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

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Abstract

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 )

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