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Fekete-Szegö inequality for certain subclass of analytic functions. (English) Zbl 1164.30345
Let \(A\) be the class of analytic functions in the unit disc \(U\) of the form \(f(z)=z+a_2z^2+a_3z^3+\cdots\). Then the Fekete-Szegö functional is defined as \(a_3-\mu a_2^2\). In this paper the authors give a sharp upper bound of the Fekete-Szegö functional over the class of functions satisfying \(\frac{(1-\alpha) z (D^nf(z))'+\alpha z(D^{n+1}f(z))'}{(1-\alpha)D^nf(z)+\alpha D^{n+1}f(z)}\prec \phi(z), \quad \alpha\geq0\), where \(D^n\) is the Sălăgean operator and \(\phi(z)\) is a univalent starlike function with respect to \(1\) which maps the unit disc \(U\) onto a region in the right half plane which is symmetric with respect to the real axis, \(\phi(0)=1,\) \(\phi'(0)>0.\)

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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