AIMS Mathematics, 2020, 5(6): 7087-7106. doi: 10.3934/math.2020454

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

The Faber polynomial expansion method and the Taylor-Maclaurin coefficient estimates of Bi-Close-to-Convex functions connected with the q-convolution

1 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
2 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China
3 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
4 Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
5 Department of Mathematics, College of Science and Arts in Badaya, Qassim University, P.O. Box: 6640, Buraidah 51452, Saudi Arabia

In this paper, we introduce a new class of analytic and bi-close-to-convex functions connected with q-convolution, which are defined in the open unit disk. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this subclass by using the Faber polynomial expansion method. Several corollaries and consequences of our main results are also briefly indicated.
  Figure/Table
  Supplementary
  Article Metrics

References

1. H. Aldweby, M. Darus, On a subclass of bi-univalent functions associated with the q-derivative operator, J. Math. Comput. Sci., 19 (2019), 58-64.    

2. M. Arif, M. Ul Haq, J. L. Liu, A subfamily of univalent functions associated with q-analogue of Noor integral operator, J. Funct. Spaces, 2018 (2018), 1-5.

3. L. de Branges, A proof of the Bieberbach conjecture, Acta Math., 154 (1985), 137-152.    

4. D. A. Brannan, J. G. Clunie, Aspects of contemporary complex analysis, Proceedings of the NATO Advanced Study Institute (University of Durham, Durham; (1979), 1-20), New York and London: Academic Press, (1980), 79-95.

5. D. A. Brannan, J. Clunie, W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 22 (1970), 476-485.    

6. D. A. Brannan, T. S. Taha, On some classes of bi-unvalent functions, In: Mathematical Analysis and Its Applications (Kuwait; (1985), 18-21) (S. M. Mazhar, A. Hamoui, N. S. Faour, Eds.), KFAS Proceedings Series, 3, Oxford: Pergamon Press (Elsevier Science Limited), (1988), 53-60; see also Studia Univ. Babeş-Bolyai Math., 31 (1986), 70-77.

7. T. Bulboacă, Differential subordinations and superordinations: Recent results, House of Scientific Book Publishers, Cluj-Napoca, 2005.

8. S. Bulut, Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions, Filomat, 30 (2016), 1567-1575.    

9. M. Çaglar, E. Deniz, Initial coefficients for a subclass of bi-univalent functions defined by Sălăgean differential operator, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 66 (2017), 85-91.

10. P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, Bd. 259, New York, Berlin, Heidelberg and Tokyo: Springer-Verlag, 1983.

11. S. M. El-Deeb, Maclaurin coefficient estimates for new subclasses of bi-univalent functions connected with a q-analogue of Bessel function, Abstr. Appl. Anal., 2020 (2020), 1-7.

12. S. M. El-Deeb, T. Bulboacă, Fekete-Szegö inequalities for certain class of analytic functions connected with q-analogue of Bessel function, J. Egyptian Math. Soc., 27 (2019), 1-11.    

13. S. M. El-Deeb, T. Bulboacă, Differential sandwich-type results for symmetric functions connected with a q-analog integral operator, Mathematics, 7 (2019), 1-17.

14. S. M. El-Deeb, T. Bulboacă, B. M. El-Matary, Maclaurin coefficient estimates of bi-univalent functions connected with the q-derivative, Mathematics, 8 (2020), 1-14.

15. S. M. El-Deeb, T. Bulboacă, J. Dziok, Pascal distribution series connected with certain subclasses of univalent functions, Kyungpook Math. J., 59 (2019), 301-314.

16. S. M. El-Deeb, H. Orhan, Fekete-Szegö problem for certain class of analytic functions connected with q-analogue of integral operator, Ann. Univ. Oradea Fasc. Math., 28 (2021). In press.

17. S. Elhaddad, M. Darus, Coefficient estimates for a subclass of bi-univalent functions defined by q-derivative operator, Mathematics, 8 (2020), 1-14.

18. G. Faber, Über polynomische Entwickelungen, Math. Ann., 57 (1903), 389-408.    

19. G. Gasper, M. Rahman, Basic hypergeometric series (with a Foreword by Richard Askey), Encyclopedia of Mathematics and Its Applications, 35, Cambridge, New York, Port Chester, Melbourne and Sydney: Cambridge University Press, 1990; 2 Eds., Encyclopedia of Mathematics and Its Applications, 96, Cambridge, London and New York: Cambridge University Press, 2004.

20. M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77-84.    

21. F. H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46 (1908), 253-281.

22. F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.

23. P. N. Kamble, M. G. Shrigan, Coefficient estimates for a subclass of bi-univalent functions defined by Sălăgean type q-calculus operator, Kyungpook Math. J., 58 (2018), 677-688.

24. W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), 169-185.

25. Q. Khan, M. Arif, M. Raza, et al. Some applications of a new integral operator in q-analog for multivalent functions, Mathematics, 7 (2019), 1-13.

26. M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63-68.    

27. S. Mahmood, N. Raza, E. S. A. Abujarad, et al. Geometric properties of certain classes of analytic functions associated with a q-integral operator, Symmetry, 11 (2019), 1-14.

