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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 517296, 4 pages
Sufficiency Criteria for a Class of -Valent Analytic Functions of Complex Order
Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan
Received 21 December 2012; Accepted 26 February 2013
Academic Editor: Fuding Xie
Copyright © 2013 Muhammad Arif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the present paper, we consider a subclass of -valent analytic functions and obtain certain simple sufficiency criteria by using three different methods for the functions belonging to this class. Many known results appear as special consequences of our work.
Let be the class of functions analytic and -valent in the open unit disk and of the form
In particular, , , and . By and , and , we mean the subclasses of which are defined, respectively, by
For , , , the previous two classes defined in (2) reduce to the well-known classes of starlike and convex, respectively.
For functions , of the form (1), we define the convolution (Hadamard product) of and by
Now we define the subclass of by
Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions, for some of the related work see [1–8]. The object of the present paper is to obtain sufficient conditions for the subclass of . We also consider some special cases of our results which lead to various interesting corollaries and relevances of some of these with other known results also being mentioned.
We will assume throughout our discussion, unless otherwise stated, that , , .
2. Preliminary Results
To obtain our main results, we need the following Lemma's.
Lemma 1 (see ). If with and satisfies the condition then
Lemma 2 (see ). If satisfing the condition where is the unique root of the equation then
Lemma 3 (see ). Let be a set in the complex plane , and suppose that is a mapping from to which satisfies for and for all real such that . If is analytic in and for all , then .
3. Main Results
Theorem 4. If satisfies then .
Proof. Let us set a function by
for . Then clearly (11) shows that .
Differentiating (11) logarithmically, we have which gives
Thus using (10), we have
Hence, using Lemma 1, we have .
From (12), we can write
Since , it implies that . Therefore, we get and this implies that .
Setting and in Theorem 4, we get the following.
Corollary 5. If satisfies then , the class of starlike functions of complex order .
Putting and in Theorem 4, we have the following.
Corollary 6. If satisfies then , the class of convex functions of complex order .
Remark 7. If we put in Corollaries 5 and 6, we get the results proved by Uyanık et al. . Furthermore, for , we obtain the results studied by Mocanu  and Nunokawa et al. , respectively. Also if we set with and in Theorem 4, we obtain the results due to Goyal et al. .
Theorem 8. If satisfies where is the unique root of (8), then .
Proof. Let be given by (11), which clearly belongs to the class .
Now differentiating (11), we have which gives
Thus using (19), we have where is the root of (8). Hence, using Lemma 2, we have .
From (20), we can write
Since , it implies that . Therefore, we get (16), and hence .
Making , with and , we have the following.
Corollary 9. If satisfies where is the unique root of (8) with , then , the class of -valent starlike functions of order .
Also if we take , with and in Theorem 8, we obtain the following result.
Corollary 10. If satisfies where is the unique root of (8) with , then , the class of -valent convex functions of order .
Theorem 12. If satisfies
Proof. Let us set
Then is analytic in with .
Taking logarithmic differentiation of (28) and then by simple computation, we obtain with
Now for all real and satisfying , we have
Reputing the values of , , , and then taking real part, we obtain where , , are given in (27).
Let . Then and , for all real and satisfying , . Using Lemma 3, we have . This implies that and hence .
If we put and in Theorem 12, we obtain the following result proved in .
Corollary 13. If satisfies then .
Corollary 14. If satisfies then .
The author is thankful to the Prof. Dr. Ihsan Ali, Vice chancellor of Abdul Wali Khan University Mardan, for providing research facilities in AWKUM.
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