Some Special Subclasses of Caratheodory's or Starlike Functions and Related Coefficient Problems

Volume 4, Issue 2, 2003

Article 45

SOME SPECIAL SUBCLASSES OF CARATHÉODORY'S OR STARLIKE FUNCTIONS AND RELATED COEFFICIENT PROBLEMS

PHILIPPOS KOULORIZOS AND NIKOLAS SAMARIS

DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF PATRAS,
UNIVERSITY CAMPUS,
GR - 265 04 PATRAS, GREECE
E-Mail: samaris@math.upatras.gr

Received 28 January, 2003; Accepted 18 April, 2003.
Communicated by: H. Silverman

ABSTRACT. Let $\mathcal{P}$ be the class of analytic functions in the unit disk ${% \scriptstyle U = \{ \vert z \vert < 1 \} }$ with

and $\Re p(z) >0$ in ${\scriptstyle U }$ . Let also ${\mathcal{S}^* }$ , $\mathcal{K}$ be the well known classes of normalized univalent starlike and convex functions respectively. For ${\Re \alpha > 0}$ we introduce the classes $\mathcal{P}% _{[\alpha]}$ , $\mathcal{S}^*_{[\alpha]}$ and $\mathcal{K}_{[\alpha]}$ which are subclasses of $\mathcal{P}$ , $\mathcal{S}^*$ and $\mathcal{K}$ respectively, being defined as follows: $p \in \mathcal{P}_{[\alpha]}$ iff $p \in \mathcal{P}$ with $p(z) \neq \alpha \forall z \in U,$ $f \in \mathcal{S}^*_{[\alpha]}$ iff $\frac{z f^\prime }{f}\in \mathcal{P}_{[\alpha]}$ and $f \in \mathcal{K}_{[\alpha]}$ iff ${1 + {\frac{ z f^{\prime\prime}(z) }{f^{\prime}(z)}}} \in \mathcal{P}_{[\alpha]}$ . In this paper we study different kind of coefficient problems for the above mentioned classes $\mathcal{P}_{[\alpha]}$ , $\mathcal{S}^*_{[\alpha]}$ and $% \mathcal{P}_{[\alpha]}$ . All the estimations obtained are the best possible.

Key words:
Coefficient problem, Carathéodory's functions, Starlike functions, Convex functions.

2000 Mathematics Subject Classification:
30C45.

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