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

Main Subjects

[1] N.I. Ahiezer, 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),