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Notes on analytic functions with a bounded positive real part
Journal of Inequalities and Applications volume 2013, Article number: 370 (2013)
For real α and β such that , we denote by the class of normalized analytic functions f such that in . We find some properties, including inclusion properties, Fekete-Szegö problem and coefficient problems of inverse functions.
Dedicated to Professor Hari M Srivastava
Let ℋ denote the class of analytic functions in the unit disc on the complex plane ℂ. Let denote the subclass of ℋ consisting of functions normalized by and . Let denote the subclass of consisting of univalent functions. Denote by and , the class of starlike functions and convex functions, respectively. It is well-known that .
We say that f is subordinate to F in , written as if and only if for some Schwarz function with and , . If is univalent in , then the subordination is equivalent to and .
We denote by , the class of Janowski starlike functions, namely, the functions satisfying the subordination equation: . Note that .
Now, we shall introduce the class of analytic functions used in the sequel.
Definition 1 Let α and β be real numbers such that . The function belongs to the class if f satisfies the following inequality:
We remark that for given α, β (), if and only if f satisfies the following two subordination equations:
since the functions and map onto the right half plane, having real part greater than α, and the left half plane, having real part smaller than β, respectively. The above class is introduced by Kuroki and Owa . They investigated coefficient estimates for and found the necessary and sufficient condition for using the following subordination.
Lemma 1 (Kuroki and Owa )
Let and . Then if and only if
Lemma 1 means that the function p defined by
maps the unit disk onto the strip domain w with . We note that the function , given by
is in the class .
2 Inclusion properties
Theorem 1 For given , let A and B be real numbers such that
Proof At first, we note that
For , we know that the following inequality holds:
Therefore, it suffices to show that α and β satisfy the following inequalities:
Using inequality (4), we can derive that
By the above inequalities (5) and (6), we can easily obtain the inequalities (3), so the proof of Theorem 1 is completed. □
Lemma 2 (Miller and Mocanu )
Let Ξ be a set in the complex plane ℂ and let b be a complex number such that . Suppose that a function satisfies the condition
for all real ρ, and all . If the function defined by is analytic in and if
then in .
Theorem 2 Let , and in . Then
Proof Write and note that for . Let p be defined by
Then p is analytic in , and
Now for all real ρ, ,
Now, we let
hence occurs at only and g satisfies
Since , we have
hence we get
This shows that . By Lemma 2, we get in , and this shows that inequality (7) holds and the proof of Theorem 2 is completed. □
Theorem 3 Let , and in . Then
Proof Note that for . Let p be defined by
Then p is analytic in , and
where ψ is given in (8). Also
Now, for all real ρ, ,
where is given (9). Since
for all , we have
This shows that . By Lemma 2, we get in , and this is equivalent to
and the proof of Theorem 3 is completed. □
Combining the above Theorems 2 and 3, we can obtain the following result:
Theorem 4 Let , and in . Then
where and is given (7) and (10).
3 Some coefficient problems
In this section, we investigate coefficient problems for functions in the class . In , Kuroki and Owa investigated the coefficient of the function p given by (2); the function p can be written as
We denote by the class of bi-univalent functions f such that and the inverse function . Srivastava et al. investigated the estimates on the initial coefficient for certain subclasses of analytic and bi-univalent functions in [3, 4]. Ali et al. have studied similar problems in .
In theorem, we shall solve the Fekete-Szegö problem for . We need the following lemma:
Lemma 3 (Keogh and Merkers )
Let be a function with positive real part in . Then, for any complex number ν,
The following result holds for the coefficient of .
Theorem 5 Let and let the function f given by be in the class . Then, for a complex number μ,
Proof Let us consider a function q given by . Then, since , we have , where
Then h is analytic and has positive real part in the open unit disk . We also have
We find from equation (12) that
which imply that
Applying Lemma 3, we can obtain
in (13), we can obtain the result as asserted. □
Using the above Theorem 5, we can get the following result.
Corollary 1 Let the function f, given by , be in the class . Also let the function , defined by
be the inverse of f. If
Proof Relations (16) and (17) give
Thus, we can get the estimate for by
immediately. An application of Theorem 5 (with ) gives the estimates for , hence the proof of Corollary 1 is completed. □
Next, we shall estimate on some initial coefficient for the bi-univalent functions .
Theorem 6 Let f be given by be in the class . Then
where and are given by (14) and (15).
Proof If , then and , where . Hence
where is given by (2). Let
Then h and k are analytic and have positive real part in . Also, we have
By suitably comparing coefficients, we get
where and are given by (14) and (15), respectively. Now, considering (20) and (22), we get
Also, from (21), (22), (23) and (24), we find that
Therefore, we have
This gives the bound on as asserted in (18). Now, further computations from (21), (23), (24) and (25) lead to
This equation, together with the well-known estimates:
lead us to inequality (19). Therefore, the proof of Theorem 6 is completed. □
Kuroki K, Owa S: Notes on new class for certain analytic functions. RIMS Kokyuroku 2011, 1772: 21–25.
Miller SS, Mocanu PT Series of Monographs and Textbooks in Pure and Applied Mathematics 225. In Differential Subordinations: Theory and Applications. Dekker, New York; 2000.
Srivastava HM, Mishra AK, Gochhayat P: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 2010, 23: 1188–1192. 10.1016/j.aml.2010.05.009
Xu QH, Gui YC, Srivastava HM: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl. Math. Lett. 2012, 25: 990–994. 10.1016/j.aml.2011.11.013
Ali RM, Lee SK, Ravichandran V, Supramaniam S: Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions. Appl. Math. Lett. 2012, 25: 344–351. 10.1016/j.aml.2011.09.012
Keogh F, Merkers E: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 1969, 20: 8–12. 10.1090/S0002-9939-1969-0232926-9
The research was supported by Kyungsung University Research Grants in 2013.
The authors declare that they have no competing interests.
The corresponding author, OSK carried out the subclasses of analytic functions studies and conceived of the study. YJS participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
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Cite this article
Sim, Y.J., Kwon, O.S. Notes on analytic functions with a bounded positive real part. J Inequal Appl 2013, 370 (2013). https://doi.org/10.1186/1029-242X-2013-370
- starlike functions
- bounded positive real part
- Fekete-Szegö problem
- bi-univalent functions