Functional Inequalities for Galué’s Generalized Modified Bessel Functions
نویسنده
چکیده
Let aIp(x) = ∑ n 0 (x/2)2n+p n!Γ(p + an + 1) be the Galué’s generalized modified Bessel function depending on parameters a = 0, 1, 2, . . . and p > −1. Consider the function aI p : R → R, defined by aI p(x) = 2pΓ(p+1)x−paIp(x). Motivated by the inequality of Lazarević, namely cosh x < ( sinh x x )3 for x = 0, in order to generalize this inequality we prove that the Turán-type, Lazarević-type inequalities [aI p+1(x)] aI p(x)aI p+2(x), [aI p(x)] [aI p+1(x)] hold for all x ∈ R. Moreover, we prove that the functions p → aI p+1(x)/aI p(x), p → [aI p(x)] are increasing on (−1,∞).
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