Certain Classes of Bi-Univalent Functions of Complex Order Associated with Quasi-Subordination Involving $(p, q)$-Derivative Operator

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DOI: 10.46793/KgJMat2004.639A


In this present paper, as applications of the post-quantum calculus known as the (p,q)-calculus, we construct a new class Dp,qk(γ, ζ,Ψ  ) of bi-univalent functions of complex order defined in the open unit disk. Coefficients inequalities and several special consequences of the results are obtained.


Coefficient bounds, Bi-univalent functions, Quasi-subordination, q-calculus, (p,q)-derivative operator.


[1]   Ş. Altınkaya and S. Yalçın, Faber polynomial coefficient bounds for a subclass of bi-univalent functions, C. R. Math. Acad. Sci. Paris 353 (2015), 1075–1080.

[2]   S. Araci, U. Duran, M. Acikgoz and H. M. Srivastava, A certain (p,q)-derivative operator and associated divided differences, J. Inequal. Appl. 301 (2016), 2016, 8 pages.

[3]    S. M. Aydoğan, Y. Kahramaner and Y. Polatoğlu, Close-to-convex functions defined by fractional operator, Appl. Math. Sci. 7 (2013), 2769–2775.

[4]   D. A. Brannan and J. G. Clunie, Aspects of contemporary complex analysis, in: Proceedings of the NATO Advanced Study Instute, University of Durham, New York, 1979.

[5]   D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Stud. Univ. Babes-Bolyai Math. 31 (1986), 70–77.

[6]   R. Chakrabarti and R. Jagannathan, A (p,q)-oscillator realization of two-parameter quantum algebras, J. Phys. A. 24 (1991), 711–718.

[7]   P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften 259, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.

[8]   G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, MA, 1990.

[9]   S. G. Hamidi and J. M. Jahangiri, Faber polynomial coefficients of bi-subordinate functions, C. R. Math. Acad. Sci. Paris 354(2016), 365–370.

[10]   T. Hayami and S. Owa, Coefficient bounds for bi-univalent functions, Pan-American Mathematical Journal 22 (2012), 15–26.

[11]   F. H. Jackson, On q-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh 46 (1908), 253–281.

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

[13]   T. H. MacGregor, Majorization by univalent functions, Duke Math. J. 34 (1967), 95–102.

[14]   A. Mohammed and M. Darus, A generalized operator involving the q-hypergeometric function, Mat. Vesnik 65 (2013), 454–465.

[15]   F. M. Sakar and H. Ö. Güney, Faber polynomial coeffıcient bounds for analytıc bi-close-to-convex functions defıned by fractional calculus, J. Fract. Calc. Appl. 9 (2018), 64–71.

[16]   E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Ration. Mech. Anal. 32 (1969), 100–112.

[17]   T. Panigarhi and G. Murugusundaramoorthy, Coefficient bounds for bi-univalent functions analytic functions associated with Hohlov operator, Proc. Jangjeon Math. Soc. 16 (2013), 91–100.

[18]   Ch. Pommerenke, Univalent Functions, Vandenhoeck and Rupercht, Gottingen, 1975.

[19]   S. D. Purohit and R. K. Raina, Certain subclass of analytic functions associated with fractional q-calculus operators, Math. Scand. 109 (2011), 55–70.

[20]   S. D. Purohit and R. K. Raina, Fractional q-calculus and certain subclass of univalent analytic functions, Mathematica (Cluj) 55 (2013), 62–74.

[21]   R. K. Raina and P. Sharma, Subordination properties of univalent functions involving a new class of operators, Electron. J. Math. Anal. Appl. 2 (2014), 37–52.

[22]   F. Y. Ren, S. Owa and S. Fukui, Some inequalities on quasi-subordinate functions, Bull. Aust. Math. Soc. 43 (1991), 317–324.

[23]   M. S. Robertson, Quasi-subordination and coefficients conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9.

[24]   G. S. Salagean, Subclasses of univalent functions, in: Proceeding of Complex Analysis - Fifth Romanian Finnish Seminar, Part 1, Bucharest, 1981, Lecture Notes in Math. 1013, Springer, Berlin, 1983, 362–372.

[25]   C. Selvaraj, G. Thirupathi and E. Umadevi, Certain classes of analytic functions involving a family of generalized differential operators, Transylvanian Journal of Mathemtics and Mechanics 9 (2017), 51–61.

[26]   P. Sharma, R. K. Raina and J. Sokol, On the convolution of a finite number of analytic functions involving a generalized Srivastava–Attiya operator, Mediterr. J. Math. 13 (2016), 1535–1553.

[27]   D. F. Sofonea, Some properties in q-calculus, Gen. Math. 16 (2008), 47–54.

[28]   H. M. Srivastava, G. Murugusundaramoorthy and N. Magesh, Certain subclasses of bi-univalent functions associated with the Hohlov operator, Appl. Math. Lett. 1 (2013), 67–73.

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

[30]   H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, in: H. M. Srivastava and S. Owa (Eds.), Univalent Functions, Fractional Calculus and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, Toronto, 1989.

[31]   A. Zireh, E. A. Adegani and S. Bulut, Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions defined by subordination, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), 487–504.