Fekete-Szegő problem for starlike and convex functions of complex order. (English) Zbl 1189.30021

Kanas, S.; Darwish, H. E.

Appl. Math. Lett. 23, No. 7, 777-782 (2010).

Summary: For a non-zero complex number \(b\), let \(\mathcal F_n (b)\) denote the class of normalized univalent functions \(f\) satisfying
\[ \text{Re\,}\left[1+\frac{1}{b}\left(\frac{z(D^nf)'(z)}{D^nf(z)}-1\right)\right]> 0 \]
in the unit disk \(\mathcal U\), where \(D^{n}f\) denotes the Ruscheweyh derivative of \(f\). Sharp bounds for the Fekete-Szegö functional \(\big|a_3 - \mu a^2_2\big|\) are obtained.

MSC:

30C45

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Keywords:

coefficient estimates; Ruscheweyh derivative; Fekete-Szegő problem; convex function of complex order; starlike function of complex order

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