Some Differential Subordinations Using Ruscheweyh Derivative and a Multiplier Transformation

Abstract

In this paper the author derives several interesting differential subordination results. These subordinations are established by means of a differential operator obtained using Ruscheweyh derivative R ^m f(z) and the multiplier transformations $I\left (m,\lambda,l\right )f(z)$, namely $RI_{m,\lambda,l}^{\alpha }$ the operator given by

$$\displaystyle\begin{array}{rcl} RI_{m,\lambda,l}^{\alpha }&: & \mathcal{A}\rightarrow \mathcal{A}, {}\\ RI_{m,\lambda,l}^{\alpha }f(z)& =& (1-\alpha )R^{m}f(z) +\alpha I\left (m,\lambda,l\right )f(z), {}\\ \end{array}$$

for z ∈ U, $m \in \mathbb{N}$, $\lambda,l \geq 0$ and

$$\displaystyle{ \mathcal{A}_{n} =\{ f \in \mathcal{H}(U): f(z) = z + a_{n+1}z^{n+1} + \cdots \,,\ z \in U\}, }$$

with $\mathcal{A}_{1} = \mathcal{A}$. A number of interesting consequences of some of these subordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided.