Some subclasses of analytic functions of complex order defined by new differential operator

Abstract

Let \hskip 2pt $\mathcal{A}(n)$ \hskip 2pt denote \hskip 2pt the \hskip 2pt class \hskip 2pt of \hskip 2pt analytic \hskip 2pt functions \hskip 2pt $f$ \hskip 2pt in \hskip 2pt the \hskip 2pt open \hskip 2pt unit \hskip 2pt disk \hskip 2pt $U=\{z:|z|<1\}$ \hskip 2pt normalized \hskip 2pt by \hskip 2pt $f(0)=f'(0)-1=0.$ \hskip 2pt In \hskip 2pt this \hskip 2pt paper, \hskip 2pt we \hskip 2pt introduce \hskip 2pt and \hskip 2pt study \hskip 2pt the \hskip 2pt classes \hskip 2pt $S_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho)$ \hskip 2pt and \hskip 2pt $R_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho)$ \hskip 2pt of \hskip 2pt functions \hskip 2pt $f\in\mathcal{A}(n)$ with $(\mu)z(D^{\mho+2}_{\lambda}(\alpha, \omega)f(z))'+(1-\mu)z(D^{\mho+1}_{\lambda}(\alpha, \omega)f(z))'\neq0$ and satisfy some conditions available in literature, where $f\in\mathcal{A}(n), \alpha, \omega, \lambda, \mu \geq0, \mho\in \mathbb{N}\cup\{0\},\,\,z\in U,$ and $D^{m}_{\lambda}(\alpha, \omega)f(z): \mathcal{A}\rightarrow \mathcal{A},$ is the linear fractional differential operator, newly defined as follows $$D^{m}_{\lambda}(\alpha, \omega)f(z) = z+ \sum\limits_{k=2}^{\infty}a_{k}(1+(k-1)\lambda \omega^{\alpha})^{m}z^{k}\cdot$$ Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion for the functions included in the classes $S_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho, \omega)$ and $R_{n, \mu}(\gamma, \alpha, \beta, \lambda, \mho, \omega)$ are given.