A new integral operator for meromorphic functions. (English) Zbl 1224.30062

Summary: In this paper, a new integral operator \[ H(f_1,f_2,\dots,f_n)(z)=\frac 1{z^2}\int^z_0 (uf_1(u))^{\gamma_1}\cdots(uf_n(u))^ {\gamma_n} du \] for \(f_i(z)\in\Sigma\) and \(z\in U^{\ast}\) is defined. In addition, a starlikeness condition is derived. Moreover, a new subclass of meromorphic functions satisfying the condition \(-\operatorname{Re}\left(\frac {zf''(z)}{f(z)}+1\right)<\beta\), where \(z\in U^{\ast}\) and \(\beta>1\), is introduced and sufficient conditions for a function to belong to this class are studied.


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