Subclass of univalent harmonic functions defined by dual convolution
© El-Ashwah; licensee Springer. 2013
Received: 6 December 2012
Accepted: 14 October 2013
Published: 12 November 2013
In the present paper, we study a subclass of univalent harmonic functions defined by convolution and integral convolution. We obtain the basic properties such as coefficient characterization and distortion theorem, extreme points and convolution condition.
A continuous function is a complex-valued harmonic function in a simply connected complex domain if both u and v are real harmonic in D. It was shown by Clunie and Sheil-Small  that such a harmonic function can be represented by , where h and g are analytic in D. Also, a necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that (see also [2–4] and ).
Clunie and Sheil-Small  investigated the class as well as its geometric subclasses and obtained some coefficient bounds.
We further consider the subclass of for h and g given by (1.2).
In this paper, we extend the results of the above classes to the classes and , we also obtain some basic properties for the class .
2 Coefficient characterization and distortion theorem
Unless otherwise mentioned, we assume throughout this paper that and are given by (1.12), , and θ is real. We begin with a sufficient condition for functions in the class .
Then f is sense-preserving, harmonic univalent in U and .
This completes the proof of Theorem 1. □
In the following theorem, it is shown that condition (2.1) is also necessary for functions , where h and g are given by (1.2).
If condition (2.4) does not hold, then the numerator in (2.6) is negative for sufficiently close to 1. Hence there exists in for which the quotient in (2.6) is negative. This contradicts the required condition for , and so the proof of Theorem 2 is completed. □
The results are sharp.
Proof We prove the left-hand side inequality for . The proof for the right-hand side inequality can be done by using similar arguments.
The following covering result follows from the left side inequality in Theorem 3.
is included in , where C is given by (2.7).
3 Extreme points
Our next theorem is on the extreme points of convex hulls of the class , denoted by .
In particular, the extreme points of the class are and , respectively.
then note that by Theorem 2, () and ().
Using Theorem 2, it is easily seen that the class is convex and closed and so . □
4 Convolution result
Using this definition, we show that the class is closed under convolution.
Theorem 5 For , let and . Then .
the right-hand side of this inequality is bounded by 1 because . Then . □
Finally, we show that is closed under convex combinations of its members.
Theorem 6 The class is closed under convex linear combination.
This condition is required by (2.4) and so . This completes the proof of Theorem 6. □
Putting in our results, we obtain the results obtained by Dixit et al. ;
Putting and in our results, we obtain the results obtained by Rosy et al. ;
Putting in our results, we obtain the results obtained by Kim et al. ;
Putting and in our results, we obtain the results obtained by Jahangiri ;
Putting and in our results, we obtain the results obtained by Jahangiri .
The author would like to express her sincere gratitude to Springer Open Accounts Team for their kind help.
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