SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND

. By employing a rened version of the P´olya type integralinequality and other techniques, the authors establish some inequalitiesand absolute monotonicity for modied Bessel functions of the rst kindwith nonnegative integer order. 1. Main resultsIt is well known that modied Bessel functions of the rst kind I ±ν (z) aresolutions of the dierential equationz 2 d 2 wdz 2 +zdwdz−z 2 +ν 2 w = 0.They are holomorphic functions of z throughout the z-plane cut along thenegative real axis, and are entire functions of ν for xed z 6= 0. When ν = ±n,I ν (z) are entire functions of z. In [1, p. 375, 9.6.7], it is listed thatI ν (z) =X ∞k=0 1k!Γ(ν +k +1)z2 2k+ν , z ∈ C, ν ∈ R\{−1,−2,...},whereΓ(z) = lim n→∞ n!n z Q nk=0 (z +k), z ∈ C\{0,−1,−2,...}is the classical gamma function, see [1, p. 255, 6.1.2].On [12, p. 63], the following three double inequalities are derived:1− z/21+ z/2 0, ν ≥ −12; Received July 18, 2015.2010 Mathematics Subject Classication. Primary 33C10; Secondary 26A48, 26D15,44A10.Key words and phrases. inequality, absolute monotonicity, complete monotonic function,modied Bessel function of the rst kind, P´olya type integral inequality.