Browse > Article
http://dx.doi.org/10.7858/eamj.2021.022

CERTAIN NEW FAMILIES FOR BI-UNIVALENT FUNCTIONS DEFINED BY A KNOWN OPERATOR  

Wanas, Abbas Kareem (Department of Mathematics, College of Science, University of Al-Qadisiyah)
Choi, Junesang (Department of Mathematics, Dongguk University)
Publication Information
East Asian mathematical journal / v.37, no.3, 2021 , pp. 319-331 More about this Journal
Abstract
In this paper, we aim to introduce two new families of analytic and bi-univalent functions associated with the Attiya's operator, which is defined by the Hadamard product of a generalized Mittag-Leffler function and analytic functions on the open unit disk. Then we estimate the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we investigate Fekete-Szegö problem for these families. Some relevant connections of certain special cases of the main results with those in several earlier works are also pointed out. Two naturally-arisen problems are given for further investigation.
Keywords
Analytic functions; Bi-univalent functions; Coefficient estimates; Generalized Mittag-Leffler functions; Fekete-Szego problem;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, Filomat 30(7) (2016), 1931-1939. DOI10.2298/FIL1607931C   DOI
2 A. Alsoboh and M. Darus, On Fekete-Szego problems for certain subclasses of analytic functions defined by differential operator involving q-Ruscheweyh operator, J. Funct. Spaces 2020 (2020), Article ID 459405, pages 6. https://doi.org/10.1155/2020/8459405   DOI
3 A. Pfluger, The Fekete-Szego inequality by a variational method, An. Acad. Scientiorum Fennicae Seria A. I., 10 (1985), 447-454.
4 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.   DOI
5 S. Roman, The Umbral Calculus, Academic Press, New York, 1984.
6 S. Altinkaya and S. Yalcin, On the Chebyshev polynomial coefficient problem of some subclasses of bi-univalent functions, Gulf J. Math. 5(3) (2017), 34-40.
7 R. K. Saxena and K. Nishimoto, N-fractional calculus of generalized Mittag-Leffler functions, J. Fract. Calc. 37 (2010), 43-52.
8 C. Abirami, N. Magesh, and J. Yamini, Initial bounds for certain classes of bi-univalent functions defined by Horadam polynomials, Abstr. Appl. Anal. 2020 (2020), Article ID 7391058, pages 8.https://doi.org/10.1155/2020/7391058   DOI
9 E. A. Adegani, S. Bulut, and A. Zireh, Coefficient estimates for a subclass of analytic biunivalent functions, Bull. Korean Math. Soc. 55(2) (2018), 405-413.https://doi.org/10.4134/BKMS.b170051   DOI
10 R. M. Ali, S. K. Lee, V. Ravichandran, and S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett. 25(3) (2012), 344-351. doi:10.1016/j.aml.2011.09.012   DOI
11 H. Arikan, H. Orhan and M. Caglar, Fekete-Szego inequality for a subclass of analytic functions defined by Komatu integral operator, AIMS Math. 5(3) (2020), 1745-1756. https://doi.org/10.3934/math.2020118   DOI
12 A. A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat 30(7) (2016), 2075-2081. DOI10.2298/FIL1607075A   DOI
13 D. A. Brannan, J. G. Clunie (Eds.), Aspects of Contemporary Complex Analysis (Proceedings of the NATO Advanced Study Institute held at the University of Durham, Durham; July 120, 1979), Academic Press, New York and London, 1980.
14 P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Band 259, Springer Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
15 B. A. Frasin, Coefficient bounds for certain classes of bi-univalent functions, Hacet. J. Math. Stat. 43(3) (2014), 383-389.
16 H. M. Srivastava, S,. Altinkaya, and S. Yalcin, Certain subclasses of bi-univalent functions associated with the Horadam polynomials, Iran. J. Sci. Technol. Trans. A Sci. 43 (2019), 1873-1879. https://doi.org/10.1007/s40995-018-0647-0   DOI
