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
[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),
[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),
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