# Starlikeness of the generalized Bessel function

Research paper by **Rosihan M. Ali, See Keong Lee, Saiful R. Mondal**

Indexed on: **02 Jul '17**Published on: **02 Jul '17**Published in: **arXiv - Mathematics - Complex Variables**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.