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Convolution Properties for Analytic Functions Defined by q−Mittag-Leffler Function

Year 2021, Volume: 16 Issue: 1, 259 - 270, 27.05.2021
https://doi.org/10.29233/sdufeffd.850713

Abstract

Recently, Mittag-Leffler function acts as beneficial role in studies of univalent functions. In this paper, motivated by some recent works, two new subclasses of univalent functions involving q−Mittag-Leffler function are introduced. Also, convolution conditions and coefficient estimates for these classes are investigated. The obtained results are presented in comparison with the results in the literature.

References

  • [1] J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,” Ann. Math., 17, 12-22, 1915-1916.
  • [2] T. M. Seoudy and M. K. Aouf, “Convolution properties for certain classes of analytic functions defined by q−derivative operator,” Abstract and Applied Analysis, Volume 2014, Article ID 846719, 7 pages.
  • [3] D. Bansal and J. K. Prajabat, “Certain geometric properties of the Mittag-Leffler functions,” Complex Var. Elliptic Eq., 61, 338-350, 2016.
  • [4] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, “Mittag-Leffler functions, related topics and applications,” Springer, New-York, 2014.
  • [5] A. W. Goodman, “Univalent functions,” Volume I and Volume II. Mariner Pub. Co. Inc. Tampa Florida, 1984.
  • [6] F. H. Jackson, “On q−functions and a certain difference operator,” Trans. Royal Soc, Edinburgh, 46, 253-281, 1909.
  • [7] F. H. Jackson, “q−difference equations,” Amer. J. Math., 32, 305-314, 1910.
  • [8] F. H. Jackson,“On q−definite integrals,” Quart. J. Pure Appl. Math., 41, 193-203, 1910.
  • [9] W. Janowski, “Some extremal problems for certain families of analytic functions I” Ann. Polon. Math,. 28, 297-326, 1973.
  • [10] V. Kac and P. Cheung, “Quantum calculus,” Springer, 2002.
  • [11] W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” In: Proceedings of the Conference on Complex Analysis, Tianjin, 157–169, 1992.
  • [12] G. M. Mittag-Leffler, “Sur la nouvelle fonction Eα(x),” C. R. Acad. Sci. Paris, 137, 554-558, 1903.
  • [13] S.K. Sharma and R. Jain, “On some properties of generalized q−Mittag-Leffler function,” Mathematica Aeterna, 4, 613–619, 2014.
  • [14] A. Wiman, “Über den fundamental satz in der teorie der functionen Eα(x) ,”Acta Math., 29, 191-201, 1905.

q−Mittag-Leffler Fonksiyonu ile Tanımlı Analitik Fonksiyonlar için Konvolüsyon Özellikleri

Year 2021, Volume: 16 Issue: 1, 259 - 270, 27.05.2021
https://doi.org/10.29233/sdufeffd.850713

Abstract

Son zamanlarda, Mittag-Leffler fonksiyonu yalınkat fonksiyonlar ile ilgili çalışmalarda önemli rol oynamaktadır. Bu makalede, yapılan son çalışmaların ışığında q−Mittag-Leffler fonksiyonu ile tanımlı yalınkat fonksiyonların iki yeni alt sınıfı tanımlanmıştır. Bu sınıflar için konvolüsyon koşulları ve katsayı tahminleri araştırılmıştır. Elde edilen bulgular literatürde olan bulgularla karşılaştırılarak sunulmuştur.

References

  • [1] J. W. Alexander, “Functions which map the interior of the unit circle upon simple regions,” Ann. Math., 17, 12-22, 1915-1916.
  • [2] T. M. Seoudy and M. K. Aouf, “Convolution properties for certain classes of analytic functions defined by q−derivative operator,” Abstract and Applied Analysis, Volume 2014, Article ID 846719, 7 pages.
  • [3] D. Bansal and J. K. Prajabat, “Certain geometric properties of the Mittag-Leffler functions,” Complex Var. Elliptic Eq., 61, 338-350, 2016.
  • [4] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, “Mittag-Leffler functions, related topics and applications,” Springer, New-York, 2014.
  • [5] A. W. Goodman, “Univalent functions,” Volume I and Volume II. Mariner Pub. Co. Inc. Tampa Florida, 1984.
  • [6] F. H. Jackson, “On q−functions and a certain difference operator,” Trans. Royal Soc, Edinburgh, 46, 253-281, 1909.
  • [7] F. H. Jackson, “q−difference equations,” Amer. J. Math., 32, 305-314, 1910.
  • [8] F. H. Jackson,“On q−definite integrals,” Quart. J. Pure Appl. Math., 41, 193-203, 1910.
  • [9] W. Janowski, “Some extremal problems for certain families of analytic functions I” Ann. Polon. Math,. 28, 297-326, 1973.
  • [10] V. Kac and P. Cheung, “Quantum calculus,” Springer, 2002.
  • [11] W. C. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” In: Proceedings of the Conference on Complex Analysis, Tianjin, 157–169, 1992.
  • [12] G. M. Mittag-Leffler, “Sur la nouvelle fonction Eα(x),” C. R. Acad. Sci. Paris, 137, 554-558, 1903.
  • [13] S.K. Sharma and R. Jain, “On some properties of generalized q−Mittag-Leffler function,” Mathematica Aeterna, 4, 613–619, 2014.
  • [14] A. Wiman, “Über den fundamental satz in der teorie der functionen Eα(x) ,”Acta Math., 29, 191-201, 1905.
There are 14 citations in total.

Details

Primary Language Turkish
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Asena Çetinkaya 0000-0002-8815-5642

Oya Mert 0000-0002-8791-3341

Publication Date May 27, 2021
Published in Issue Year 2021 Volume: 16 Issue: 1

Cite

IEEE A. Çetinkaya and O. Mert, “q−Mittag-Leffler Fonksiyonu ile Tanımlı Analitik Fonksiyonlar için Konvolüsyon Özellikleri”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 16, no. 1, pp. 259–270, 2021, doi: 10.29233/sdufeffd.850713.