On boundedness and compactness of a generalized Srivastava–Owa fractional derivative operator

Theorem 1.1 (Bieberbach’s Conjecture)

If the function $f(z)$ defined by (1.1) is in the class ${\mathcal{S}}^{\ast }$ then $|a_{\kappa }|⩽\kappa$ for all $\kappa ⩾2$ and if it is in the class $\mathcal{K}$ then $|a_{\kappa }|⩽1$ for all $\kappa ⩾2$ (Duren, 1983).

The Fox–Wright ${}_p{\mathrm{\Psi }}_q$ generalization of the hypergeometric ${}_p{F}_q$ function is defined as:

By usage the Hadamard product technique, Srivastava (2007) provided families of analytic and univalent functions associated with the Fox–Wright generalized hypergeometric functions ${}_p{\mathrm{\Psi }}_q$ in the open unit disk $\mathbb{U}$ .

Definition 2.1 (Ibrahim, 2011)

The generalized Srivastava–Owa fractional derivative of $f(z)$ of order $\beta$ is defined by

The generalized fractional differential of $f(z)$ of two parameters $\beta$ and $\tau$ is defined by

For $f(z)\in \mathcal{A}$ , we have

• i. when $\gamma =0$ in (2.3), is reduced to the classical known one (1.4) and

• ii. when $\tau =\beta$ in (2.3), we have ${\mathfrak{T}}_z^{\beta \text{,}\beta \text{,}\gamma }f(z)=f(z)\text{.}$

Let $0⩽\beta -\tau <1$ for some $0<\beta ⩽1\text{;}0<\tau ⩽1$ and $\upsilon \in N$ , then we have

Let $f(z)=z{(1-z)}^{-\alpha }\text{,}\alpha ⩾1$ and $z\in \mathbb{U}$ , then $$\begin{array}{cc}\hfill & {\mathfrak{T}}_z^{\beta \text{,}\tau \text{,}\gamma }f(z)=\frac{{(\gamma +1)}^{\beta -\tau }\mathrm{\Gamma }(\tau )}{\mathrm{\Gamma }(\beta )}{z}^{(1-\beta +\tau )\gamma +1}{}_3{\mathrm{\Psi }}_2\hfill \\ \hfill & \times \left[\left.\begin{array}{c}(1\text{,}1)\text{,}(\alpha +1\text{,}1)\text{,}(1+\frac{\beta }{\gamma +1}\text{,}\frac{1}{\gamma +1})\hfill \\ (2\text{,}1)\text{,}(1-\beta +\tau +\frac{\beta }{\gamma +1}\text{,}\frac{1}{\gamma +1})\hfill \end{array}\right|z\right]\text{,}\hfill \end{array}$$ when $\alpha =2$ , the equality holds true for the Koebe function.

Let $f(z)=z{}_1{F}_1(\alpha \text{,}\lambda \text{;}z)\text{,}0<\beta ⩽1$ and $0<\tau ⩽1$ such that $0⩽\beta -\tau <1$ . Then $${\mathfrak{T}}_z^{\beta \text{,}\tau \text{,}\gamma }\left\{z{}_1{F}_1(\alpha \text{,}\lambda \text{;}z)\right\}=\frac{\alpha {(\gamma +1)}^{\beta -\tau }\mathrm{\Gamma }(\tau )}{\lambda \mathrm{\Gamma }(1+\tau )}{z}^{(1-\beta +\tau )\gamma +1}{\displaystyle \sum _{\kappa =0}^{\infty }}\frac{{(1)}_{\kappa }{(\alpha +1)}_{\kappa }}{{(\lambda +1)}_{\kappa }{(2)}_{\kappa }}\frac{B(\frac{\kappa +\beta }{\gamma +1}+1\text{,}1-\beta +\tau )}{B(\beta \text{,}1-\beta +\tau )}\frac{z^{\kappa }}{{(1)}_{\kappa }}\text{,}$$ where ${}_1{F}_1(\alpha \text{,}\lambda \text{;}z)$ is the confluent hypergeometric function (see Kilbas et al., 2006).

Let $0<\beta ⩽1$ and $0<\tau ⩽1$ such that $0⩽\beta -\tau <1$ . Then, we obtain $$\begin{array}{cc}\hfill & {\mathfrak{T}}_z^{\beta \text{,}\tau \text{,}\gamma }\left\{z{}_p{\mathrm{\Psi }}_q(z)\left[\left.\begin{array}{c}{(a_j\text{,}{A}_j)}_{1\text{,}p}\hfill \\ {(b_j\text{,}{B}_j)}_{1\text{,}q}\hfill \end{array}\right|z\right]\right\}=\frac{{(\gamma +1)}^{\beta -\tau }\mathrm{\Gamma }(\tau )}{\mathrm{\Gamma }(1+\tau )}{z}^{(1-\beta +\tau )\gamma +1}\hfill \\ \hfill & \times {\displaystyle \sum _{\kappa =0}^{\infty }}\left(\frac{{\mathrm{\Pi }}_{j=1}^p\mathrm{\Gamma }(a_j\text{,}(\kappa +1)A_j)}{{\mathrm{\Pi }}_{j=1}^q\mathrm{\Gamma }(b_j+(\kappa +1)B_j)\mathrm{\Gamma }(\kappa +2)}\frac{B(\frac{\kappa +\beta }{\gamma +1}+1\text{,}1-\beta +\tau )}{B(\beta \text{,}1-\beta +\tau )}{z}^{\kappa }\right)\hfill \end{array}$$ where ${}_p{\mathrm{\Psi }}_q(z)$ is given by (1.5), for all $|z|<1$ .

