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Fekete-Szegő problem for subclasses of generalized uniformly starlike functions with respect to symmetric points

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Czech Digital Mathematics Library
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Abstract

summary:The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^{2}+st+t^{2}<3$, $s\neq t$ and $s+t\neq 2$, or in the region $s^{2}+st+t^{2}>3,$ $s\neq t$ and $s+t\neq 2$) for certain normalized analytic functions $f(z)$ belonging to $k\text {\rm -UST}_{\lambda ,\mu }^{n}(s,t,\gamma )$ which satisfy the condition \begin {equation*} \Re \bigg \{ \frac {(s-t)z ( D_{\lambda ,\mu }^{n}f(z))'} {D_{\lambda ,\mu }^{n}f(sz)-D_{\lambda ,\mu }^{n}f(tz)}\bigg \} >k \biggl \vert \frac {(s-t)z ( D_{\lambda ,\mu }^{n}f(z))'}{D_{\lambda ,\mu }^{n}f(sz)-D_{\lambda ,\mu }^{n}f(tz)}{-1} \biggr \vert +\gamma , \quad z\in \mathcal {U} . \end {equation*} Also certain applications of the main result a class of functions defined by the Hadamard product (or convolution) are given. As a special case of this result, the Fekete-Szegő inequality for a class of functions defined through fractional derivatives is obtained.

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