This paper explores a new perspective to determine the sufficient conditions on the parameters \(\zeta \) and r such that the normalized generalized Bessel functions defined by \(\begin{aligned} \textrm{U}_{\zeta ,r}(z)=z+\sum _{k=1}^{\infty }\frac{(-r)^{k}}{4^{k}(1)_{k}(\zeta )_{k}}z^{k+1},\;z\in \mathbb {D}, \end{aligned}\) belongs to the classes of lemniscate starlike and convex functions. The technique we used for obtaining our results is based on using the Cauchy’s product as well as some new inequalities that can be proved by different methods. Our findings improve some of the results recently given for a specific domain, like it could be seen in the paper.
