On certain properties of functions associated with a nonlinear operator

Abstract

In this article, we develop a nonlinear operator $\mathfrak{\Im}% _{\alpha,\gamma}^{\sigma}\left( f\right) \left( z\right) :$% \[ \mathfrak{\Im}_{\alpha,\gamma}^{\sigma}\left( h\right) \left( z\right) :=\left( 1-\alpha \right) \sigma+\alpha \phi \left( z\right) -\frac{\gamma z\phi^{\prime}\left( z\right) }{\phi \left( z\right) }-\left( 1-\alpha-\gamma \right) \frac{z\left( \phi \left( z\right) +z\phi^{\prime }\left( z\right) \right) ^{\prime}}{\phi \left( z\right) +z\phi^{\prime }\left( z\right) }, \] based on the functional $\phi \left( z\right) :=\left( \frac{z}{f\left( z\right) }\right) ^{\sigma}f^{\prime}\left( z\right) :z\in \mathbb{E},$ the open unit disk, $\alpha,\gamma \in \mathbb{R},$ $\sigma \in \left[ -1,1\right] $ \ and find conditions on the functional $\left( \frac{z}{f\left( z\right) }\right) ^{\sigma}f^{\prime}\left( z\right) $ so that it is a filtration. Moreover, we define a family $\mathcal{R}_{\sigma}\left( \alpha ,\gamma \right) $ and study bounds on Fekete-Szeg\"{o} functional $% %TCIMACRO{\tciFourier}% %BeginExpansion \mathcal{F}% %EndExpansion _{f}\left( \eta \right) $ along with some inclusions and different related results. These results can be further extended to symmetric, conjugate symmetric and other related setting in the present formulations.