New Class of Specific Functions with Fractional Derivatives

Let ${\mathcal{A}}_n$ be the class of specific analytic functions $f\left(z\right)=z+{\displaystyle \sum _{k=1}^{\infty }}{a}_{1+{\displaystyle \frac{k}{n}}}{z}^{1+{\displaystyle \frac{k}{n}}}(n\in \mathbb{N}=\{1,2,3,\dots \})$ in the open unit disk $\mathbb{U}.$ For $f\left(z\right)\in {\mathcal{A}}_n,$ fractional derivatives $D_z^{\lambda }f\left(z\right)$ and $D_z^{j+\lambda }f\left(z\right)$$(0\le \lambda <1,j\in \mathbb{N})$ are defined by using Gamma functions. Applying such fractional derivatives, we introduce a new subclass ${\mathcal{A}}_n(j,\lambda ,\alpha ,\beta )$ of ${\mathcal{A}}_n.$ In this paper, we establish sufficient conditions for $f\left(z\right)$ for ${\mathcal{A}}_n(j,\lambda ,\alpha ,\beta ),$ coefficient inequalities for $|a_{1+{\displaystyle \frac{1}{n}}}|$ and $|a_{1+{\displaystyle \frac{k}{n}}}|$$(k=2,3,4,\dots )$ of $f\left(z\right)\in {\mathcal{A}}_n(j,\lambda ,\alpha ,\beta ),$ and some interesting argument properties of fractional derivatives for $f\left(z\right)\in {\mathcal{A}}_n$ through an example.