The paper deals with the modified Bessel function of the first kind and order \(p\), denoted by \(I_{p}(x)\), \(x\in R\), \(p\neq -1,-2,\dots\) and the functions \(\mathcal{I}_{p}(x)=2^{p}\Gamma (p+1)x^{-p}I_{p}(x)\), \(\gamma _{p}(x)=\mathcal{I}_{p}(\sqrt{x})\) and \(v_{p}(x)=2(p+1){{\gamma _{p}(x^{2})}\over {\gamma _{p+1}(x^{2})}}\). The authors prove logarithmic convexity (logarithmic concavity), monotonicity and inequalities obeyed by the above functions. This paper is a continuation of \textit{E. Neuman}'s paper in [J. Math. Anal. Appl. 171, 532--536 (1992; Zbl 0769.33004)].