Research Article | Open Access
Ding-Gong Yang, Jin-Lin Liu, "A Class of Analytic Functions with Missing Coefficients", Abstract and Applied Analysis, vol. 2011, Article ID 456729, 16 pages, 2011. https://doi.org/10.1155/2011/456729
A Class of Analytic Functions with Missing Coefficients
Abstract
Let and denote the class of functions of the form which are analytic in the open unit disk and satisfy the following subordination condition , for, for. We obtain sharp bounds on , and coefficient estimates for functions belonging to the class . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.
1. Introduction
Let denote the class of functions of the form which are analytic in the open unit disk . Let and denote the subclasses of whose members are univalent and starlike, respectively.
For functions and analytic in , we say that is subordinate to in and we write , if there exists an analytic function in such that Furthermore, if the function is univalent in , then Throughout our present discussion, we assume that We introduce the following subclass of .
Definition 1.1. A function is said to be in the class if it satisfies where The classes have been studied by several authors (see [1–5]). Recently, Gao and Zhou [6] showed some mapping properties of the following subclass of : Note that For further information of the above classes (with ) and related analytic function classes, see Srivastava et al. [7], Yang and Liu [8], Kim [9], and Kim and Srivastava [10].
In this paper, we obtain sharp bounds on , and coefficient estimates for functions belonging to the class . Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also presented. One can see that the methods used in [6] do not work for the more general class than .
2. The bounds on , , and in
In this section, we let where and With (2.1), it is easily seen that the function given by (1.6) can be expressed as
Theorem 2.1. Let . Then, for , The bounds in (2.4) are sharp for the function defined by
Proof. The analytic function given by (1.6) is convex (univalent) in (cf. [11]) and satisfies . Thus, for ,
Let . Then, we can write
where is analytic and for . By the Schwarz lemma, we know that . It follows from (2.7) that
which leads to
or to
Since
we deduce from (2.6) and (2.10) that
Now, by using (2.3) and (2.12), we can obtain (2.4).
Furthermore, for the function defined by (2.5), we find that
Hence, and from (2.13), we see that the bounds in (2.4) are the best possible.
Hereafter, we write
Corollary 2.2. Let . Then, for , The results are sharp.
Proof. For , it follows from (2.12) (used in the proof of Theorem 2.1) that for . From these, we have the desired results.
The bounds in (2.16) and (2.17) are sharp for the function
Theorem 2.3. Let . Then, for , The results are sharp.
Proof. Noting that an application of Theorem 2.1 yields (2.20). Furthermore, the results are sharp for the function defined by (2.5).
Corollary 2.4. Let . Then, for , The results are sharp for the function defined by (2.19).
Proof. For , it follows from (2.6) and (2.10) (with ) that for and . Making use of (2.21) and (2.23), we can obtain (2.22).
Theorem 2.5. Let and and ). If then , where the symbol stands for the familiar Hadamard product (or convolution) of two analytic functions in .
Proof. Since and ), it follows from Corollary 2.4 (with ) and (2.24) that
Thus, has the Herglotz representation
where is a probability measure on the unit circle and .
For , we have
where
In view of the function is convex (univalent) in , we deduce from (2.26) to (2.28) that
This shows that .
Corollary 2.6. Let , and Then, .
Proof. By taking , and , (2.24) in Theorem 2.5 becomes that is, Hence, the desired result follows as a special case from Theorem 2.5.
Remark 2.7. R. Singh and S. Singh [4, Theorem 3] proved that, if and belong to , then . Obviously, for Corollary 2.6 generalizes and improves Theorem 3 in [4].
Theorem 2.8. Let and . Then, for , The result is sharp, with the extremal function defined by (2.5).
Proof. It is well known that for and , Since , we have and so (2.35) leads to By virtue of (1.6), (2.10), and (2.36), we have for and . Now, by using (2.3), (2.21) and (2.37), we can obtain (2.34).
Theorem 2.9. Let Then, The result is sharp for each .
Proof. It is known (cf. [12]) that, if
where is analytic in and is analytic and convex univalent in , then .
By (2.38), we have
where
and is given by (1.6). Since the function is analytic and convex univalent in , it follows from (2.41) that
which gives (2.39).
Next, we consider the function
It is easy to verify that
The proof of Theorem 2.9 is completed.
