In this paper, we establish several monotonicity results concerning the modified Lommel function, the modified Bessel function of the first kind, and the modified Struve function of the first kind. Specifically, we establish necessary and sufficient conditions for the monotonicity of the function \(x\mapsto {\mathcal {L}}_{\mu _1,\nu _1}(x)/{\mathcal {L}}_{\mu _2,\nu _2}(x)\) under the assumptions \(\mu _1 > -3\) , \(\mu _2 > -3\) , \(|\nu _1| \le \mu _1 + 3\) , and \(|\nu _2| \le \mu _2 + 3\) , where \({\mathcal {L}}_{\mu ,\nu }(x) = \frac{x^{-\mu -1} 2^{1-\mu }}{ \Gamma \left( (\mu - \nu + 1)/2\right) \Gamma \left( (\mu + \nu + 1)/2\right) } t_{\mu , \nu }(x)\) and \(t_{\mu ,\nu }(x)\) denotes the modified Lommel function. This result fills a research gap in the earlier works by Mondal (2019) and Gaunt (2022). As corollaries, we obtain several monotonicity results for the modified Bessel and Struve functions of the first kind, including known results as well as new findings. Moreover, we present necessary and sufficient conditions for the function \(y \mapsto \int _0^y e^{-\beta x} {\mathcal {L}}_{\mu _1,\nu _1}(x) \text {d} x/\int _0^y e^{-\beta x} {\mathcal {L}}_{\mu _2,\nu _2}(x) \text {d} x\) to be monotonic and establish several related properties for it, where \(\beta >0\) .
