Eine Verschärfung der Abschätzung des Restes Taylorscher Näherungspolynome. (An improvement of the estimate of the remainder term of Taylor polynomials) (Q1076923)

The paper begins with a new short proof of (I) Let P(t) in \(R^ d\) be a function of t having n continuous derivatives for \(a\leq t\leq x\). Then P(x)\(\in conv K\), where \[ K=\{\sum^{n-1}_{j=0}\frac{(x-a)^ j}{j!}P^{(j)}(a)+\frac{(x-a)^ n}{n!}P^{(n)}(t),\quad a\leq t\leq x\}. \] For applying (I) let f(t) be a real function such that the point \(((t- a)^{n+1},f(t))\) fulfills the conditions of (I). Then (I) gives a sharper estimate of the nth remainder term of f(x) than the Lagrange remainder formula. If \(f^ n(t)\) is also convex in \(a\leq t\leq x\), then f(x)\(\in [c,d]\), where \[ c=\sum^{n-1}_{j=0}\frac{(x-a)^ j}{j!}f^{(j)}(a)+\frac{(x-a)^ n}{n\quad !}f^{(n)}(\frac{na+x}{n+1}), \] \[ d=\sum^{n-1}_{j=0}\frac{(x-a)^ j}{j!}f^{(j)}(a)+\frac{(x-a)^ n}{n\quad !}\cdot \frac{nf^{(n)}(a)+f^{(n)}(x)}{n+1}. \]