On Classes of Functions Related to Starlike Functions with Respect to Symmetric Conjugate Points Defined by a Fractional Differential Operator

Let A denote the class of analytic functions f, in the open unit disk E = {z : |z| < 1}, normalized by f(0) = f′(0) − 1 = 0. In this paper, we introduce and study the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ST^{n,\alpha}_{\lambda,m}(h)}$$\end{document} of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f\in A}$$\end{document}, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}$$\end{document}, satisfying\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{z\left(D^{n,\alpha}_\lambda f(z)\right)'}{D^{n,\alpha}_\lambda f_m(z)}\prec h(z),\quad z\in E,$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n\in \mathbb N_0, =\mathbb N\cup \{0\}, 0\leq \alpha <1 ,\lambda \geq 0, m \in\mathbb N,h}$$\end{document} is a convex function in E with h(0) = 1, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D^{n,\alpha}_\lambda:A\rightarrow A}$$\end{document}, is the linear fractional differential operator, newly defined as follows \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{n,\alpha}_\lambda f(z) = z + \sum^\infty_{k=2}\Psi_{k,n}(\alpha,\lambda)a_k z^k.$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi_{k,n}(\alpha,\lambda) = \left( \frac{\Gamma(k+1)\Gamma(2-\alpha)}{\Gamma(k+1-\alpha)}(1+\lambda(k-1))\right)^n,$$\end{document}and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_m(z) = \frac{1}{2m} \sum^{m-1}_{k=0}\left[w^{-k} f(w^k z) + w^k\overline{f(w^k \overline{z})}\right],\quad w = \exp\left(\frac{2\pi i}{m}\right).$$\end{document}For special values of the functions h and the parameters n, α, m and λ, we get known classes of starlike functions with respect to symmetric conjugate points. Inclusion relations, convolution properties and other results are given. Another related class is also defined.