  In this paper our aim is to show some new inequalities of Redheffer type for
Bessel and modified Bessel functions of the first kind. The key tools in our
proofs are some classical results on the monotonicity of quotients of
differentiable functions as well as on the monotonicity of quotients of two
power series. We use also some known results on the quotients of Bessel and
modified Bessel functions of the first kind, and by using the monotonicity of
the Dirichlet eta function we prove a sharp inequality for the tangent
function. At the end of the paper a conjecture is stated, which may be of
interest for further research.
