1. Introduction and Definitions
Let denote the class of analytic functions of the form
(1)
which are analytic in the open unit disk and normalized by . Let S be the subclass of consisting of analytic univalent function of the form (1.1).
The study of normalised analytic univalent functions is enhanced by the used of operators, mostly, differential and integral operators. In this study, we have implored the used of convulation of well known differential operators to defined our class. For more works on operators see [1] [2] [3] .
Definition 1
Let T denotes the subclass of S consisting of functions of the form
(2)
Further we define the class by
(3)
Definition 2 ( [4] )
For and of the form (1.1) and , the operator is defined by
i.e.
(4)
Definition 2 [3] Let , the operator is defined by
Thus it is obvious to see from above that
(5)
where
Thus by convolution as earlier defined by [5] we have
(6)
We now defined a class , which consist of functions such that the following inequality is satisfy
Motivated here by the works of [1] [6] , we characterize our class using well know existing geometric properties.
2. Properties of the Class
2.1. Coefficient Inequality
Theorem 2.1.
let . Then if and only if
(7)
where
Proof:
Supposed the inequality (7) holds true and , then we have
But by maximun modullus principle, establishing our desired result.
Conversely,
Let , then
(8)
Then
Recall that , thus we have
(9)
Choose z on the real axis and let . Then we have
(10)
This yields;
(11)
This establishes our proof.
Corollary 2.1.
If then
(12)
equality is attained for
(13)
We shall state the growth and distortion theorems for the class The results of which follow easily on applying Theorem 2.1, therefore, we deem it necessary to omit the trivial proofs.
2.2. Growth and Distortion Theorems
Theorem 2.2.
Let the function then for
Theorem 2.3.
Let the function then for
When
we obtain a sharp result.
2.3. Radii of Close-to-Convexity, Starlikeness and Convexity
Theorem 2.4.
Let the function , then is close-to-convex of order in
where
The result obtained is sharp.
Proof.
It is sufficient to show that for Thus we can write
Therefore if
(14)
But we have from theorem 2.1. that
(15)
Relating (14) and (15) we have our desired result.
Theorem.2.5.
Let the function , then is starlike of order , in
where
The result obtain here is sharp.
Proof.
We must show that for . Equivalently, we have
(16)
But we have from theorem 2.1. that
(17)
Relating (16) and (17) will have our desired result.
Theorem 2.6.
Let the function , then is convex of order , in
where
The result obtain here is sharp.
Proof.
By using the technique of theorem 2.5 we easily show that this holds for . The analogous details of theorem 2.5 are thus omitted, hence the proof.
3. Integral Operator
Theorem 3.1.
Let the function defined by (2) be in the class and let be a real number such that . Then the function defined by
(18)
also belong to the class
Proof.
From the representation and definition of we have that
(19)
where
(20)
Thus we have
(21)
(22)
since . By theorem 1.1 . This establishes our proof.
[1] Khairnar, S.M. and Meena, M. (2008) Properties of a Class of Analytic and Univalent Functions Using Ruscheweyh Derivative. International Journal of Contemporary Mathematical Sciences, 3, 967-976.
[2] Salagean, G.S. (1981) Subclasses of Univalent Functions. Complex Analysis: Fifth Romanian-Finnish Seminar, Part I Bucharest, Lecture Notes in Mathematics 1013, Springer-Verlag.
[3] Ruscheweyh, S. (1975) New Criteria for Univalent Functions. Proceedings of the AMS—American Mathematical Society, 49, 109-115.
https://doi.org/10.1090/S0002-9939-1975-0367176-1
[4] Al-Oboudi, F.M. (2004) On Univalent Functions Defined by a Generalized Salagean Operator. Indian Journal of Pure and Applied Mathematics, 25-28, 1429-1436.
https://doi.org/10.1155/S0161171204108090
[5] Lupas, A.A. (2011) Certain Differential Superordination Using a Generalized Salagean and Ruscheweyh Operator. Acta Universitatis Apulensis, 25, 31-40.
[6] Esa, G.H. and Darus, M. (2007) Application of Fractional Calculus Operators to a Certain Class of Univalent Functions with Negative Coefficients. International Mathematical Forum, 57, 2807-2814.
https://doi.org/10.12988/imf.2007.07253