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mmr 2020, 6(3): 441-448 |
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Kargar R, Ebadian A, Kanzi N. Sharp Estimates of Logarithmic Coefficients of Certain Class of Analytic Functions. mmr 2020; 6 (3) :441-448
URL: http://mmr.khu.ac.ir/article-1-2752-en.html
URL: http://mmr.khu.ac.ir/article-1-2752-en.html
1- Department of mathematics, Faculty of Science, Urmia University, Urmia, Iran.
2- ,n.kanzi@pnu.ac.ir
2- ,
Abstract: (2434 Views)
Introduction
Let be the open unit disc in the complex plane and be the class of all functions of which are analytic and normalized in The subclass of consisting of all univalent functions in is denoted by We say that a function is said to be starlike function if and only if for all We denote by the class of all satrlike functions in If and are two of the functions in then we say that is subordinate to written or if there exists a Schwartz function such that for all Furthermore, if the function is univalent in then we have the following equivalence: Also for and their Hadamard product (or convolution) is defined by The logarithmic coefficients of , denoted by , are defined by These coefficients play an important role for various estimates in the theory of univalent functions. For example, consider the Koebe function where It is easy to see that the above function has logarithmic coefficients where and Also for the function we have and the sharp estimates and hold. We remark that the Fekete-Szego theorem is used. For , the problem seems much harder and no significant upper bounds for when appear to be known. Moreover, the problem of finding the sharp upper bound for for is still open for . The sharp upper bounds for modulus of logarithmic coefficients are known for functions in very few subclasses of . For functions in the class it is easy to prove that for and the equality holds for the Koebe function. The celebrated de Branges' inequalities (the former Milin conjecture) for univalent functions state that where with the equality if and only if De Branges used this inequality to prove the celebrated Bieberbach conjecture. Moreover, the de Branges' inequalities have also been the source of many other interesting inequalities involving logarithmic coefficients of such as Let denote the class of functions and satisfying the following subordination relation where .
Material and methods
In this paper, first we obtain a subordination relation for the class and by making use of this relation we give two sharp estimates for the logarithmic coefficients of the function
Results and discussion
We obtain two sharp estimates for the logarithmic coefficients of the function
Conclusion
The following conclusions were drawn from this research.
Let be the open unit disc in the complex plane and be the class of all functions of which are analytic and normalized in The subclass of consisting of all univalent functions in is denoted by We say that a function is said to be starlike function if and only if for all We denote by the class of all satrlike functions in If and are two of the functions in then we say that is subordinate to written or if there exists a Schwartz function such that for all Furthermore, if the function is univalent in then we have the following equivalence: Also for and their Hadamard product (or convolution) is defined by The logarithmic coefficients of , denoted by , are defined by These coefficients play an important role for various estimates in the theory of univalent functions. For example, consider the Koebe function where It is easy to see that the above function has logarithmic coefficients where and Also for the function we have and the sharp estimates and hold. We remark that the Fekete-Szego theorem is used. For , the problem seems much harder and no significant upper bounds for when appear to be known. Moreover, the problem of finding the sharp upper bound for for is still open for . The sharp upper bounds for modulus of logarithmic coefficients are known for functions in very few subclasses of . For functions in the class it is easy to prove that for and the equality holds for the Koebe function. The celebrated de Branges' inequalities (the former Milin conjecture) for univalent functions state that where with the equality if and only if De Branges used this inequality to prove the celebrated Bieberbach conjecture. Moreover, the de Branges' inequalities have also been the source of many other interesting inequalities involving logarithmic coefficients of such as Let denote the class of functions and satisfying the following subordination relation where .
Material and methods
In this paper, first we obtain a subordination relation for the class and by making use of this relation we give two sharp estimates for the logarithmic coefficients of the function
Results and discussion
We obtain two sharp estimates for the logarithmic coefficients of the function
Conclusion
The following conclusions were drawn from this research.
- Logarithmic coefficients of the function are estimated.
Keywords: Univalent functions, Starlikeness, Subordination, Logarithmic coefficients, Hadamard product.
Type of Study: S |
Subject:
alg
Received: 2018/03/4 | Revised: 2021/01/9 | Accepted: 2019/03/2 | Published: 2020/11/30 | ePublished: 2020/11/30
Received: 2018/03/4 | Revised: 2021/01/9 | Accepted: 2019/03/2 | Published: 2020/11/30 | ePublished: 2020/11/30
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