Opuscula Math. 31, no. 4 (2011), 651-668
http://dx.doi.org/10.7494/OpMath.2011.31.4.651

Opuscula Mathematica

# On some classes of meromorphic functions defined by subordination and superordination

Alina Totoi

Abstract. Let $$p\in \mathbb{N}^*$$ and $$\beta,\gamma\in \mathbb{C}$$ with $$\beta\neq 0$$ and let $$\Sigma_p$$ denote the class of meromorphic functions of the form $$g(z)=\frac{a_{-p}}{z^p}+a_0+a_1 z+\ldots,\,z\in \dot U$$, $$a_{-p}\neq 0$$. We consider the integral operator $$J_{p,\beta,\gamma}:K_{p,\beta,\gamma}\subset\Sigma_p\to \Sigma_p$$ defined by $J_{p,\beta,\gamma}(g)(z)=\left[\frac{\gamma-p\beta}{z^\gamma }\int_0^zg^{\beta}(t) t^{\gamma-1}dt\right]^{\frac{1}{\beta}},\,g\in K_{p,\beta,\gamma},\,z\in \dot U.$ We introduce some new subclasses of the class $$\Sigma_p$$, associated with subordination and superordination, such that, in some particular cases, these new subclasses are the well-known classes of meromorphic starlike functions and we study the properties of these subclasses with respect to the operator $$J_{p,\beta,\gamma}$$.

Keywords: meromorphic functions, integral operators, subordination, superordination.

Mathematics Subject Classification: 30C45.

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