On uniformly univalent functions with respect to symmetrical points
© Noor; licensee Springer. 2014
Received: 28 March 2014
Accepted: 13 June 2014
Published: 22 July 2014
In this paper, we define and study some new subclasses of starlike and close-to-convex functions with respect to symmetrical points. These functions map the open unit disc onto certain conic regions in the right half plane. Some basic properties, a necessary condition, and coefficient and arc length problems are investigated. The mapping properties of the functions in these classes are studied under a certain linear operator.
which are analytic in the open unit disc . Let S, K, , and C be the subclasses of A which consist of univalent, close-to-convex, starlike (with respect to origin), and convex functions, respectively. For recent developments, extensions, and applications, see [1–25] and the references therein.
Uniformly starlike and convex functions were first introduced by Goodman  and then studied by various other authors. If , we say f is subordinate to g in E, written as or , if there exists a Schwarz function such that for .
For , the class consists of functions analytic in E with such that for , and, with , we obtain the well-known class P of Carathéodory functions with positive real part.
For fixed k, represents the conic regions bounded, successively, by the imaginary axis (), the right branch of a hyperbolic () and a parabola (). When , the domain becomes a bounded domain being the interior of the ellipse.
These functions are univalent in E and belong to the class P. Using the subordination concept, we define the class as follows.
Let be analytic in E with . Then if, and only if, in E and are given by (1.2).
It can easily be seen that the analytic function , with , belongs to the class if in E.
So we can write , .
It is also well known  that if, and only if, .
We now define the following.
Also, for , the class reduces to .
The class is defined as follows.
Definition 1.2 Let . Then if, and only if for .
where consists of close-to-convex functions with respect to symmetrical starlike functions.
From the definition, it is clear that consists of univalent functions.
For , and , reduces to the class .
2 Preliminary results
We shall need the following lemmas to prove our main results.
Lemma 2.1 
Then , .
and let for . Then implies that for .
This proves that for . □
The following lemma is an easy extension of a result proved in .
and is the best dominant of (2.1).
3 The class
In this section, we shall study some basic properties of the class .
belongs to in E.
In particular is an odd starlike function of order in E.
Since is a convex set, it follows that and thus in E. □
As a special case, we note that, for , in E, and hence . We now discuss a geometric property for . Here we investigate the behavior of the inclusion of the tangent at a point to the image of the circle , , , under the mapping by means of a function f from the class .
So, the integral on the left side of the last inequality characterizes the increment of the angle of the inclination of the tangent to the curve between the points and for .
We have the following necessary condition for .
where σ is given by (1.3) and .
Proof Since , and .
Using (3.4) and (3.5) in (3.2), we obtain the required result. □
- 1.For , , , it follows from Theorem 3.2 that
Theorem 3.3 (Integral representation)
where , .
and the result follows when we integrate. □
When , , we obtain the result for the class given in .
We now study the class under a certain integral operator.
Then belongs to in E.
Since , , and . Therefore it can easily be verified that is -valently starlike in E.
say, where and D is -valently starlike.
This proves that in E. □
where , , and . Then belongs to for .
When , , we obtain a generalized form of the Bernardi operator; see . Also for , , and , we have the well-known integral operator studied by Libera  who showed that it preserves the geometric properties of convexity, starlikeness, and close-to-convexity.
Thus and the proof is complete. □
4 The class
Here we shall study some properties of the class which consists of k-uniformly close-to-convex functions.
Let denote the length of the image of the circle under f. We prove the following.
where and σ is given by (1.3), and is a constant depending only on k, β.
and , .
where C and are constants depending only on and σ. This completes the proof. □
We now discuss the growth rate of coefficients of .
where is a constant depending only on σ and and σ, are as given in Theorem 4.1.
With , we use Theorem 4.1 and obtain the required result. □
Then in E. That is, the class is preserved under the integral operator (4.4).
This proves that in E. □
We study a partial converse of the above result as follows.
We note that and . So there exists . The right hand side of (4.9) is positive for , where is given by (4.6). This implies that for and the proof is complete. □
For , . Then , defined by (4.5) belongs to for .
When and (that is, ), then belongs to the same class for . This result has been proved by Livingston  for convex and starlike functions.
The author would like to thank editor and anonymous referee for their valuable suggestions. The author is grateful to Dr. SM Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan for providing an excellent research and academic environment. This research is supported by HEC NRPU project No: 20-1966/R&D/11-2553, titled, Research unit od Academic Excellence in Geometric Function Theory and Applications.
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