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.


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