Vol. 42, No. 2, 1972

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Inequalities involving f_p and f(n)q for f with n zeros

James Edwin Brink

Vol. 42 (1972), No. 2, 289–311

Let  p denote the Lp-norm. This paper determines the smallest possible constants C which satisfy

∥f∥p ≦ C ⋅(b− a)s∥f (n)∥q

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.

Mathematical Subject Classification 2000
Primary: 26A84
Secondary: 34B99
Received: 11 February 1971
Revised: 16 June 1971
Published: 1 August 1972
James Edwin Brink