On the successive derivatives of functions in a quasianalytic class

G. V. Badalyan

Izvestiya. Mathematics , Tome 2 (1968) no. 4, pp. 745-779

Résumé

In this paper we investigate the question of the signs in the sequence $\{(-1)^n\varphi_n(u)\}$, where $\varphi_0(u)=\varphi(u)$, $\varphi_1(u)=\varphi'(u)$, $\dots$, $$ \varphi_{k+1}(u)=\varphi{k+1}(u)_\gamma=\biggl(\frac{\varphi_k(u)}{u^{\gamma_k-\gamma_{k-1}-1}}\biggr)', \quad k=1,2,\dots, $$ $0=\gamma_0\gamma_1\leqslant\gamma_2\leqslant\dots\leqslant\gamma_n\leqslant\dots\to\infty$, when the real function $\varphi(t)$ belongs to a certain quasianalytic class in the sense of Carleman (according to the classification suggested by the author). A particular corollary of the result given in the paper is the correctness of Borel's hypothesis that there cannot exist a quasianalytic function $f(x)$ all of whose derivatives are positive at a given point in the domain of quasianalyticity of the function, except when the function is analytic.