Document Type : Research Paper

**Authors**

Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Pakistan.

**Abstract**

The main objective of this investigation is to introduce certain new subclasses of the class $\Sigma $ of bi-univalent functions by using concept of conic domain. Furthermore, we find non-sharp estimates on the first two Taylor-Maclaurin coefficients $ \left \vert a_{2}\right \vert $ and $\left \vert a_{3}\right \vert $ for functions in these new subclasses. We consider various corollaries and consequences of our main results. We also point out relevant connections to some of the earlier known developments.

**Keywords**

- Univalent function
- Analytic function
- Bi-univalent function
- Subordination between analytic functions
- Starlike and strongly starlike functions
- Conic domain

**Main Subjects**

*Elements of theory of elliptic functions*, Moscow, 1970.

[2] G.D. Anderson, M.K. Vamanamurthy, and M.K. Vourinen,

*Conformal invariants, inequalities and quasiconformal maps*, Wiley-Interscience, 1997.

[3] M. Arif, J. Dziok, M. Raza, and J. Sokol,

*On products of multivalent close-to-star functions*, J. Ineq. appl., 2015 (2015), pp. 1-14.

[4] S.Z.H. Bukhari, M. Nazir, and M. Raza,

*Some generalisations of analytic functions with respect to 2k-symmetric conjugate points*, Maejo Int. J. Sci. Technol., 2016, pp. 10, 1-12.

[5] P.L. Duren, Univalent Functions,

*Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Science),*vol. 259, Springer-Verlag, New York, Berlin, 1983.

[6] B.A. Frasin,

*Coefficient bounds for certain classes of bi-univalent functions*, Hacettepe J. Math. Stat., 43 (2014), pp. 383-389.

[7] S. Hussain, N. Khan, S. Khan, and Q.Z. Ahmad,

*On a subclass of analytic and bi-univalent functions*, Southeast Asian Bull. Math., article in press.

[8] S. Kanas and A. Wisniowska,

*Conic regions and k-uniform convexity*, J. Comput. Appl. Math., 105 (1999), pp. 327-336.

[9] S. Kanas and A. Wisniowska,

*Conic domains and starlike functions*, Rev. Roumaine Math. Pures Appl., 45 (2000), pp. 647-657.

[10] S. Kanas,

*Coefficient estimates in subclasses of the Caratheodory class related to conical domains*, Acta Math. Univ. Comenian., 74 (2005), pp. 149-161.

[11] N. Khan, B. Khan, Q.Z. Ahmad, and S. Ahmad,

*Some Convolution Properties of Multivalent Analytic Functions*, AIMS Math., 2 (2017), pp. 260-268.

[12] N. Khan, Q.Z. Ahmad, T. Khalid, and B. Khan,

*Results on spirallike $p$-valent functions*, AIMS Math., 3 (2018), pp. 12-20.

[13] N. Khan, A. Khan, Q.Z. Ahmad, B. Khan, and S. Khan,

*Study of multivalent spirallike Bazilevic functions AIMS Math.*, 3 (2018), pp. 353-364.

[14] K.I. Noor, N. Khan, M. Darus, Q.Z. Ahmad, and B. Khan,

*Some properties of analytic functions associated with conic type regions*, Intern. J. Anal. Appl., 16 (2018), pp. 689-701.

[15] K.I. Noor,

*On a generalization of uniformly convex and related functions*, Comput. Math. Appl., 61 (2011), pp. 117-125.

[16] K.I. Noor, M. Arif, and M.W. Ul-Haq,

*On $k$-uniformly close-to-convex functions of complex order*, Appl. Math. Comput., 215 (2009), pp. 629-635.

[17] K.I. Noor, Q.Z. Ahmad, and M.A. Noor,

*On some subclasses of analytic functions defined by fractional derivative in the conic regions*, Appl. Math. Inf., Sci., 9 (2015), pp. 8-19.

[18] K.I. Noor, J. Sokol, and Q.Z. Ahmad,

*Applications of conic type regions to subclasses of meromorphic univalent functions with respect to symmetric points*, RACSAM, 2016, pp. 1-14.

[19] K.I. Noor, Q.Z. Ahmad, and N. Khan,

*On some subclasses of meromorphic functions defined by fractional derivative operator,*Italian J. Pure. App Math., (2017), pp. 1-8.

[20] K.I. Noor and N. Khan,

*Some convolution properties of a subclass of p-valent functions*, Maejo Int. J. Sci. Technol., 9 (2015), pp. 181-192.

[21] M. Nunokawa, S. Hussain, N. Khan, and Q.Z. Ahmad,

*A subclass of analytic functions related with conic domain*, J. Clas. Anal., 9 (2016), pp. 137-149.

[22] M. Obradovic and S. Owa,

*Some sufficient conditions for strongly starlikeness*, Int. J. Math. Math. Sci., 24 (2000), pp. 643-647.

[23] M. Raza, M. U Din, and S.N. Malik,

*Certain geometric properties of normalized wright functions*, J. Func. Spaces, 2016 (2016), 9 pages.

[24] W. Rogosinski,

*On the coefficients of subordinate functions,*Proc. Lond. Math. Soc., 48 (1943), pp. 48-82.

[25] H.M. Srivastava, A.K. Mishra, and P. Gochhayat,

*Certain subclasses of analytic and bi-univalent functions*, Appl. Math. Lett., 23 (2010), pp. 1188-1192.

[26] H.M. Srivastava, S. Bulut, M. Caglar, and N. Yagmur,

*Coefficient estimates for a general subclass of analytic and bi-univalent functions*, Filomat, 27 (2013), pp. 831-842.

[27] H.M. Srivastava, and D. Bansal,

*Coefficient estimates for a subclass of analytic and bi-univalent functions*, J. Egyptian Math. Soc., 23 (2015), pp. 242-246.

[28] H.M. Srivastava, G. Murugusundaramoorthy, and N. Magesh,

*Certain subclasses of bi-univalent functions associated with the Hohlov operator*, Global J. Math. Anal., 1 (2013), pp. 67-73.

[29] W.Ul-Haq and S. Manzar,

*Coefficient Estimates for Certain Subfamilies of Close-to-Convex Functions of Complex Order*, Filomat, 30 (2016), pp. 99-103.

[30] W. Ul-Haq, A. Nazneen, and N. Rehman,

*Coefficient estimates for certain subfamilies of close-to-convex functions of complex order*, Filomat, 28 (2014), pp. 1139-1142.

[31] W. Ul-Haq, A. Nazneen, M. Arif, and N. Rehman,

*Coefficient estimate of certain subfamily of close to convex functions*, J. Comput. Anal. Appl., 16 (2013), pp. 133-138.

[32] W. Ul-Haq and S. Mahmmod,

*Certain properties of a subfamily of close-to-convex functions related to conic regions*, Abst. Appl. Anal., Article ID: 847287, 2013 (2013), 6 pp.

[33] Q.-H. Xu, H.-G. Xiao, and H. M. Srivastava,

*A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems*, Appl. Math. Comput., 218 (2012),