This paper studies the geometric properties of a normalized form of the Lommel function of the first kind, including starlikeness, convexity, and close-to-convexity, employing a relatively simple and efficient algorithm. Our approach relies on properties of the generalized function 1 \(\begin{aligned} \varphi _{k_1,k_2,c}(z) = z + \sum _{n=2}^{\infty } \frac{(-c)^{n-1}}{4^{n-1}(k_1)_{n-1}(k_2)_{n-1}} \, z^{n}, \;\, z \in {\mathbb {C}} \end{aligned}\) where \(k_1 = (a-b+3)/2\) and \(k_2 = (a+b+3)/2\) with \(a \pm b\) being non-negative odd integers and \(c \in {\mathbb {C}}\) . The function (1) can be defined as the classical convolution product of \(z/(1-cz)\) , \(z \in {\mathbb {C}}\) , and the normalized form of the Lommel function of the first kind. The results complement and, in some cases, improve existing results in the literature concerning the orders of starlikeness and convexity associated with the generalized function defined by (1) in the open unit disk \({\mathbb {U}} = \{z \in {\mathbb {C}} :|z|<1\}\) . We then compare the properties of the specified functions with those of more commonly used functions of the same class. We determine conditions for \((k_1, k_2)\) and c for which \(f \in \mathcal {R}(\beta ) = \left\{ f \in \mathcal {A}({\mathbb {U}}) :\operatorname {Re}f^{\prime }(z)> \beta , \, z \in {\mathbb {U}} \right\}\) , \(0 \le \beta <1\) , indicates that the generalized convolution \(\varphi _{k_1,k_2,c} *f\) belongs to \(\mathcal {H}^{\infty }({\mathbb {U}})\) . Here, \(\mathcal {A}({\mathbb {U}})\) denotes the class of normalized analytic functions in \({\mathbb {U}}\) and \(\mathcal {H}^{\infty }({\mathbb {U}})\) is the space of bounded analytic functions on \(\mathcal {A}({\mathbb {U}})\) . We also obtain sufficient conditions for the expansion coefficients of \(f \in \mathcal {A}({\mathbb {U}})\) to belong to some subclasses of univalent functions.
