Opuscula Mathematica (Jan 2011)

On some classes of meromorphic functions defined by subordination and superordination

  • Alina Totoi

DOI
https://doi.org/10.7494/opmath.2011.31.4.651
Journal volume & issue
Vol. 31, no. 4
pp. 651 – 668

Abstract

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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 \(J_{p,\beta,\gamma}\).

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