (15) if .
Rewriting the inequality, we get .
1): A function is said to be in the class if it satisfies

∑
This completes the proof of the theorem.