On Subclass of Bazilevic Function B1(®), It's Distortion and the Fekete-Szego Problem
Abstrak
In this paper we present the distortion and the Fekete-SzegÄo problem of subclass of Bazilevi·c functions, B1(®). First, we present the result of Singh concerning the sharp value of the coe±cients for B1(®), ja2j, ja3j and ja4j. Second, we give a solution of the Fekete-SzegÄo problem, i.e. an estimate of ja3 ¡ ¹a 2 2j for any real and complex numbers ¹ where a2 and a3 are the coe±cients of functions f in B1(®), where B1(®) is defined by (2), i.e. for each ® > 0 and for z 2 D, Re f 0 (z) f(z) z ®¡1 > 0. These results are sharp for the functions f0 defined by (3) for any real number ¹ which satisfies ¹ < (1 ¡ ®)=2, or ¹ ¸ (4 + 3® + ® )=[2(2 + ®)] and for any complex number ¹ which satisfies j3 + ® ¡ 2¹(2 + ®)j ¸ (1 + ®) 2 . These results are sharp for the functions f1 defined by (4) for the other real and complex numbers ¹. Next, we use similar methods to get estimates for linear expressions involving higher coefficients of function in B1(®).
Referensi
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M. Darus and D.K.Thomas, On the Fekete-SzegÄ o theorem for close-to-convex functions, Math.Japonica Vol.47 No.1 (1998),125-132.
M. Fekete and G. SzegÄ o, Eine Bemerkung Ä uber ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85-89.
M. Jahangiri, On the coe±cients of powers of a class of Bazilevi· c functions, Indian J. pure appl. Math. 17 (9)(1986), 1140-1144.
F.R. Keogh and E.P. Merkes, A coe±cient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8-12.
Marjono, Contributions to the theoremry of Bazilevi· c functions, Dissertation University of Graz;1999.
R. Singh, On Bazilevi· c Functions, Proc. Amer. Math. Soc. 38 (1973), 261-271.
