On Subclass of Analytic Functions Defined by q-Analogue of Modified Tremblay Fractional Derivative Operator

Abstract

In this research, by using the principle of quantum calculus, we introduce a modified fractional derivative operator $\mathcal{T}^{\xi,\digamma}_{q,\varsigma}$ of the analytic functions in the open unit disc $\diamondsuit=\{\varsigma:\varsigma\in\mathbb{C},|\varsigma|<1\}$. The operator $\mathcal{T}^{\xi,\digamma}_{q,\varsigma}$ can then be used to introduce a new subclass of analytic functions $\mathcal{D}\bigoplus(\vartheta,\digamma,d,\xi,\gamma;q)$. We present the necessary conditions for functions belonging to the subclass $\mathcal{D}\bigoplus(\vartheta,\digamma,d,\xi,\gamma;q) $.\\ Furthermore, we discuss a growth and distortion bounds, the convolution condition, and the radii of starlikeness. In addition, we present neighbourhoods problems involving the $\mathfrak{q}$-analogue of a modified Tremblay operator for functions in the introduced class $\mathcal{D}\bigoplus(\vartheta,\digamma,d,\xi,\gamma;q)$.

Keywords

• Analytic functions
• Starlikeness
• Radii of starlikeness
• Neighbourhoods Problems
• Fractional Operator
• q-calculus

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