Based on a family of bi-univalent functions introduced through the Faber polynomial expansions and Noor integral operator

F. Müge Sakar; Arzu Akgül; F. Müge Sakar; Arzu Akgül

doi:10.3934/math.2022287

AIMS Mathematics

2022, Volume 7, Issue 4: 5146-5155. doi: 10.3934/math.2022287

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Based on a family of bi-univalent functions introduced through the Faber polynomial expansions and Noor integral operator

F. Müge Sakar ^{1
,
,},
Arzu Akgül ²

1.
Department of Management, Faculty of Economics and Administrative Sciences, Dicle University, Diyarbakır, Turkey
2.
Department of Mathematics, Faculty of Arts and Science, Umuttepe Campus, Kocaeli University, Kocaeli, Turkey

Received: 14 December 2020 Revised: 10 August 2021 Accepted: 11 August 2021 Published: 04 January 2022
MSC : 30C45, 30C50

In this study, by using $ q $-analogue of Noor integral operator, we present an analytic and bi-univalent functions family in $ \mathfrak{D} $. We also derive upper coefficient bounds and some important inequalities for the functions in this family by using the Faber polynomial expansions. Furthermore, some relevant corollaries are also presented.
- bi-univalent functions,
- Faber polynomials,
- $ q $-analogue of Noor integral operator,
- coefficient inequalities
Citation: F. Müge Sakar, Arzu Akgül. Based on a family of bi-univalent functions introduced through the Faber polynomial expansions and Noor integral operator[J]. AIMS Mathematics, 2022, 7(4): 5146-5155. doi: 10.3934/math.2022287

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Abstract

In this study, by using $ q $-analogue of Noor integral operator, we present an analytic and bi-univalent functions family in $ \mathfrak{D} $. We also derive upper coefficient bounds and some important inequalities for the functions in this family by using the Faber polynomial expansions. Furthermore, some relevant corollaries are also presented.

References

[1]	H. Airault, A. Bouali, Differential calculus on the Faber polynomials, B. Sci. Math., 130 (2006), 179–222. http://dx.doi.org/10.1016/j.bulsci.2005.10.002 doi: 10.1016/j.bulsci.2005.10.002
[2]	H. Airault, J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, B. Sci. Math., 126 (2002), 343–367. http://dx.doi.org/10.1016/S0007-4497(02)01115-6 doi: 10.1016/S0007-4497(02)01115-6
[3]	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 (2019), 2275–2286. http://dx.doi.org/10.3906/mat-1905-17 doi: 10.3906/mat-1905-17
[4]	H. Aldweby, M. Darus, A subclass of harmonic univalent functions associated with q-analogue of Dziok-Srivastava operator, International Scholarly Research Notices, 2013 (2013), 382312. http://dx.doi.org/10.1155/2013/382312 doi: 10.1155/2013/382312
[5]	R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25 (2012), 344–351. http://dx.doi.org/10.1016/j.aml.2011.09.012 doi: 10.1016/j.aml.2011.09.012
[6]	Ş. Altınkaya, Bounds for a new subclass of bi-univalent functions subordinate to the Fibonacci numbers, Turk. J. Math., 44 (2020), 553–560. http://dx.doi.org/10.3906/mat-1910-41 doi: 10.3906/mat-1910-41
[7]	Ş. Altınkaya, S. Yalçın, S. Çakmak, A subclass of bi-univalent functions based on the Faber polynomial expansions and the Fibonacci numbers, Mathematics, 7 (2019), 160. http://dx.doi.org/10.3390/math7020160 doi: 10.3390/math7020160
[8]	M. Arif, M. Ul Haq, J.-L. Liu, Subfamily of univalent functions associated with q-analogue of Noor integral operator, J. Funct. Space., 2018 (2018), 3818915. http://dx.doi.org/10.1155/2018/3818915 doi: 10.1155/2018/3818915
[9]	D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, In: S. M. Mazhar, A. Hamoui, N. S. Faour, Editors, Mathematical analysis and its applications, Oxford: Pergamon Press, Elsevier Science Limited, 1988, 53–60.
[10]	E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Class. Anal., 2 (2013), 49–60. http://dx.doi.org/10.7153/jca-02-05 doi: 10.7153/jca-02-05
[11]	P. L. Duren, Univalent functions, New York, Berlin, Heidelberg and Tokyo: Springer-Verlag, 1983.
[12]	G. Faber, Über polynomische entwickelungen, Math. Ann., 57 (1903), 389–408. http://dx.doi.org/10.1007/BF01444293 doi: 10.1007/BF01444293
[13]	B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24 (2011), 1569–1573. http://dx.doi.org/10.1016/j.aml.2011.03.048 doi: 10.1016/j.aml.2011.03.048
[14]	H. Grunsky, Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Z., 45 (1939), 29–61. http://dx.doi.org/10.1007/BF01580272 doi: 10.1007/BF01580272
[15]	J. M. Jahangiri, S. G. Hamidi, Coefficient estimates for certain classes of bi-univalent functions, Int. J. Math. Math. Sci., 2013 (2013), 190560. http://dx.doi.org/10.1155/2013/190560 doi: 10.1155/2013/190560
[16]	P. A. A. Laura, A survey of modern applıcatıons of the method of conformal mapping, Rev. Unión Mat. Argent., 27 (1975), 167–179.
[17]	M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63–68.
[18]	M. 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. http://dx.doi.org/10.1007/BF00247676 doi: 10.1007/BF00247676
[19]	K. I. Noor, On new classes of integral operators, J. Natur. Geom., 16 (1999), 71–80.
[20]	K. I. Noor, M. A. Noor, On integral operators, J. Math. Anal. Appl., 238 (1999), 341–352. http://dx.doi.org/10.1006/jmaa.1999.6501
[21]	Z. G. Peng, Q. Q. Han, On the coefficients of several classes of bi-univalent functions, Acta Math. Sci., 34 (2014), 228–240. http://dx.doi.org/10.1016/S0252-9602(13)60140-X doi: 10.1016/S0252-9602(13)60140-X
[22]	H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: Univalent functions; fractional calculus, and their applications, New York, Chichester, Brisbane and Toronto: Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, 1989,329–354.
[23]	H. M. Srivastava, Some inequalities and other results associated with certain subclasses of univalent and bi-univalent analytic functions, In: Nonlinear analysis, New York: Springer, 2012,607–630. http://dx.doi.org/10.1007/978-1-4614-3498-6_38
[24]	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. Sci., 44 (2020), 327–344. http://dx.doi.org/10.1007/s40995-019-00815-0 doi: 10.1007/s40995-019-00815-0
[25]	T. S. Taha, Topics in univalent function theory, Ph.D. Thesis of University of London, 1981.
[26]	P. G. Todorov, On the Faber polynomials of the univalent functions of class $\sum$, J. Math. Anal. Appl., 162 (1991), 268–276. http://dx.doi.org/10.1016/0022-247X(91)90193-4 doi: 10.1016/0022-247X(91)90193-4

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