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Cesáro partial sums of certain analytic functions
Journal of Inequalities and Applications volume 2013, Article number: 51 (2013)
Abstract
The aim of the present paper is to consider geometric properties such as starlikeness and convexity of the Cesáro partial sums of certain analytic functions in the open unit disk. By using the Cesáro partial sums, we improve some recent results including the radius of convexity.
AMS Subject Classification:30C45.
1 Introduction
Let be a unit disk in the complex plane ℂ and let ℋ denote the space of all analytic functions on U. Here we suppose that ℋ is a topological vector space endowed with the topology of uniform convergence over compact subsets of U. Also, for and , let be the subspace of ℋ consisting of functions of the form . Further, let and denote the class of univalent functions in . A function is called starlike if is a starlike domain with respect to the origin, and the class of univalent starlike functions is denoted by . It is called convex , if is a convex domain. Each univalent starlike function f is characterized by the analytic condition
Also, it is known that is starlike if and only if f is convex, which is characterized by the analytic condition
For a function , we introduce the partial sum of by
For the partial sums of , Szegö [1] showed that if , then for and for . Owa [2] considered the starlikeness and convexity of partial sums,
of certain functions in the unit disk. Moreover, Darus and Ibrahim [3] determined the conditions under which the partial sums of functions of bounded turning are also of bounded turning.
In this paper, we consider the Cesáro partial sums, it is showed that this kind of partial sums preserve the properties of the analytic functions in the unit disk. Robertson [4] showed that if is univalent, then also all the Cesáro sums are univalent in the unit disk. Moreover, if its ordinary partial sums (1) is univalent in U, then the Cesáro sums are univalent. By employing the concept of the subordination, these results were extended by Ruscheweyha and Salinas [5]. The classical Cesáro means play an important role in geometric function theory (see [6–10]).
From the partial sum
with , we construct the Cesáro means of by
where
Our aim is to consider geometric properties such as starlikeness and convexity of the Cesáro partial sums of certain analytic functions in the open unit disk.
2 Main results
We define the function which is a partial sum of by
Theorem 1 The function satisfies
for
Furthermore, for
Proof
Noting that
it follows that for , we obtain
Moreover, we also observe that
Now assume that
for
This completes the proof. □
Remark 2 For example, the values , and imply the radius of starlikeness of is , and for the same values, the radius of starlikeness of the ordinary partial sums is (see [2]).
Next, we derive the radius of convexity.
Theorem 3 The function satisfies
for
Furthermore, for
Proof
A computation gives
Therefore, for , we obtain
Moreover, we impose
Now, consider that
for
This completes the proof. □
Remark 4 In view of Theorem 3, for example, the values , and pose the radius of convexity of is and for the same values, the radius of convexity of the ordinary partial sums is (see [2]).
Next, we assume special ordinary partial sums depending so that their coefficients satisfy the relation .
Theorem 5 Assume the partial sum
Then the function .
Proof We consider α such that
This implies that
that is,
By letting , we define the function as follows:
Logarithmic derivative of yields
where
Now, for all and , the function has unique real negative zeros in the interval . This leads to the fact that has unique positive real zeros for all distributed in the interval . Therefore, we will calculate α in . It is easy to check that is decreasing for in the interval . Moreover, we have
We conclude that for ,
thus . This completes the proof. □
By letting in Theorem 5, we have the following result.
Corollary 6 The Cesáro partial sums
of the function are starlike of order .
Theorem 7 Assume the partial sum as in Theorem 5. Then the function .
Proof We consider α such that
This implies that
therefore, a computation gives
thus
By putting , we define the function as follows:
Logarithmic derivative of yields
where
Now, for all and , the function has unique real negative zeros in the interval . This leads to the fact that has unique positive real zeros for all in the interval . So, we calculate α in the interval . A computation yields is decreasing for in the interval . Thus, we have
which implies . This completes the proof. □
Theorem 8 Assume the Cesáro partial sum
of the function
Then the function for all .
Proof We consider α such that
This implies that
therefore, a computation gives
thus
By putting , we define the function as follows:
Logarithmic derivative of yields
The function has a unique real negative zero in the interval for all which is around . This leads to the fact that has a unique positive real zero in the interval around . A computation yields is decreasing in the interval and assuming its maximums at and . Thus, we have
which implies . This completes the proof. □
Note that some other results related to partial sums can be found in [11–15].
References
Szego G: Zur theorie der schlichten abbilungen. Math. Ann. 1928, 100: 188–211. 10.1007/BF01448843
Owa S: Partial sums of certain analytic functions. Int. J. Math. Math. Sci. 2001, 25(12):771–775. 10.1155/S0161171201005099
Darus M, Ibrahim RW: Partial sums of analytic functions of bounded turning with applications. Comput. Appl. Math. 2010, 29(1):81–88.
Robertson MS: On the univalency of Cesáro sums of univalent functions. Bull. Am. Math. Soc. 1936, 42: 241–243. 10.1090/S0002-9904-1936-06279-8
Ruscheweyha S, Salinas LC: Subordination by Cesáro means. Complex Var. Elliptic Equ. 1993, 3(21):279–285.
Splina LT: On certain applications of the Hadamard product. Appl. Math. Comput. 2008, 199: 653–662. 10.1016/j.amc.2007.10.031
Darus M, Ibrahim RW: On Cesáro means for Fox-Wright functions. J. Math. Stat. 2008, 4(3):156–160.
Darus M, Ibrahim RW: On some properties of differential operator. Acta Didact. Napoc. 2009, 2(2):1–6.
Darus M, Ibrahim RW: Coefficient inequalities for concave Cesáro operator of non-concave analytic functions. Eur. J. Pure Appl. Math. 2010, 3(6):1086–1092.
Srivastava HM, Darus M, Ibrahim RW: Classes of analytic functions with fractional powers defined by means of a certain linear operator. Integral Transforms Spec. Funct. 2011, 1(22):17–28.
Frasin BA: Generalization of partial sums of certain analytic and univalent functions. Appl. Math. Lett. 2008, 21: 735–741. 10.1016/j.aml.2007.08.002
Wang Z-G, Liu Z-H, Catas A: On neighborhoods and partial sums of certain meromorphic multivalent functions. Appl. Math. Lett. 2011, 24: 864–868. 10.1016/j.aml.2010.12.033
Murugusundaramoorthy G, Uma K, Darus M: Partial sums of generalized class of analytic functions involving Hurwitz-Lerch zeta function. Abstr. Appl. Anal. 2011., 2011: Article ID 849250. doi:10.1155/2011/849250
Ghanim F, Darus M: Partial sums of certain new subclasses for meromorphic functions. Far East J. Math. Sci. 2011, 55(2):181–195.
Ibrahim RW, Darus M: Partial sums for certain classes of meromorphic functions. Tamkang J. Math. 2010, 41(1):39–49.
Acknowledgements
The work was fully supported by UKM-DLP-2011-050 and LRGS/TD/2011/UKM/ICT/03/02. The authors also would like to thank the referees for giving some suggestions for improving the work.
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Both authors jointly worked on deriving the results and approved the final manuscript.
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Ibrahim, R.W., Darus, M. Cesáro partial sums of certain analytic functions. J Inequal Appl 2013, 51 (2013). https://doi.org/10.1186/1029-242X-2013-51
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DOI: https://doi.org/10.1186/1029-242X-2013-51
Keywords
- analytic function
- univalent functions
- starlike functions
- unit disk
- Cesáro partial sums
- partial sums