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Sabordinasyon ve Fibonacci Sayılar Dizisi ile Tanımlanan Kendisi ve Tersi Yalınkat Fonksiyonların Yeni Bir Alt Sınıfı için Katsayı Eşitsizlikleri

Year 2021, Volume: 16 Issue: 1, 308 - 318, 27.05.2021
https://doi.org/10.29233/sdufeffd.740915

Abstract

Bu çalışmada, geometrik fonksiyonlar teorisinin en önemli araştırma konularından biri olan ve son yıllarda oldukça popüler hale gelen kendisi ve tersi yalınkat fonksiyonlar araştırılmıştır. Fonksiyonlar üzerine yapılan araştırmalarda sınır belirleme çalışması alışılagelen bir durumdur. Bu bağlamda, ilk olarak kendisi ve tersi yalınkat fonksiyonlar sınıfının D={z∈C: |z|<1} açık birim diskinde yeni bir alt sınıfı tanımlanmıştır. Bu alt sınıf tanımlanırken kompleks değerli fonksiyonlar için geliştirilen Komatu integral operatörü ve sabordinasyon prensibi kullanılmıştır. Daha sonra Fibonacci sayı dizisi ile reel kısmı pozitif olan fonksiyonlar arasındaki ilişki verilmiştir. Bu ilişki Bulgular bölümü için temel teşkil etmektedir. Tanımlanan sınıfa ait fonksiyonların ilk iki Taylor Maclaurin katsayıları a2 ve a3 ün modülleri için üst sınırlar araştırılmıştır. Son olarak yine bu sınıfa ait fonksiyonlar için Fekete-Szegö eşitsizlikleri elde edilmiştir. Elde edilen bulgular literatürdeki sonuçlar ile karşılaştırılmıştır.

Supporting Institution

TÜBİTAK

Project Number

118F543

Thanks

Bu çalışma, Şahsene ALTINKAYA’nın yürütücüsü, Meryem YILDIZ’ın bursiyeri olduğu TÜBİTAK 118F543 nolu proje ile desteklenmektedir.

