Certain inequalities for the modified Bessel-type function

Abstract

Abstract We establish some new inequalities for the modified Bessel-type function λν,σ(β)(x) $\lambda _{\nu ,\sigma }^{(\beta )} (x )$ 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 Kν(x) $K_{\nu } (x )$ and Mills’ ratio M(x) $\mathsf{M} (x )$ as worthwhile applications of our main results.

Keywords

• Čebyšev inequality
• Generalized Hurwitz–Lerch zeta function
• Hölder’s inequality
• Incomplete gamma function
• Mills’ ratio