TY - JOUR

T1 - Amos-type bounds for modified Bessel function ratios.

AU - Hornik, Kurt

AU - Grün, Bettina

PY - 2013/10/1

Y1 - 2013/10/1

N2 - We systematically investigate lower and upper bounds for the modified Bessel function ratio Rν=Iν+1/Iν by functions of the form View the MathML source in case Rν is positive for all t>0, or equivalently, where ν≥−1 or ν is a negative integer. For ν≥−1, we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If ν≥−1/2, the minimal elements are tangent to Rν in exactly one point 0≤t≤∞, and have Rν as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if ν is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively.

AB - We systematically investigate lower and upper bounds for the modified Bessel function ratio Rν=Iν+1/Iν by functions of the form View the MathML source in case Rν is positive for all t>0, or equivalently, where ν≥−1 or ν is a negative integer. For ν≥−1, we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If ν≥−1/2, the minimal elements are tangent to Rν in exactly one point 0≤t≤∞, and have Rν as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if ν is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively.

UR - http://www.sciencedirect.com/science/article/pii/S0022247X13005374?np=y

U2 - 10.1016/j.jmaa.2013.05.070

DO - 10.1016/j.jmaa.2013.05.070

M3 - Journal article

VL - 408

SP - 91

EP - 101

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -