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 \({ST^{n,\alpha}_{\lambda,m}(h)}\) of functions \({f\in A}\), with \({\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}\), satisfying $$\frac{z\left(D^{n,\alpha}_\lambda f(z)\right)'}{D^{n,\alpha}_\lambda f_m(z)}\prec h(z),\quad z\in E,$$ where \({n\in \mathbb N_0, =\mathbb N\cup \{0\}, 0\leq \alpha <1 ,\lambda \geq 0, m \in\mathbb N,h}\) is a convex function in E with h(0) = 1, and \({D^{n,\alpha}_\lambda:A\rightarrow A}\), is the linear fractional differential operator, newly defined as follows $$D^{n,\alpha}_\lambda f(z) = z + \sum^\infty_{k=2}\Psi_{k,n}(\alpha,\lambda)a_k z^k.$$ where $$\Psi_{k,n}(\alpha,\lambda) = \left( \frac{\Gamma(k+1)\Gamma(2-\alpha)}{\Gamma(k+1-\alpha)}(1+\lambda(k-1))\right)^n,$$ and $$f_m(z) = \frac{1}{2m} \sum^{m-1}_{k=0}\left[w^{-k} f(w^k z) + w^k\overline{f(w^k \overline{z})}\right],\quad w = \exp\left(\frac{2\pi i}{m}\right).$$ 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.