3. The Univalency and Starlikeness of
Theorem 3.1. if and only if
Proof. Let and (3.1) be satisfied. Then, by (2.16) in Corollary 2.2, we see that . Thus, is close-to-convex and univalent in .
On the other hand, if
then the function defined by (2.19) satisfies and
as . Hence, there exists a point such that . This implies that is not univalent in and so the theorem is proved.
Theorem 3.2. Let (3.1) in Theorem 3.1 be satisfied. If and then .
Proof. We first show that
Equation (3.5) is obvious when . For , we have
where
Since and
is a monotonically decreasing sequence. Therefore, the inequality (3.5) follows from (3.6).
Let . Then,
Define in by
In view of (3.1) in Theorem 3.1 is satisfied, the function is univalent in , and so is analytic in . Also it follows from (3.10) that
We want to prove now that for . Suppose that there exists a point such that Then, applying a result of Miller and Mocanu [13, Theorem 4], we have For , we deduce from Corollaries 2.2 and 2.4, (3.1), (3.5), (3.11), (3.13), and (3.4) that But this contradicts (3.9) at . Therefore, we must have and the proof of Theorem 3.2 is completed.
Remark 3.3. In [6, Theorem 4(ii)], the authors gave the following: if and is the solution of the equation then for . However, this result is not true because the series in (3.15) diverges.
4. The Radius of Convexity
Theorem 4.1. Let belong to the class defined by and . Then, where is the root in of the equation The result is sharp.
Proof. For , we can write where is analytic and in . Differentiating both sides of (4.4) logarithmically, we arrive at Put and . Then, (4.4) implies that With the help of the Carathéodory inequality it follows from (4.5) and (4.6) that where and because of (4.6) and (4.7). In view of (4.10) and (4.11), we see that
Let us now calculate the minimum value of on the closed interval . Noting that and (4.7), we deduce from (4.12) that where Also and . Suppose that . Then, Hence, by virtue of the mathematical induction, we have for all and . This implies that In view of (4.14) and (4.18), we see that Further it follows from (4.9), (4.12), and (4.19) that where and Note that and . If we let denote the root in of the equation , then (4.20) yields the desired result (4.2).
To see that the bound is the best possible, we consider the function It is clear that for , which shows that the bound cannot be increased.
Setting , Theorem 4.1 reduces to the following result.
Corollary 4.2. Let and . Then, is convex of order in The result is sharp.
References
- R. M. Ali, “On a subclass of starlike functions,” The Rocky Mountain Journal of Mathematics, vol. 24, no. 2, pp. 447–451, 1994. View at: Publisher Site | Google Scholar
- P. N. Chichra, “New subclasses of the class of close-to-convex functions,” Proceedings of the American Mathematical Society, vol. 62, no. 1, pp. 37–43, 1977. View at: Google Scholar
- H. Silverman, “A class of bounded starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol. 17, no. 2, pp. 249–252, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- R. Singh and S. Singh, “Convolution properties of a class of starlike functions,” Proceedings of the American Mathematical Society, vol. 106, no. 1, pp. 145–152, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- S. Ponnusamy and V. Singh, “Criteria for strongly starlike functions,” Complex Variables, vol. 34, no. 3, pp. 267–291, 1997. View at: Google Scholar | Zentralblatt MATH
- C. Y. Gao and S. Q. Zhou, “Certain subclass of starlike functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 176–182, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
- H. M. Srivastava, N. E. Xu, and D. G. Yang, “Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 686–700, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
- D. G. Yang and J. L. Liu, “On a class of analytic functions with missing coefficients,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3473–3481, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- Y. C. Kim, “Mapping properties of differential inequalities related to univalent functions,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 272–279, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
- Y. C. Kim and H. M. Srivastava, “Some applications of a differential subordination,” International Journal of Mathematics and Mathematical Sciences, vol. 22, no. 3, pp. 649–654, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
- H. M. Srivastava, D. G. Yang, and N. E. Xu, “Some subclasses of meromorphically multivalent functions associated with a linear operator,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 11–23, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
- P. L. Duren, Univalent Functions, vol. 259 of Fundamental Principles of Mathematical Sciences, Springer, New York, NY, USA, 1983.
- S. S. Miller and P. T. Mocanu, “Second order differential inequalities in the complex plane,” Journal of Mathematical Analysis and Applications, vol. 65, no. 2, pp. 289–305, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright
Copyright © 2011 Ding-Gong Yang and Jin-Lin Liu. 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.