Starlikeness of the generalized Bessel function

Abstract

For a fixed $a \in \{1, 2, 3, \ldots\},$ the radius of starlikeness of positive order is obtained for each of the normalized analytic functions \begin{align*} \mathtt{f}_{a, \nu}(z)&:= \bigg(2^{a \nu-a+1} a^{-\frac{a(a\nu-a+1)}{2}} \Gamma(a \nu+1) {}_a\mathtt{B}_{2a-1, a \nu-a+1, 1}(a^{a/2} z)\bigg)^{\tfrac{1}{a \nu-a+1}},\\ \mathtt{g}_{a, \nu}(z)&:= 2^{a \nu-a+1} a^{-\frac{a}{2}(a\nu-a+1)} \Gamma(a \nu+1) z^{a-a\nu} {}_a\mathtt{B}_{2a-1, a \nu-a+1, 1}(a^{a/2} z),\\ \mathtt{h}_{a, \nu}(z)&:= 2^{a \nu-a+1} a^{-\frac{a}{2}(a\nu-a+1)} \Gamma(a \nu+1) z^{\frac{1}{2}(1+a-a\nu)} {}_a\mathtt{B}_{2a-1, a \nu-a+1, 1}(a^{a/2} \sqrt{z}) \end{align*} in the unit disk, where ${}_a\mathtt{B}_{b, p, c}$ is the generalized Bessel function \begin{align*} {}_a\mathtt{B}_{b, p, c}(z):= \sum_{k=0}^\infty \frac{(-c)^k}{k! \; \mathrm{\Gamma}{\left( a k +p+\frac{b+1}{2}\right)} } \left(\frac{z}{2}\right)^{2k+p}. \end{align*} The best range on $\nu$ is also obtained for a fixed $a$ to ensure the functions $\mathtt{f}_{a, \nu}$ and $\mathtt{g}_{a, \nu}$ are starlike of positive order in the unit disk. When $a=1,$ the results obtained reduced to earlier known results.