We remark that there is no smooth flat function f (x) on [0,∞) such that for a fixed interval (0, α) (α > 0) on which the derivative of any higher order is positive

Kazuo MASUDA

Yoshihiko MITSUMATSU

application/pdfWe remark that there is no smooth flat function f (x) on [0,∞) such that for a fixed interval (0, α) (α > 0) on which the derivative of any higher order is positive.othe

MASUDA, Kazuo

23329

MITSUMATSU, Yoshihiko

23350

Chuo University Academic Information Repository / 中央大学学術リポジトリ

CHUO MATH NO.113(2015)DEPARTMENT OF MATHEMATICSCHUO UNIVERSITYBUNKYOKU TOKYO JAPANA small remark on flat functionsbyKazuo MASUDAand Yoshihiko MITSUMATSUSept. 22 , 2015DEPARTME   THEMATICSBUNKYOKU  JAPANver. 1.1, September 11, 2015A small remark on flat functionsKazuo MASUDA and Yoshihiko MITSUMATSUAbstractWe remark that there is no smooth flat function f (x) on [0,¥)such that for a fixed interval (0, a) (a > 0) on which the derivative ofany higher order is positive.We consider smooth functions on the half line [0,¥) which are flat atthe origin, namely of class C¥ and any derivative f (n)(x) converges to 0when x ! 0+ 0. Eventually it is equivalent to say that f extends to thewhole real line as a smooth function by defining f (x) = 0 for x < 0. In thistiny note wemake a small remark on the asymptotics of higher derivativesaround the origin.Among non-tirivial flat functions the most well-known might be theone which is defined as follows.f (0) = 0 and f (x) = e 1x for x > 0If we imagine its graph, of course it seems smooth enough, and it canbe extended as constantly 0 on ( ¥, 0] as a smooth function on the realline R. Its first derivative is positive on (0,¥), but the second derivativevanishes at x = 12 = x2 and the third vanishes at x3 =1 1/p32 < x2, and soon. That is, setting xn = minfx ; f (n)(x) = 0, x > 0g for n = 2, 3, 4, . . . ,it is clear that fxng is strictly decreasing, and in fact limn!¥ xn = 0. Moreover, if we fix any interval [0, a) (a > 0), f (n)(x) tend to behave more andmore wildly when n ! ¥ on the interval.Also, if we take g0(x) = f (x)(sin( 1x ) + 1) andgn(x) =Z x0Z tn 10  Z t10g0(t0)dt0    dtn 2dtn 1 ,then for n = 1, 2, 3,    , gn(x) is positive on (0,¥) and is flat at x = 0,and apparently g(k)n (x) > 0 when x > 0 for 0  k  n  1 but there is nointerval (0, a) on which g(n)n (x) is positive.They present some features of a general property of positive flat func-tions, which is formulated as follows.Theorem 1 Let f (x) be a smooth function on [0,¥) which is flat at x =0. Then there does not exist positive real number a such that for any higherderivatives (i.e., for 8n 2N), f (n)(x) > 0 holds on (0, a].Lemma 2 Let n be an integer and g(x) be a function on [0, 1] of classCn+1 with the following properties.1(1) g(k)(0) = 0 for k = 0, ... , n, and g(1) = 1,(2) g(n+1)(x) > 0 for x > 0.Then g(x) < xn holds on (0, 1).Proof of Theorem. The theorem is easily deduced from the lemma by con-tradiction. Assume for some a > 0 that f (x) is smooth on [0, a], is flat atx = 0, and that its n-th derivative is positive on (0, a] for any n 2 N. Weadjust the function f into g(x) = f (a) 1 f (ax). Then g(x) satisfies the con-dition of the lemma for any n 2 N. Therefore g(x)  0 on [0, 1), and weobtained a contradiction. Proof of Lemma. It is enough to show that g(x)/xn is increasing on [0, 1].Asddxg(x)xn=xg0(x)  ng(x)xn+1, it is also sufficient to show that the nu-merator xg0(x)  ng(x) is positive on (0, 1).Then because (xg0(x)   ng(x))(n) = xg(n+1)(x) is positive on (0, 1]from our condition, we see successively that each k-th derivative (xg0(x) ng(x))(k) = xg(k+1)  (n  k)g(k)(x) vanishes at x = 0 and therefore is pos-itive on (0, 1] for k = n  1, n  2, . . . , k = 0. This completes the proof. Acknowledgement: The second author proposed the problem. The firstauthor proved it and the second one simplified the proof. The authors aregrateful to Hiroki Kodama for the colaborations in simplifying the proof.Kazuo MASUDAmasuda@math.titech.ac.jpYoshihiko MITSUMATSUDepartment of Mathematics,Chuo University1-13-27 Kasuga Bunkyo-ku,Tokyo, 112-8551,Japanyoshi@math.chuo-u.ac.jp2

A small remark on flat functions

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A small remark on flat functions

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