On a class of spiral-like functions with respect to a boundary point related to subordination
© Haji Mohd and Darus; licensee Springer 2013
Received: 29 December 2012
Accepted: 1 May 2013
Published: 31 May 2013
For , φ a starlike univalent function, the class of functions f that are spiral-like with respect to a boundary point satisfying the subordination
is investigated. The integral representation, growth and distortion theorem are proved by relating these functions with Ma and Minda starlike functions. Some earlier results are shown to be a special case of the results obtained.
Dedicated to Professor Hari M Srivastava
1 Introduction and motivation
This was partially proved in . It was only in 1984 that the characterization was validated by Lyzzaik . Todorov  associated this class with a functional and obtained a structured formula and coefficient estimates in the year 1986. Later, Silverman and Silvia  gave a full description of the class of univalent functions on , the image of which is star-shaped, with respect to a boundary point. Since then, this class of starlike functions with respect to a boundary point has gained notable interest among geometric function theorist and also other researchers. Among them, Abdullah et al.  studied the properties of functions in this class. The distortion results for starlike functions with respect to a boundary point were obtained in [6, 7]. The dynamical characterizations of functions starlike with respect to a boundary point can be found in . In the year 2001, Lecko  gave another representation of starlike functions with respect to a boundary point. Also, Lecko and Lyzzaik obtained different characterizations of this class in .
Following the studies on the class of starlike functions, many authors extensively studied the class of spiral-like functions. For recent work on the class of spiral-like functions, see . Later, there was interest towards the class of spiral-like functions with respect to a boundary point. See [12–15]. Aharonov et al.  gave a comprehensive definition for spiral-shaped domains with respect to a boundary point.
Definition 1.1 A simply connected domain , , is called a spiral-shaped domain with respect to a boundary point if there is a number with such that, for any point , the curve , , is contained in Ω.
It was also showed in  (see also ) that each spiral-like function with respect to a boundary point is a complex power of starlike function with respect to a boundary point. In particular, if in Definition 1.1, then Ω is called a star-shaped domain with respect to a boundary point. The following was proved in the same.
Conversely, if f is a univalent function with and satisfies (1.1) for some , then f is a spiral-like function with respect to a boundary point.
Elin  then considered the class of spiral-like functions of order β () with respect to a boundary point and obtained interesting results including the distortion and covering theorems.
where φ is an analytic function with a positive real part in , is symmetric with respect to the real axis and starlike with respect to and . A function is called Ma and Minda starlike (with respect to φ). The class consisting of starlike functions of order β, and the class of Janowski starlike functions are special cases of when and for , respectively.
For (), denote the class by . For , and , denote by .
The class defined by subordination is investigated to obtain representation, estimates for f and and subordination conditions. We obtained some interesting result in a wider context and our approach is mainly based on .
2 Representation for the class
The following result provides an integral representation of functions belonging to the class .
The desired result follows from this. The converse follows easily. □
3 Estimates for f and in the class
and hence the desired result follows. □
The result then follows easily as the relation (3.3) holds. □
Combining (3.5) and (3.6), the desired results follows. □
We have the following corollaries as (3.4) holds.
4 Necessary and sufficient condition
and hence the desired result follows. □
The following corollaries hold for is convex univalent on .
Let , and in Corollary 4.1 and hence we have the result.
Corollary 4.2 
Theorem 4.2 as well as Corollaries 4.3 and 4.4 below are respectively special cases of Theorem 4.1 and Corollaries 4.1 and 4.2 when and . However, we prove the below without the convexity assumption on φ.
Let be defined as in (3.2) and hence we arrive at the desired conclusion. □
When , and , the above corollary reduces to the following result.
Corollary 4.4 
5 Coefficient estimate for
We denote the class satisfying the above subordination as .
and hence from the above relation the desired results are obtained. □
Corollary 5.1 When , our results coincide with [, Corollary 2.3].
Remark 5.1 All the results for the special case when or the class starlike with respect to a boundary point defined by subordination were presented at the 8th International Symposium on GFTA, 27-31 August 2012, Ohrid, Republic of Macedonia and thereafter published as .
The work here is partially supported by MOHE:LRGS/TD/2011/UKM/ICT/03/02.
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