Bounds for the first positive zero of a mixed Bessel function

摘要

The present article is concerned with lower and upper bounds of the first positive zero of the function Hν(z, α) = αJν(z) + zJ′ν(z), Where Jgn(z) is the ordinary Bessel function of order ν > −1 and J′ν(z) is the derivative of Jν(z). A lower bound found here improves and extends the range of validity of the order ν, of a lower bound found in a previous work [8]. Also, two upper bounds given here improve a previously known upper bound [8]. In the particular case α = 0, these bounds lead to lower and upper bounds for the first positive zero j′ν,1 of J′ν(z) which improve well-known bounds in the literature.