References

  • [1] J. W. Alexander, “Function which map the interior of the unit circle upon simple regions,” Ann. Math., Second Series, 17, 12-22, 1915.
  • [2] O. P. Ahuja, A. Çetinkaya, N. Bohra, “On a class of q-bi-univalent functions of complex order related to shell-like curves connected with the Fibonacci numbers,” Honam Mathematical J., 42, 319-330, 2020.
  • [3] A. Akgül, “(p,q)-Lucas polynomial coefficient inequalities of the bi-univalent function class,” Turk. J. Math., 43, 2170-2176, 2019.
  • [4] A. Akgül, F. M. Sakar, “A certain subclass of bi-univalent analytic functions introduced by means of the q-analogue of Noor integral operator and Horadam polynomials,” Turk. J. Math., 43, 2275-2286, 2019.
  • [5] A. Akgül, “The Fekete–Szegö coefficient inequalitiy for a new class of m-fold symmetric bi-univalent functions satisfying subordination condition,” Honam Mathematical J., 70, 733-748, 2018.
  • [6] A. Akgül, “New subclasses of analytic and bi-univalent functions involving a new integral operator defined by polylogarithm function,” Theory Appl. Math. Comput. Sci., 7, 31-40, 2017.
  • [7] R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramanian, “Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions,” Appl. Math. Lett., 25, 344-351, 2012.
  • [8] Ş. Altınkaya, S. Yalçın, S. Çakmak, “A Subclass of bi-univalent functions based on the Faber polynomial expansions and the Fibonacci numbers,” Mathematics, 7, 1-9, 2019.
  • [9] D. A. Brannan, T. S. Taha, “On some classes of bi-univalent functions,” Stud. Univ. Babeş-Bolyai Math., 31, 70-77, 1986.
  • [10] J. Dziok, R. K. Raina, J. Sokól, “On α-convex functions related to shell-like functions connected with Fibonacci numbers,” Appl. Math. Comput., 218, 996-1002, 2011.
  • [11] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, 1983.
  • [12] M. Fekete, G. Szegö, “Eine bemerkung über ungerade schlichte funktionen,” J. London Math. Soc., [s1-8 (2)], 85-89, 1993.
  • [13] U. Grenander, G. Szegö, Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences Univ, California Press, Berkeley, 1958.
  • [14] H. Ö. Güney, G. Murugusundaramoorthy, J. Sokol, “Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers,” Acta Univ. Sapientiae Mathematica, 10, 70-84, 2018.
  • [15] T. Hayami, S. Owa, “Coefficient bounds for bi-univalent functions,” Pan Amer. Math. J., 22, 15-26, 2012.
  • [16] I. B. Jung, Y. C. Kim, H. M. Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” J. Math. Anal. Appl., 176, 138-147, 1993.
  • [17] Y. Komatu, “ On analytic prolongation of a family of operators,” Math. (Cluj), 32, 141-145, 1990.
  • [18] M. Lewin, “On a coefficient problem for bi-univalent functions,” Proc. Amer. Math. Soc., 18, 63-68, 1967.
  • [19] R. J. Libera, “Some classes of regular univalent functions,” Proc. Amer. Math. Soc., 16, 755-758, 1965.
  • [20] W. C. Ma, D. Minda, “A unified treatment of some special classes of univalent functions,” in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, pp. 157-169, Conf. Proc. Lecture Notes Anal. I, Int. Press, Cambridge, MA, 1994.
  • [21] E. Netanyahu, “ The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1,” Arch. Rational Mech. Anal., 32, 100-112, 1969.
  • [22] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
  • [23] G. S. Sălăgean, “Subclasses of univalent functions,” in: Complex Analysis-Fifth Romanian Finish Seminar, Bucharest, 1983, pp. 362-372.
  • [24] H. M. Srivastava, S. Sümer Eker, S. G. Hamidi, J. M. Jahangiri, “Faber polynomial coefficient estimates for Bi-univalent funnctions defined by the Tremblay fractional derivative operator,” Bull. Iran. Math. Soc., 44, 149-157, 2018.
  • [25] H. M. Srivastava, A. K. Mishra, P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions,” Appl. Math. Lett., 23, 1188-1192, 2010.
  • [26] H. M. Srivastava, F. M. Sakar, H. Ö. Güney, “Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination,” Filomat, 32, 1313-1322, 2018.

Coefficient Inequalities for Bi-univalent Functions Defined by Subordination and A Sequence of Fibonacci Numbers

Year 2021, Volume: 16 Issue: 1, 308 - 318, 27.05.2021
https://doi.org/10.29233/sdufeffd.740915

Abstract

In this work, bi-univalent functions, which are one of the most important research areas of Geometric Function Theory and which are still very popular in recent years, have been investigated. For the studies of functions, it is customary to determine the bound. In this direction, firstly, we introduce a new subclass of bi-univalent functions in the open unit disk D={z∈C: |z|<1}. For this purpose, the Komatu integral operator developed for complex functions and subordination princible are used. Afterwards, the relation between the Fibonacci number sequence and the functions with the positive real part is given. This relation is a fundamental role for Results section. The upper bounds of the first two Taylor-Maclaurin coefficients are investigated. Finally, we derive Fekete-Szegö inequalities for functions belonging to this newly-defined class. The obtained results are compared with studies in the literature.

