Coefficient Estimates for Function with Respect to Symmetric Points Based on Shell-Like Curves Defined by Convolution
M. İbrahim
Abstract. Making use of the Hadamard product(or convolution), we find
some estimates to symmetric points, related to shell-like curves connected with
Fibonacci numbers. We determine the initial Taylor-Maclaurin coefficients
| a2 | and | a3 | for functions belong to Bi-univalent functions of the Bazileviˇc
type of order γ. Further we determine the Fekete-Szeg¨o result for the function
class. References. [1] A. Alamoush and M. Darus, New criteria for certain classes containing generalised differential operator, Journal of Quality Measurement and Analysis. 9 (2), 59-71, 2013.[2] F. M. Al-Oboudi, On univalent functions defined by a generalized S˘al˘agean operator, Int. J. Math. Math. Sci. 2004, no. 25-28, 1429–1436. MR20850 [3] M. Darus and R. W. Ibrahim, New classes containing generalization of differential operator, Appl. Math. Sci. (Ruse) 3 (2009), no. 49-52, 2507–2515. MR2557999 [4] S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115. MR0367176 [5] G. S˘al˘agean, Subclasses of univalent functions, in Complex analysis—fifth RomanianFinnish seminar, Part 1 (Bucharest, 1981), 362–372, Lecture Notes in Math., 1013, Springer, Berlin. MR0738107 [6] A. G.Alamoush, M. Darus, On coefficient estimates for new generalized subclasses of biunivalent functions, AIP Conference Proceedings 1614, 844, 2014. [7] G. Murugusundaramoorthy, C. Selvaraj and O. S. Babu, Coefficient estimate of Ma-Minda type bi-Bazilevi functions of complex order involving Srivastava-Attiya operator, Electron. J. Math. Anal. Appl. 2 (2014), no. 2, 31–41. MR3311245 [8] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in z < 1, Arch. Rational Mech. Anal. 32 (1969), 100–112. MR0235110 [9] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68. MR0206255 [10] I. E. Bazilevic, On a case of integrability in quadratures of the Loewner-Kufarev equation,Mat. Sb., Vol 37 (79),1955, pp. 471-476. [11] R. K. Raina and J. Sok´o l, Fekete-Szeg¨o problem for some starlike functions related to shell-like curves, Math. Slovaca 66 (2016), no. 1, 135–140. MR3510858 [12] J. Dziok, R. K. Raina and J. Sok´o l, On α-convex functions related to shell-like functions connected with Fibonacci numbers, Appl. Math. Comput. 218 (2011), no. 3, 996–1002. MR2831345 [13] C. Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrb¨ucher, Band XXV, Vandenhoeck & Ruprecht, G¨ottingen, 1975. MR0507768 [14] M. Fekete and G. Szeg¨o, Eine Bemerkung Uber Ungerade Schlichte Funktionen, J. London Math. Soc. 8 (1933), no. 2, 85–89. MR1574865 [15] P. Zaprawa, On the Fekete-Szeg¨o problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 1, 169–178. MR3178538 |
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