Document Type : Research Paper
Author
Dicle University, Department of Mathematics, Science Faculty, TR-21280 Diyarbakir, Turkey.
Abstract
In this paper, we define and investigate a new class of bi-Bazilevic functions related to shell-like curves connected with Fibonacci numbers. Furthermore, we find estimates of first two coefficients of functions belonging to this class. Also, we give the Fekete-Szegoinequality for this function class.
Keywords
Main Subjects
[1] I.E. Bazilevic, On a case of integrability in quadratures of the Lowner-Kufarev equation, Math. Sb., 37(1955), pp. 471-476.
[2] D.A. Brannan, J. Clunie, and W.E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 22 (1970), pp. 476-485.
[3] D.A. Brannan and T.S.Taha, On some classes of bi-univalent functions, Stud. Univ.Babes-Bolyai Math., 31 (1986), pp. 70-77.
[4] P.L. Duren, Univalent Functions, In: Grundlehren der Mathematischen Wissenschaften, Band 259, New York, Berlin, Heidelberg and Tokyo, Springer-Verlag, 1983.
[5] J. Dziok, R.K. Raina, and J. Sokol, On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Math. and Computer Modelling, 57 (2013), pp. 1203-1211.
[6] J. Dziok, R.K. Raina, and J. Sokol, On $alpha-$convex functions related to a shell-like curve connected with Fibonacci numbers, Appl. Math. Comp., 218 (2011), pp. 996-1002.
[7] M. Fekete and G. Szego, Eine Bemerkunguber ungerade schlichte Functionen, J. London Math. Soc., 8 (1933), pp. 85-89.
[8] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), pp. 63-68.
[9] X.F. Li and A.P. Wang, Two new subclasses of bi-univalent functions, International Mathematical Forum, 7 (2012), pp. 1495--1504.
[10] Ch. Pommerenke, Univalent Functions, Math. Math, Lehrbucher, Vandenhoeck and Ruprecht, Gottingen, 1975.
[11] R.K. Raina and J. Sokol, Fekete-Szego problem for some starlike functions related to shell-like curves, Math. Slovaca, 66 (2016), pp. 135-140.
[12] J. Sokol, On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis, 175 (1999), pp. 111-116.
[13] 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.
[14] 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.
[15] H.M. Srivastava, S. Gaboury, and F. Ghanim, Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat., 28(2017), pp. 693-706.
[16] H.M. Srivastava, S. Gaboury, and F. Ghanim, Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Math. Sci. Ser. B Engl. Ed., 36(2016), pp. 863-871.
[17] H.M. Srivastava, S. Gaboury, and F. Ghanim, Coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Univ. Apulensis Math. Inform., 23(2015), pp. 153-164.
[18] 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.
[19] Srivastava, S. Sivasubramanian, and R. Sivakumar, Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi Math. J., 7(2014), pp. 1-10.
[20] H.M. Srivastava, S. Sumer Eker, and M. Ali Rosihan, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29(2015), pp. 1839-1845.
[21] H. Tang, H.M. Srivastava, S. Sivasubramanian, and P. Gurusamy, The Fekete-Szego functional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal., 10 (2016), pp. 1063-1092.
[22] QH Xu, YC Gui, and H.M. Srivastava, Coefficinet estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25 (2012), pp. 990-994.
[23] P. Zaprawa, On the Fekete-Szego problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin, 21(2014), pp.169-178.
[2] D.A. Brannan, J. Clunie, and W.E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math., 22 (1970), pp. 476-485.
[3] D.A. Brannan and T.S.Taha, On some classes of bi-univalent functions, Stud. Univ.Babes-Bolyai Math., 31 (1986), pp. 70-77.
[4] P.L. Duren, Univalent Functions, In: Grundlehren der Mathematischen Wissenschaften, Band 259, New York, Berlin, Heidelberg and Tokyo, Springer-Verlag, 1983.
[5] J. Dziok, R.K. Raina, and J. Sokol, On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Math. and Computer Modelling, 57 (2013), pp. 1203-1211.
[6] J. Dziok, R.K. Raina, and J. Sokol, On $alpha-$convex functions related to a shell-like curve connected with Fibonacci numbers, Appl. Math. Comp., 218 (2011), pp. 996-1002.
[7] M. Fekete and G. Szego, Eine Bemerkunguber ungerade schlichte Functionen, J. London Math. Soc., 8 (1933), pp. 85-89.
[8] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), pp. 63-68.
[9] X.F. Li and A.P. Wang, Two new subclasses of bi-univalent functions, International Mathematical Forum, 7 (2012), pp. 1495--1504.
[10] Ch. Pommerenke, Univalent Functions, Math. Math, Lehrbucher, Vandenhoeck and Ruprecht, Gottingen, 1975.
[11] R.K. Raina and J. Sokol, Fekete-Szego problem for some starlike functions related to shell-like curves, Math. Slovaca, 66 (2016), pp. 135-140.
[12] J. Sokol, On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis, 175 (1999), pp. 111-116.
[13] 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.
[14] 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.
[15] H.M. Srivastava, S. Gaboury, and F. Ghanim, Coefficient estimates for some general subclasses of analytic and bi-univalent functions, Afr. Mat., 28(2017), pp. 693-706.
[16] H.M. Srivastava, S. Gaboury, and F. Ghanim, Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Math. Sci. Ser. B Engl. Ed., 36(2016), pp. 863-871.
[17] H.M. Srivastava, S. Gaboury, and F. Ghanim, Coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Univ. Apulensis Math. Inform., 23(2015), pp. 153-164.
[18] 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.
[19] Srivastava, S. Sivasubramanian, and R. Sivakumar, Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi Math. J., 7(2014), pp. 1-10.
[20] H.M. Srivastava, S. Sumer Eker, and M. Ali Rosihan, Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29(2015), pp. 1839-1845.
[21] H. Tang, H.M. Srivastava, S. Sivasubramanian, and P. Gurusamy, The Fekete-Szego functional problems for some subclasses of m-fold symmetric bi-univalent functions, J. Math. Inequal., 10 (2016), pp. 1063-1092.
[22] QH Xu, YC Gui, and H.M. Srivastava, Coefficinet estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25 (2012), pp. 990-994.
[23] P. Zaprawa, On the Fekete-Szego problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin, 21(2014), pp.169-178.
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