17 H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
18 A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill Book Company, New York, Toronto and London, 1955.
19 A. G. Al-Amoush, Certain subclasses of bi-univalent functions involving the Poisson distribution associated with Horadam polynomials, Malaya J. Mat. 7(4) (2019), 618-624.https://doi.org/10.26637/MJM0704/0003   DOI
20 M. Fekete and G. Szego, Eine bemerkung uber ungerade schlichte funktionen, J. London Math. Soc. s1-8(2) (1933), 85-89. https://doi.org/10.1112/jlms/s1-8.2.85   DOI
21 B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24(9) (2011), 1569-1573. https://doi.org/10.1016/j.aml.2011.03.048   DOI
22 H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernal, Appl. Math. Comp. 211(1) (2009), 198-210. https://doi.org/10.1016/j.amc.2009.01.055   DOI
23 H. M. Srivastava, S. S. Eker, S. G. Hamidi, and J. M. Jahangiri, Faber polynomial coefficient estimates for bi-univalent functions defined by the Tremblay fractional derivative operator, Bull. Iranian Math. Soc. 44(1) (2018), 149-157.https://doi.org/10.1007/s41980-018-0011-3   DOI
24 H. M. Srivastava, A. K. Mishra, and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188-1192. https://doi.org/10.1016/j.aml.2010.05.009   DOI
25 H. M. Srivastava, N. Raza, E. S. A. AbuJarad, G. Srivastava and M. H. AbuJarad, Fekete-Szego inequality for classes of (p, q)-starlike and (p, q)-convex functions, Rev. Real Acad. Cienc. Exactas Fis. Natur. Ser. A Mat. (RACSAM)., 113 (2019), 3563-3584. https://doi.org/10.1007/s13398-019-00713-5   DOI
26 H. M. Srivastava and 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(3) (2019), 493-503. https://doi.org/10.5666/KMJ.2019.59.3.493   DOI
27 S. R. Swamy, A. K. Wanas and Y. Sailaja, Some special families of holomorphic and Salagean type bi-univalent functions associated with (m,n)-Lucas polynomials, Commun. Math. Appl. 11(4) (2020), 563-574. https://doi.org/10.26713/cma.v11i4.1411
28 E. Szatmari and S. Altinkaya, Coefficient estimates and Fekete-Szego inequality for a class of analytic functions satisfying subordinate condition associated with Chebyshev polynomials, Acta Univ. Sapientiae, Mathematica, 11(2) (2019), 430-436. https://doi.org/10.2478/ausm-2019-0031   DOI
29 N. U. Khan, , M. Aman, T. Usman, and J. Choi, Legendre-Gould Hopper-based Sheffer polynomials and operational methods, Symmetry 12 (2020), ID 2051, 17 pages. doi: 10.3390/sym12122051   DOI
30 H. O. Guney, G. Murugusundaramoorthy and J. Sokol, Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers, Acta Univ. Sapient. Math. 10(1) (2018), 70-84.https://doi.org/10.2478/ausm-2018-0006   DOI
31 M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18(1) (1967), 63-68.   DOI
32 G. M. Mittag-Leffler, Sur la nouvelle function, C. R. Acad. Sci. Paris, 137 (1903), 554-558.
33 G. M. Mittag-Leffler, Sur la representation analytique dune function monogene (cinquieme note), Acta Math. 29 (1905), 101-181. https://projecteuclid.org/euclid.acta/1485887138   DOI
34 N. Magesh and S. Bulut, Chebyshev polynomial coefficient estimates for a class of analytic bi-univalent functions related to pseudo-starlike functions, Afr. Mat. 29(1-2) (2018), 203-209.https://doi.org/10.1007/s13370-017-0535-3   DOI
35 A. K. Wanas and A. L. Alina, Applications of Horadam polynomials on Bazilevic bi-univalent function satisfying subordinate conditions, J. Phys.: Conf. Ser. 1294 (2019), 032003.https://doi.org/10.1088/1742-6596/1294/3/032003   DOI
36 H. M. Srivastava, S. Gaboury and F. Ghanim, Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat. 28 (2017), 693-706. https://doi.org/10.1007/s13370-016-0478-0   DOI
37 A. K. Wanas, Applications of Chebyshev polynomials on λ-pseudo bi-starlike and λ-pseudo bi-convex functions with respect to symmetrical points, TWMS J. App. Eng. Math. 10(3) (2020), 568-573.
38 A. K. Wanas, Coefficient estimates of bi-starlike and bi-convex functions with respect to symmetrical points associated with the second kind Chebyshev polynomials, Earthline J. Math. Sci. 3 (2020), 191-198.https://doi.org/10.34198/ejms.3220.191198   DOI
39 A. K. Wanas and A. H. Majeed, Chebyshev polynomial bounded for analytic and biunivalent functions with respect to symmetric conjugate points, Appl. Math. E-Notes 19 (2019), 14-21. https://www.emis.de/journals/AMEN/2019/AMEN-180201
40 A. K. Wanas and S. Yalcin, Initial coefficient estimates for a new subclasses of analytic and m-fold symmetric bi-univalent functions, Malaya J. Mat. 7 (2019), 472-476.https://doi.org/10.26637/MJM0703/0018   DOI
41 H. M. Srivastava, A. K. Wanas and G. Murugusundaramoorthy, Certain family of bi-univalent functions associated with Pascal distribution series based on Horadam polynomials, Surv. Math. Appl. 16 (2021), 193-205.
42 A. K. Wanas and S. Yalcin, Coefficient estimates and Fekete-Szego inequality for family of bi-univalent functions defined by the second kind Chebyshev polynomial, Int. J. Open Problems. Compt. Math. 13(2) (2020), 25-33.
43 A. K. Wanas, Bounds for initial Maclaurin coefficients for a new subclasses of analytic and m-fold symmetric bi-univalent functions, TWMS J. App. Eng. Math. 10(2) (2020), 305-311. https://dergipark.org.tr/tr/download/article-file/1181240
44 P. Zaprawa, On the Fekete-Szego problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin, 21 (2014), 169-178. https://doi.org/10.36045/bbms/1394544302   DOI