Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

In this paper, we introduce some monotonicity rules for the ratio of integrals. Furthermore, we demonstrate that the function - T ν , α , β ( s ) is completely monotonic in s and absolutely monotonic in ν if and only if β ≥ 1 , where T ν , α , β ( s ) = K ν 2 ( s ) - β K ν - α ( s ) K ν + α ( s ) defined on s > 0 and K ν ( s ) is the modified Bessel function of the second kind of order ν . Finally, we determine the necessary and sufficient conditions for the functions s ↦ T μ , α , 1 ( s ) / T ν , α , 1 ( s ) , s ↦ ( T μ , α , 1 ( s ) + T ν , α , 1 ( s ) ) / ( 2 T ( μ + ν ) / 2 , α , 1 ( s )