Abstract: For $0\le\alpha< 1$ and $0<\beta\le 1$. Let $K_{sh}(\alpha,\beta)$ be the class of normalized close-to-convex functions defined in the open unit disc $D$ by
\left|\arg\left(\dfrac{zf'(z)}{g(z)}\right)\right|\le\dfrac{\pi\alpha}{2},
such that $g\in S^{*}(\beta),$ the class of analytic normalized starlike functions of order $\beta$, i.e. for $z\in D$,
\Re\left(\dfrac{zg'(z)}{g(z)}\right)>\beta.
For $f\in K_{sh}(\alpha,\beta)$ and given by $f(z)=z+a_{2}z^2+a_{3}z^3+\cdots,$ some sharp bounds are obtained for the Fekete - Szego functional $|a_{3}-\mu a_{2}^2|$ when $\mu$ is real.
Keywords: Fekete - Szego theorem, close-to-convex functions, starlike, convex.
Classification (MSC2000): 30C45
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