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)-βK 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))/(2T (μ+ν)/2,α,1 (s)), and s↦d n 1 dν n 1 T ν,α,1 (s)/d n 2 dν n 2 T ν,α,1 (s) to be monotonic in s∈(0,∞) by employing the monotonicity rules.

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