On the Extension of a Functional Inequality of S. Bernstein to Non-analytic Functions
نویسندگان
چکیده
We wish to demonstrate here the following elementary inequality of the differential calculus. If a function satisfies the conditions {ƒ(*)} ^ 1 and {/(x)} 2 + {F-»(x)} ^ 1 for all x and for some positive integer n, then the latter inequality is valid also when n is replaced by any smaller positive integer. Tha t such an inequality might be true is suggested by the validity of a similar but more specialized inequality concerning trigonometric polynomials. Thus,if P(x) =^2o {a„cos (vx/N) +6,s in (vx/N)} and if {P(x)} ^ 1 for all x, it has been proved that {P<*> (x)} + {P^~ (x)} ^ 1 , (fe = l, 2, 3, • • • ). This theorem, a refinement of a theorem of S. Bernstein, has been proved by several different methods, and generalizations have been given which prove that the inequality is true for a wider class of analytic functions. I t will be shown here that this theorem is a rapid deduction from the elementary inequality given above. Moreover, this method of proof serves to distinguish those features of Bernstein's theorem arising from the characteristic properties of trigonometric polynomials from those which are merely properties of the differential coefficient. The second part of this paper is concerned with finding the functions which cause the inequality to become an equality at some point. For example, if {jf(#)} 2 + {/0*0 } S 1 and {ƒ(*)} ^ 1, we find that the equality ƒ'(O) = 1 necessitates t h a t / ( x ) = s i n x. I t is to be noted
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