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On a class of spiral-like functions with respect to a boundary point related to subordination
Journal of Inequalities and Applications volume 2013, Article number: 274 (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
Let be an open unit disk of the complex plane ℂ and let be a class of analytic functions f normalized by and . Let be an interior or a boundary point of a set in ℂ. The set is starlike with respect to if the line segment joining to every other point in lies in the interior of . If a function maps onto a starlike domain with respect to origin, then f is a starlike function. The class of starlike functions with respect to origin is denoted by . Analytically,
Robertson  took a leap forward with the characterization of the class and defined the class of starlike functions with respect to a boundary point. Geometrically, it is the characterization of a function such that is starlike with respect to the boundary point and lies in a half-plane. The analytic description given by Robertson was
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.
Theorem 1.1 Let f be an analytic function with , , and let it be a spiral-like function with respect to a boundary point. Then there exists a number such that
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.
On the other hand, Ma and Minda  gave a unified presentation of the class starlike using the method of subordination. For two functions h and g in , the function h is subordinate to g, written
if there exists a function , with and , such that . In particular, if the function g is univalent in , then is equivalent to and . A function is starlike if is subordinated to . Ma and Minda  introduced the class
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.
Definition 1.2 Let , and . Also, let φ be an analytic function with a positive real part , let be symmetric with respect to the real axis and starlike with respect to and . The function if the subordination
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 .
Theorem 2.1 The function if and only if there exists p satisfying such that
Proof Let . Then define by
Then implies that . Rewriting the above equation as
and integrating from 0 to z, it follows that
An exponentiation gives
The desired result follows from this. The converse follows easily. □
3 Estimates for f and in the class
Theorem 3.1 Let be an analytic function with , satisfying the equation . If , then
Proof Define the function by
Since f is univalent and , it is clear that in . Therefore, the function h is well defined and analytic in . A computation shows that
Hence we have the relation if and only if . Ma and Minda [, Corollary 1′] have shown that for ,
Using this inequality for h in (3.2) gives
and hence the desired result follows. □
If and hence
In particular, for , the inequality reduces to the following inequality :
Theorem 3.2 Let and . Then, for ,
Proof For a function , in the paper [, Corollary 3′] it is shown that
The result then follows easily as the relation (3.3) holds. □
Corollary 3.1 If , then for ,
Corollary 3.2 If , then for
Theorem 3.3 Let and
For , if then
Proof By Definition 1.2, for , we have
When (3.4) holds, the above subordination indicates that
This shows that
For , Theorem 3.1 gives
Combining (3.5) and (3.6), the desired results follows. □
We have the following corollaries as (3.4) holds.
Corollary 3.3 Let . For , let
For , let
For , if then
Corollary 3.4 Let ,
For , if then
4 Necessary and sufficient condition
Theorem 4.1 Let φ be a convex univalent function defined on . The function if and only if for all , ,
Proof Ruscheweyh [, Theorem 1] showed that for φ a convex univalent function, F as in the hypothesis and
if and only if for all , ,
From the relation (3.3), we know that if and only if . Substituting (3.2) in (4.1), we have
and hence the desired result follows. □
The following corollaries hold for is convex univalent on .
Corollary 4.1 The function if and only if for all , ,
Let , and in Corollary 4.1 and hence we have the result.
Corollary 4.2 
The function if and only if for all , ,
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.
Theorem 4.2 If , then
Proof Clearly . If , then
Therefore by [, Theorem 1′]
Let be defined as in (3.2) and hence we arrive at the desired conclusion. □
Corollary 4.3 If then
When , and , the above corollary reduces to the following result.
Corollary 4.4 
5 Coefficient estimate for
In particular, when , (1.2) becomes
We denote the class satisfying the above subordination as .
Theorem 5.1 Let . If , then the coefficients , , satisfy the following inequalities:
Proof Define the function by
Then a computation shows that
Since , we have
or there is an analytic function such that
we see that
In view of the well-known inequality , we have
for and as in the hypothesis. Todorov in  shows that for
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 .
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The work here is partially supported by MOHE:LRGS/TD/2011/UKM/ICT/03/02.
The authors declare that they have no competing interests.
The first author MHM is currently a PhD student under supervision of the second author MD and jointly worked on deriving the results. All authors read and approved the final manuscript.
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Haji Mohd, M., Darus, M. On a class of spiral-like functions with respect to a boundary point related to subordination. J Inequal Appl 2013, 274 (2013). https://doi.org/10.1186/1029-242X-2013-274
- starlike with respect to boundary point
- spiral-like with respect to a boundary point