Article信州大学工学部紀要 66: 1-5 (1989)departmental bulletin pape

SAKAI,  Yuji

English

SAKAI  Yuji

Shinshu University Institutional Repository

      Journal of the Faculty of Engineering, Shinshu University, No. 66, l989 1                         fitiiMJi<S}fiIe}EiYSfiEas gg66g   Convex Functions and g-Rearrangements on Interuals                                by                            Yuji SAKAI*                        (Received May 31, l989)     Two problems are dealt with : (1) Is a rearrangement of a convex function  on I fii! (O, a) always a convex function?; (2) Is it possible to give conditions for  (1), if necessary?     A new concept called generators on Lebesgue space is first introduced and  the equimeasurable rearrangement fA of a function f with respect to a generator  is defined, resuking generalization of the decreasing rearrangement f* of I Then  the transmission function of a generator is defined, and the relation between the  convexities of functions f and fA is studied, It is proved that f* is convex when-  ever f is convex on L Conditions for generators are obtained which make fA convex  whenever f is convex.                 1. