Bounds for functions involving ratios of modified Bessel functions

We give bounds for the functions f ν ( x ) = I ν + 1 ( x ) x I ν ( x ) and g ν ( x ) = I ν ′ ( x ) I ν ( x ) , using the Mittag-Leffler expansion. Some of the obtained bounds are already known and other are sharper than the known or extend the validity of ν.