Let L defined by
(Lf)(z) = pmzmf(m)(z) +⋯+ p1zf′(z) + p0f(z) be an Euler-type differential
operator with positive coefficients pj and let A0 denote the class of functions
analytic in |z| < 1 which satisfy f(0) = 0. For example, the function g(z) = cz
belongs to A0, if c is a constant. Clearly (Lg)(z) = cz(p1+ p0) and hence,
if |(Lg)(z)|≤ 1 for |z| < 1, we must have |g(z)|≤ 1∕λ for |z| < 1, where
λ = p0+ p1. Here it is shown that the same result holds for all f ∈ A0,
provided p0≤ 2p2, and that the latter condition is sharp. Our result solves, in
sharpened and generalized form, a problem that has been open since 1978. An
important aid in the proof is a recent theorem of Brown and Hewitt, which
improves a well-known criterion of Vietoris for positivity of certain trigonometric
sums.
