On monotonic and analytic functions in \(C^{\infty }\)
This paper presents simple sufficient conditions for analyticity of a real function, given on some open interval in \(\mathbb R\). The basic result in this field is the following well-known theorem of S. Bernstein. Theorem B. Let \(I=(-d,d)\). If \(f\) is a regularly monotonic function on \(I\) (i.e., for each \(n\geq 0\) \(f^{(n)}\) does not change sign on \(I\)), then for any \(x\in I\) we have \[ f(x) = \sum ^{\infty }_{n=0} \frac {f^{(n)}(0)}{n!} x^n. \tag \(*\) \] Now, the author generalizes Theorem B, e.g., as follows. Theorem I. Let \(I=(-d,d)\). Further, let \(\{c_n\}^{\infty }_0\) be a sequence of real numbers such that the sequence \(\{\frac {c_nd^n}{n!}\}^{\infty }_0\) is bounded and the functions \(f^{(n)}-c_n, \;n = 0,1,\ldots,\) do not change sign on \(I\). Then for any \(x\in I\) \((*)\) holds. From Theorem I three similar results are derived. All proofs are elementary (by Taylor's remainder theorem), but far from triviality.
scientific article; zbMATH DE number 1449453