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Fekete-Szegö inequality for a certain class of analytic functions. (English) Zbl 1174.30009

Let 𝒜 denote the class of analytic functions f in the unit disc Δ of the form

f(z)=z+ k=2 a k z k ,

and let 𝒮 denote the subclass of 𝒜 consisting of univalent functions. Let ϕ be an analytic function in Δ with positive real part, ϕ(0)=1, ϕ ' (0)>0 which maps Δ onto a region starlike with respect to 1 and symmetric with respect to the real axis. Then let S * (ϕ) be the class of functions f𝒮 with zf ' (z) f(z)ϕ(z), and let C(ϕ) be the class of functions f𝒮 with 1+zf '' (z) f ' (z)ϕ(z), where denotes subordination between analytic functions. These classes were introduced and studied by W. Ma and D. Minda [Li, Zhong (ed.) et al., Proceedings of the conference on complex analysis, held June 19–23, 1992 at the Nankai Institute of Mathematics, Tianjin, China. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Anal. 1, 157–169 (1994; Zbl 0823.30007)].

In this paper the authors consider the more general class M α (ϕ) which is defined as follows. For α0 let M α (ϕ) be the class of functions f𝒜 with zf ' (z) f(z)+αz 2 f '' (z) f(z)ϕ(z). For fM α (ϕ) the authors prove a sharp coefficient estimate for |a 3 -μa 2 2 | in terms of α, μ and the Taylor coefficients B 1 and B 2 of ϕ. As an application of the main result, they prove a respective estimate for a class of analytic functions which are defined by convolution (Hadamard product), and as a special case they obtain such an estimate for a class of functions defined by fractional derivatives.

MSC:
30C45Special classes of univalent and multivalent functions