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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, $$\phi(0)=1,$$ $$\phi'(0)>0.$$

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