New results on coefficient estimates for subclasses of bi-univalent functions related by a new integral operator

Document Type : Research Paper

Authors

1 Department of Mathematics, College of Education for Girls, University of Kufa, Najaf, Iraq

2 Department of Mathematics, College of Science, University of Al-Qadisiyah, Diwaniyah, Iraq

Abstract

In the present paper, we introduce two new subclasses of the function class $\sum$ of bi-univalent functions defined in the open unit disc $U$. Furthermore, we find estimates on the coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses.

Keywords

[1] R. Abd Al-Sajjad and W.G. Atshan, Certain analytic function sandwich theorems involving operator defined by Mittag-Leffler function, AIP Conf. Proc. 2398 (2022), 060065.
[2] S.A. Al-Ameedee, W.G. Atshan and F.A. Al-Maamori, Second Hankel determinant for certain subclasses of biunivalent functions, J. Phys.: Conf. Ser. 1664 (2020), 012044.
[3] S.A. Al-Ameedee, W.G. Atshan and F.A. Al-Maamori, Coefficients estimates of bi-univalent functions defined by new subclass function, J. Phys.: Conf. Ser. 1530 (2020), 012105.
[4] S.A. Al-Ameedee, W.G. Atshan and F.A. Al-Maamori, On sandwich results of univalent functions defined by a linear operator, J. Interdiscip. Math. 23 (2020), no. 4, 803–809.
[5] S.A. Al-Ameedee, W.G. Atshan and F.A. Al-Maamori, Some new results of differential subordinations for Higherorder derivatives of multivalent functions, J. Phys.: Conf. Ser. 1804 (2021), 012111.
[6] W.G. Atshan and A.A.R. Ali, On some sandwich theorems of analytic functions involving Noor-S˜al˜agean operator, Adv. Math.: Sci. J. 9 (2020), no. 10, 8455–8467.
[7] W.G. Atshan and A.A.R. Ali, On sandwich theorems results for certain univalent functions defined by generalized operators, Iraqi J. Sci. 62 (2021), no. 7, 2376–2383.
[8] W.G. Atshan and R.A. Al-Sajjad, Some applications of quasi-subordination for bi-univalent functions using Jackson’s convolution operator, Iraqi J. Sci. 63 (2022), no. 10, 4417–4428.
[9] W.G. Atshan, A.H. Battor and A.F. Abaas, Some sandwich theorems for meromorphic univalent functions defined by new integral operator, J. Interdiscip. Math. 24 (2021), no. 3, 579–591.
[10] W.G. Atshan and S.R. Kulkarni, On application of differential subordination for certain subclass of meromorphically p valent functions with positive coefficients defined by linear operator, J. Inequal. Pure Appl. Math. 10 (2009), no. 2, 11.
[11] W.G. Atshan, I.A.R. Rahman and A.A. Lupas, Some results of new subclasses for bi-univalent functions using quasi-subordination, Symmetry 13 (2021), no. 9, p. 1653.
[12] W.G. Atshan, S. Yalcin and R.A. Hadi, Coefficient estimates for special subclasses of k-fold symmetric bi-univalent functions, Math. Appl. 9 (2020), no. 2, 83–90.
[13] D.A. Brannan, J. Clunie and W.E. Kirwan, Coefficient estimates for a class of starlike functions, Canad. J. Math. 22 (1970), 476–485.
[14] D.A. Brannan and T.S. Taha, On some classes of bi-univalent functions, Stud. Univ. Babes-Bolyai Math. 31 (1986), no. 2, 70–77.
[15] S. Bulut, Coefficient estimates for a class of analytic and bi-univalent functions, Novi. Sad. J. Math. 43 (2013), 59–65.
[16] N.E. Cho, O.S. Kwon and S. Owa, Certain subclasses of Sakaguchi functions, SEA Bull. Math. 17 (1993), 121–126.
[17] P.L. Duren, Univalent functions, Springer Science & Business Media, 2001.
[18] B.A. Frasin and M.K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), 1569–1573.
[19] I.A. Kadum, W.G. Atshan and A.T. Hameed, Sandwich theorems for a new class of complete homogeneous symmetric functions by using cyclic operator, Symmetry 14 (2022), no. 10, 2223.
[20] S. Kanas and H.E. Darwish, Fekete-Szego problem for starlike and convex functions of complex order, Appl. Math. Lett. 23 (2010), 777–782.
[21] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68.
[22] W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proc. Conf. Complex Anal., Tianjin, 1992, pp. 157–169.
[23] B.K. Mihsin, W.G. Atshan and S.S. Alhily, On new sandwich results of univalent functions defined by a linear operator, Iraqi J. Sci. 63 (2022), no. 12, 5467–5475.
[24] M.H. Mohd and M. Darus, Fekete-Szeg¨o problems for quasi-subordination classes, Abstr. Appl. Anal. 2012 (2012).
[25] G. Murugusundaramoorthy, N. Magesh and V. Prameela, Coefficient bounds for certain subclasses of bi-univalent functions, Abstr. Appl. Anal. 2013 (2013), 573017.
[26] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of an univalent functions in: |z| < 1, Arch. Rational Mech. Anal. 32 (1969), 100–112.
[27] C. Pommerenke, Univalent functions, Vandenhoeck and Rupercht, Gottingen, Germany, 1975.
[28] M.A. Sabri, W.G. Atshan and E. El-Seidy, On sandwich-type results for a subclass of certain univalent functions using a new Hadamard product operator, Symmetry 14 (2022), no. 5, 931.
[29] H.M. Srivastava, A.K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188–1192.
[30] T.S. Taha, Topics in univalent function theory, Ph.D. Thesis, University of London, London, UK, 1981.
[31] S.D. Theyab, W.G. Atshan and H.K. Abdullah, On some sandwich results of univalent functions related by differential operator, Iraqi J. Sci. 63 (2022), no. 11, 4928–4936.
[32] S.D. Theyab, W.G. Atshan, A.A. Lupas and H.K. Abdullah, New results on higher-order differential subordination and superordination for univalent analytic functions using a new operator, Symmetry 14 (2022), no. 8, 1–12.
[33] S. Yalcin, W.G. Atshan and H.Z. Hassan, Coefficients assessment for certain subclasses of bi-univalent functions related with quasi-subordination, Pub. Inst. Math. 108 (2020), no. 122, 155–162.
Volume 14, Issue 4
April 2023
Pages 47-54
  • Receive Date: 03 November 2022
  • Revise Date: 15 January 2023
  • Accept Date: 11 February 2023