Norm inequalities for a certain class of ∞ functions

In 1936 the author showed that the function sin(π(x+1)/4) is the entire function of least exponential type (=π/4) among all entire functionsf(z) with the property thatf ⁽ⁿ⁾(z) vanishes somewhere in the real interval −1, 1] (n=0, 1,2,…). Now more precise results of this kind are obtained by working within the class ∞−1, 1]. For Paul Montel on his 95th birthday