Let $0<\beta ⩽1$ and $0<\tau ⩽1$ such that $0⩽\beta -\tau <1$ . Then $$\begin{array}{cc}\hfill & {\mathfrak{T}}_z^{\beta \text{,}\tau \text{,}\gamma }\left\{z{\mathrm{\Omega }}_{\alpha \text{,}\lambda \text{,}\rho }(z\text{,}s\text{,}a)\right\}=\frac{\alpha \lambda {(\gamma +1)}^{\beta -\tau }\mathrm{\Gamma }(\tau )}{\rho \mathrm{\Gamma }(\beta )\mathrm{\Gamma }(1-\beta +\tau )}{z}^{(1-\beta +\tau )\gamma +1}\hfill \\ \hfill & \times {\displaystyle \sum _{\kappa =0}^{\infty }}\left(\frac{{(\alpha +1)}_{\kappa }{(\lambda +1)}_{\kappa }}{{(\rho +1)}_{\kappa }\mathrm{\Gamma }(\kappa +2)}\times B\left(\frac{\kappa +\beta }{\gamma +1}+1\text{,}1-\beta +\tau \right)·\frac{z^{\kappa }}{{(\kappa +1+a)}^s}\right)\hfill \end{array}$$ where ${\mathrm{\Omega }}_{\alpha \text{,}\lambda \text{,}\rho }(z\text{,}s\text{,}a)$ is the extended general Hurwitz-Lerch Zeta function (see Bateman and Erdélyi, 1953; Lin and Srivastava, 2004).

Let $f\in \mathcal{S}$ . If

• (i) for $0<\beta ⩽1\text{,}0<\tau \le 1$ such that $0⩽\beta -\tau <1$ and

• (ii) ${\rho }_i>0\text{,}i=1\text{,}\dots \text{,}p$ and ${\lambda }_j>0\text{,}j=1\text{,}\dots \text{,}q$ ; $p⩽q+1$ ,

Let the condition (i) as the Theorem 3.1 be satisfied. If $0⩽\beta -\tau <1$ , $${}_2{\mathrm{\Psi }}_1\left[\left.\begin{array}{c}(2\text{,}1)\text{,}\left(1+\frac{\beta }{\gamma +1}\text{,}\frac{1}{\gamma +1}\right)\hfill \\ \left(1-\beta +\tau +\frac{\beta }{\gamma +1}\text{,}\frac{1}{\gamma +1}\right)\hfill \end{array}\right|1\right]<2\left(\frac{\mathrm{\Gamma }(\frac{\beta }{\gamma +1}+1)}{\mathrm{\Gamma }(\frac{\beta }{\gamma +1}+1-\beta +\tau )}\right)$$ then ${\mathrm{\Theta }}^{\beta \text{,}\tau \text{,}\gamma }:\mathcal{K}\to \mathcal{K}$ .

A holomorphic function $f\in \mathcal{H}(\mathbb{U})$ is said to be in Bloch space $\mathbb{B}$ whenever $\Vert f{\Vert }_{\mathbb{B}}={\sup }_{z\in \mathbb{U}}\left(1-|z{|}^2\right)|f^{\prime }(z)|<\infty$ and the little Bloch space ${\mathbb{B}}_0$ is given as follows $${\displaystyle \underset{|z|\to 1-}{\lim }}\left(1-|z{|}^2\right)|f^{\prime }(z)|=0\text{.}$$

Let $w:[0.1)\to [0\text{,}\infty )$ and f be an analytic function on open unit disk $\mathbb{U}$ which is said to be in the weighted Bloch space ${\mathbb{B}}_w$ if $$\left(1-|z|\right)|f^{\prime }(z)|<hw(1-|z|)\text{,}z\in \mathbb{U}$$ for some $h>0$ . Note that, if $w=1$ then ${\mathbb{B}}_1\equiv \mathbb{B}$ . Further, the weighted $\mu$ -Bloch space ${\mathbb{B}}_w^{\mu }$ , covering of all $f\in {\mathbb{B}}_w^{\mu }$ defined by $${\displaystyle \underset{|z|\to -1}{\lim }}\frac{{(1-|z|)}^{\mu }|f^{\prime }(z)|}{w(1-|z|)}=0$$ and

It is easy to note that if an analytic function $g(z)\in {\mathbb{B}}_w^{\mu }$ , then

(Ruscheweyh, 1982) Let f and g be two analytic functions. Then $$z{(g*f)}^{\prime }(z)≔g(z)*{\mathit{zf}}^{\prime }(z)\iff {(g*f)}^{\prime }(z)=\frac{g(z)}{z}*f^{\prime }(z)\text{.}$$

Let f be an analytic function on the open unit disk $\mathbb{U}$ , and ${\mathbb{B}}_w^{\mu }\text{;}w:[0\text{,}1)\to [0\text{,}\infty )$ . Then $$f\in {\mathbb{B}}_w^{\mu }\iff {\mathrm{\Theta }}^{\beta \text{,}\tau \text{,}\gamma }f\in {\mathbb{B}}_w^{\mu }\text{.}$$

Let f be an analytic function on open unit disk $\mathbb{U}$ , and ${\mathbb{B}}_w^{\mu }\text{;}w:[0\text{,}1)\to [0\text{,}\infty )$ , then the operator ${\mathrm{\Theta }}^{\beta \text{,}\tau \text{,}\gamma }f:{\mathbb{B}}_w\to {\mathbb{B}}_w^{\mu }$ is compact.