28. S. S. Miller, P. T. Mocanu, Differential subordination: Theory and applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, New York and Basel: Marcel Dekker Incorporated, 225 (2000).

29. 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 (1969), 100-112.

30. S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal., 2014 (2014), 1-3.

31. F. M. Sakar, H. Ö. Güeny, Coefficient bounds for a new subclass of analytic bi-close-to-convex functions by making use of Faber polynomial expansion, Turkish J. Math., 41 (2017), 888-895.    

32. L. Shi, Q. Khan, G. Srivastava, et al. A study of multivalent q-starlike functions connected with circular domain, Mathematics, 7 (2019), 1-12.

33. H. M. Srivastava, Certain q-polynomial expansions for functions of several variables, I and II, IMA J. Appl. Math., 30 (1983), 315-323; ibid. 33 (1984), 205-209.    

34. H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: Univalent functions, fractional calculus, and their applications (H. M. Srivastava, S. Owa, Eds.), New York, Chichester, Brisbane and Toronto: Halsted Press (Ellis Horwood Limited, Chichester) and John Wiley and Sons, (1989), 329-354.

35. H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A: Sci., 44 (2020), 327-344.    

36. H. M. Srivastava, Q. Z. Ahmad, N. Khan, et al. Some applications of higher-order derivatives involving certain subclasses of analytic and multivalent functions, J. Nonlinear Var. Anal., 2 (2018), 343-353.

37. H. M. Srivastava, Ş. Altinkaya, S. Yalçn, Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator, Filomat, 32 (2018), 503-516.    

38. H. M. Srivastava, Ş. Altınkaya, S. Yalçin, Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran. J. Sci. Technol. Trans. A: Sci., 43 (2019), 1873-1879.    

39. H. M. Srivastava, S. Arjika, A. S. Kelil, Some homogeneous q-difference operators and the associated generalized Hahn polynomials, Appl. Set-Valued Anal. Optim., 1 (2019), 187-201.

40. H. M. Srivastava, D. Bansal, Close-to-convexity of a certain family of q-Mittag-Leffler functions, J. Nonlinear Var. Anal., 1 (2017), 61-69.

41. H. M. Srivastava, S. S. Eker, R. M. Ali, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29 (2015), 1839-1845.    

42. H. M. Srivastava, S. S. Eker, S. G. Hamidi, et al. Faber polynomial coefficient estimates for biunivalent functions defined by the Tremblay fractional derivative operator, Bull. Iran. Math. Soc., 44 (2018), 149-157.    

43. H. M. Srivastava, S. M. El-Deeb, A certain class of analytic functions of complex order connected with a q-analogue of integral operators, Miskolc Math. Notes, 21 (2020), 417-433.    

44. H. M. Srivastava, S. Gaboury, F. Ghanim, Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma-Minda type, Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat., (RACSAM), 112 (2018), 1157-1168.

45. H. M. Srivastava, P. W. Karlsson, Multiple Gaussian Hypergeometric Series, New York, Chichester, Brisbane and Toronto: Halsted Press (Ellis Horwood Limited, Chichester) and John Wiley and Sons, 1985.

46. H. M. Srivastava, B. Khan, N. Khan, et al. Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48 (2019), 407-425.    

47. H. M. Srivastava, S. Khan, Q. Z. Ahmad, et al. The Faber polynomial expansion method and its application to the general coefficient problem for some subclasses of bi-univalent functions associated with a certain q-integral operator, Stud. Univ. Babeş-Bolyai Math., 63 (2018), 419- 436.

48. H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188-1192.    

49. H. M. Srivastava, A. Motamednezhad, E. A. Adegan, Faber polynomial coefficient estimates for biunivalent functions defined by using differential subordination and a certain fractional derivative operator, Mathematics, 8 (2020), 1-12.

50. 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, 34 (2018), 1313- 1322.

51. H. M. Srivastava, N. Raza, E. S. A. AbuJarad, et al. Fekete-Szegö inequality for classes of (p, q)- starlike and (p, q)-convex functions, Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat., (RACSAM), 113 (2019), 3563-3584.

52. H. M. Srivastava, M. Tahir, B. Khan, et al. Some general classes of q-starlike functions associated with the Janowski functions, Symmetry, 11 (2019), 1-14.

53. H. M. Srivastava, M. Tahir, B. Khan, et al. Some general families of q-starlike functions associated with the Janowski functions, Filomat, 33 (2019), 2613-2626.    

54. H. M. Srivastava, A. K. Wanas, Initial Maclaurin coefficient bounds for new subclasses of analytic and m-fold symmetric bi-univalent functions defined by a linear combination, Kyungpook Math. J., 59 (2019), 493-503.

55. H. Tang, Q. Khan, M. Arif, et al. Some applications of a new integral operator in q-analog for multivalent functions, Mathematics, 7 (2019), 1-13.

56. H. E. Ö. Uçar, Coefficient inequality for q-starlike functions, Appl. Math. Comput., 276 (2016), 122-126.

57. B. Wongsaijai, N. Sukantamala, Certain properties of some families of generalized starlike functions with respect to q-calculus, Abstr. Appl. Anal., 2016 (2016), 1-8.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved