Abstract
We establish some new inequalities for the modified Bessel-type function lambda _{nu ,sigma }^{(beta )} (x ) studied by Glaeske et al. [in J. Comput. Appl. Math. 118(1–2):151–168, 2000] as the kernel of an integral transformation that modifies Krätzel’s integral transformation. The inequalities obtained are closely related to the generalized Hurwitz–Lerch zeta function and complementary incomplete gamma function. We also deduce some useful inequalities for the modified Bessel function of the second kind K_{nu } (x ) and Mills’ ratio mathsf{M} (x ) as worthwhile applications of our main results.
Highlights
Introduction and motivationIn a series of papers [2,3,4], E
Where λ(νn)(x) is a Bessel-type function defined by λ(νn)(x)
We prove some auxiliary results which are required in the proofs of our main results
Summary
We establish some inequalities for the function λ(νβ,σ) (x) defined by λ(νβ,σ). 3 involve the complementary incomplete gamma function Γ (a, z) and the generalized Hurwitz–Lerch zeta function Φμ∗(z, s, a). Their definitions and various properties are presented in Sect. The generalized Hurwitz–Lerch zeta function Φμ∗(z, s, a) was introduced in [18] (see [19,20,21] and [22]) and is of the form: Φμ∗ (z, s, a) =. We prefer here to examine a simpler case of determining the asymptotic behavior of the function defined by (10) and the result is included in the following proposition.
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