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A class of multivalent functions with negative coefficients defined by convolution. (English) Zbl 1105.30002

Summary: For a given \(p\)-valent analytic function \(g\) with positive coefficients in the open unit disk \(\Delta\), we study a class of functions \(f(z)=z^p-\sum^\infty_{n =m}a_nz^n\) \((a_n\geq 0)\) satisfying \[ \frac 1p \text{Re}\left(\frac{z(f*q)'(z)} {(f*g)(z)} \right)\geq\alpha\quad (0\leq\alpha<1;\;z \in\Delta). \] Coefficient inequalities, distortion and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special cases.

MSC:

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