Let ∥∥p denote the
Lp-norm. This paper determines the smallest possible constants C which
satisfy
for certain classes of n-times continuously differentiable functions having n zeros on
some interval [a,b]. Particular interest is placed on functions having α zeros at a and
n − α zeros at b. It is shown that smallest possible constants exist for all positive
integers n, for all extended real numbers p and q not less than one, and for
α = 0,… , n providing the exponent s is chosen properly. Moreover, these
constants can be used to determine best possible constants when the n zeros
are restricted only by the condition that α are at a and β < n − α are at
b.
