Dual Approach to the Generalization of Extended Bessel Function

Abstract

In this paper, we discuss the geometrical interpretation of generalized Bessel function, which is defined as:
\begin{equation*}
_{k}H_{\xi,b}(z) = z \ . \ _{k}h_{\xi,b}(z) \ = z^{2} + \sum\limits_{r=1}^{\infty}\frac{(-b)^{r}  \ z^{r+2}}{r! \ 4^{r} \ k^{r} \ (\xi)_{r,k}}
\end{equation*}
where $\xi = v+k \in (0,+\infty)$, $k \in \mathbb{R}^{+}, \quad v>-k ,\quad b \in \mathbb{R}$.
The generalization of Pochammer's symbol in the form of inequality:
\begin{equation*}
(q)_{r,k} > q(q+\beta)^{r-1}
\end{equation*}
for $q>0, k \in \mathbb{R}^{+}, 0\leq \beta \leq \beta_{0} = \sqrt{2} \simeq1.4142..., r\in \mathbb{N} \backslash \{1,2\}$, which is proved by using the generalization of Lemma \cite{Bulboaca}. This has been proved by many authors by using different methods. Using this inequality to analyse the order of starlikeness and convexity. We  prove this lemma by the same technique used by Zayed and Bulboaca (partial derivative and two-variable extremum technique). Also, we prove the geometrical interpretation of Generalized Bessel $k$-function for different values of $k$. Providing some examples for better understanding of the reader regarding our approach.