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1. It has been shown by S. Bernstein that if a function which is defined in an open interval is infinitely often differentiable (belongs to C°°) and each of its derivatives is of constant sign then the function is analytic in the interval^). As a possible generalization of this result Polya raised the following question (2): If in an open interval a function is infinitely often differentiable and no derivative changes sign more than a given number of times is the function analytic in the interval} The present paper contains an answer to this question and to several related questions. These results are consequences of an inequality (Theorem I of the present paper) which relates the magnitude of the first derivative of a function to the number of times some higher derivative changes sign. All functions considered in this paper will be supposed real for real values of the variable. In the case in which f(x) is infinitely often differentiable over the interval ( — oo , cc) let A7* be the maximum number of sign changes of f(k)(x) over any interval of length a, where a is some arbitrary but fixed positive number. Polya and Wiener(3) made the additional hypothesis that f(x) is periodic and found that if 7^ = 0(1) then f(x) is a trigonometric polynomial, while if Nk = o(k112) then/(x) is an entire function. Szegö obtained a new proof of these results and indeed obtained the sharper theorem(4) that if A7* < A/log k for large k then/(x) is an entire function. Hille(5) has obtained analogous theorems for a very general class of differential operators and functions which satisfy appropriate conditions. Let us consider for the moment functions which are bounded and infinitely often differentiable over (—«>, °°), but are otherwise completely general. It is shown in the present paper that if 2V* = 0(1) then/(x) is an entire function of exponential type, and if Nk = 0(logyk), y<l, then/(x) is an entire function(6). In the case in which TV* is bounded this is a generalization of

On the oscillation of differential transforms. III. Oscillations of the derivative of a function