In the article, we prove that the double inequalities hold for all if and only if and if , where is the modified Bessel function of the second kind. As applications, we provide bounds for with and present the necessary and sufficient condition such that the function is strictly increasing (decreasing) on .

In the article, we prove that the double inequality holds for all , we present the best possible constants λ and μ such that for all , and we find the value of in the interval such that for and for , where is the classical gamma function, is Euler-Mascheroni constant and  .

Abstract. Let Iν (x) be the modified Bessel functions of the first kind of order ν, and Sp,ν (x) = Wν (x) 2

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