MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 152 OCTOBER 1980, PAGES 1191-1193 On Determination of Best-Possible Constants in Integral Inequalities Involving Derivatives By Beny Neta Abstract. This paper is concerned with the numerical approximation of the best possible constants yn k in the inequality ll^ll2 < Tñjfcíll^ll2 + llf (n)ll2}- where HI2 = ¡Ô \F(x)\2 dx. A list of all constants yn k for n < 10 is given. 1. Introduction. This paper utilizes the algorithm given in [1] to numerically approximate the best possible constants ynk, 1 <k < n, for n < 10 in the inequality: (*) ||F<*>||2 <T~1fc{ll/qi2+ |1F^>H2}S where || • || denotes the L2 [0, °°) norm. The function F has a locally absolutely con- tinuous (n - l)st derivative. The inequality (1) is equivalent to (2) ||F<*>|| < Mnk \\F\\(n-k)ln |[F(«)||fcAl) where (3) M2 =y-1 f^A] +( -*_] Mn,k yn,k\ k J *\n-kj see [1]. interest in inequalities (1) and (2) increased because of their close connection with problems of best approximation of the differentiation operator by bounded oper- ators; see [2], [3], [4], [5], and with the problem of best approximation of one class of functions by another; see [4], [6], [7]. In the next section we shall give lower and upper bounds for the best possible constants yn k and Mn k for n < 10. 2. Numerical Results. In this section the best possible constants yn k and Mn k are listed. 721 =1, see [1]. 731 = 732 = V^ -2V2 = .555669, see [1]. Received October 31, 1978; revised February 1, 1980. AMS (MOS) subject classifications (1970). Primary 46E30, 26A84; Secondary 47E05, 34B05, 65D20. © 1980 American Mathematical Society 002 5-571 8/80 /0000-01 60/$01.75 1191 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use