Two-sided bounds uniform in the real argument and the index for modified Bessel functions (Q1974359)

Two-sided bounds are derived for the modified Bessel functions and the functions \(a_\nu(x)=xI_\nu'(x)/I_\nu(x)\) and \(b_\nu(x)=xK_\nu'(x)/K_\nu(x)\) for \(x>0\), \(\nu\geq 0\), except for some neighborhoods of the point \((x,\nu)=(0,0)\). The bounds are obtained by using the Riccati equation for \(a_\nu(x)\), \(b_\nu(x)\), and a general theorem on inequalities for solutions of a type of differential equations.