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  Volume 4, Issue 2, Article 45
Some Special Subclasses of Carathéodory's or Starlike Functions and Related Coefficient Problems

    Authors: Philippos Koulorizos, Nikolas Samaris,  
    Keywords: Coefficient problem, Carathéodory's functions, Starlike functions, Convex functions.  
    Date Received: 28/01/03  
    Date Accepted: 18/04/03  
    Subject Codes:


    Editors: Herb Silverman,  

Let $mathcal{P}$ be the class of analytic functions in the unit disk ${% scriptstyle U = { vert z vert  1 } }$ with $p(0) = 0$ 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^{primeprime}(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.

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