Project Number

118F543

References

  • [1] J. W. Alexander, “Function which map the interior of the unit circle upon simple regions,” Ann. Math., Second Series, 17, 12-22, 1915.
  • [2] O. P. Ahuja, A. Çetinkaya, N. Bohra, “On a class of q-bi-univalent functions of complex order related to shell-like curves connected with the Fibonacci numbers,” Honam Mathematical J., 42, 319-330, 2020.
  • [3] A. Akgül, “(p,q)-Lucas polynomial coefficient inequalities of the bi-univalent function class,” Turk. J. Math., 43, 2170-2176, 2019.
  • [4] A. Akgül, F. M. Sakar, “A certain subclass of bi-univalent analytic functions introduced by means of the q-analogue of Noor integral operator and Horadam polynomials,” Turk. J. Math., 43, 2275-2286, 2019.
  • [5] A. Akgül, “The Fekete–Szegö coefficient inequalitiy for a new class of m-fold symmetric bi-univalent functions satisfying subordination condition,” Honam Mathematical J., 70, 733-748, 2018.
  • [6] A. Akgül, “New subclasses of analytic and bi-univalent functions involving a new integral operator defined by polylogarithm function,” Theory Appl. Math. Comput. Sci., 7, 31-40, 2017.
  • [7] R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramanian, “Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions,” Appl. Math. Lett., 25, 344-351, 2012.
  • [8] Ş. Altınkaya, S. Yalçın, S. Çakmak, “A Subclass of bi-univalent functions based on the Faber polynomial expansions and the Fibonacci numbers,” Mathematics, 7, 1-9, 2019.
  • [9] D. A. Brannan, T. S. Taha, “On some classes of bi-univalent functions,” Stud. Univ. Babeş-Bolyai Math., 31, 70-77, 1986.
  • [10] J. Dziok, R. K. Raina, J. Sokól, “On α-convex functions related to shell-like functions connected with Fibonacci numbers,” Appl. Math. Comput., 218, 996-1002, 2011.
  • [11] P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, 1983.
  • [12] M. Fekete, G. Szegö, “Eine bemerkung über ungerade schlichte funktionen,” J. London Math. Soc., [s1-8 (2)], 85-89, 1993.
  • [13] U. Grenander, G. Szegö, Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences Univ, California Press, Berkeley, 1958.
  • [14] H. Ö. Güney, G. Murugusundaramoorthy, J. Sokol, “Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers,” Acta Univ. Sapientiae Mathematica, 10, 70-84, 2018.
  • [15] T. Hayami, S. Owa, “Coefficient bounds for bi-univalent functions,” Pan Amer. Math. J., 22, 15-26, 2012.
  • [16] I. B. Jung, Y. C. Kim, H. M. Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” J. Math. Anal. Appl., 176, 138-147, 1993.
  • [17] Y. Komatu, “ On analytic prolongation of a family of operators,” Math. (Cluj), 32, 141-145, 1990.
  • [18] M. Lewin, “On a coefficient problem for bi-univalent functions,” Proc. Amer. Math. Soc., 18, 63-68, 1967.
  • [19] R. J. Libera, “Some classes of regular univalent functions,” Proc. Amer. Math. Soc., 16, 755-758, 1965.
  • [20] W. C. Ma, D. Minda, “A unified treatment of some special classes of univalent functions,” in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, pp. 157-169, Conf. Proc. Lecture Notes Anal. I, Int. Press, Cambridge, MA, 1994.
  • [21] E. Netanyahu, “ The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1,” Arch. Rational Mech. Anal., 32, 100-112, 1969.
  • [22] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
  • [23] G. S. Sălăgean, “Subclasses of univalent functions,” in: Complex Analysis-Fifth Romanian Finish Seminar, Bucharest, 1983, pp. 362-372.
  • [24] H. M. Srivastava, S. Sümer Eker, S. G. Hamidi, J. M. Jahangiri, “Faber polynomial coefficient estimates for Bi-univalent funnctions defined by the Tremblay fractional derivative operator,” Bull. Iran. Math. Soc., 44, 149-157, 2018.
  • [25] H. M. Srivastava, A. K. Mishra, P. Gochhayat, “Certain subclasses of analytic and bi-univalent functions,” Appl. Math. Lett., 23, 1188-1192, 2010.
  • [26] H. M. Srivastava, F. M. Sakar, H. Ö. Güney, “Some general coefficient estimates for a new class of analytic and bi-univalent functions defined by a linear combination,” Filomat, 32, 1313-1322, 2018.
There are 26 citations in total.

Details

Primary Language Turkish
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Meryem Yıldız 0000-0002-7594-7552

Şahsene Altınkaya 0000-0002-7950-8450

Project Number 118F543
Publication Date May 27, 2021
Published in Issue Year 2021 Volume: 16 Issue: 1

Cite

IEEE M. Yıldız and Ş. Altınkaya, “Sabordinasyon ve Fibonacci Sayılar Dizisi ile Tanımlanan Kendisi ve Tersi Yalınkat Fonksiyonların Yeni Bir Alt Sınıfı için Katsayı Eşitsizlikleri”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 16, no. 1, pp. 308–318, 2021, doi: 10.29233/sdufeffd.740915.