Sufficient conditions for analyticity (Q1322621)

The author generalizes the Bernstein's theorem on absolutely monotonic functions replacing derivatives \({d^ n\over dx^ n} f\) by operators \[ {d^ n\over dx^ n} (\varphi^{(n)} (x- a) f) \] or by operators \[ L_ n f= \varphi^{- n}(x) \textstyle{{d\over dx}} [\varphi^ n(x) L_{n- 1} f], \] where \(\varphi\) belongs to a special class of entire functions.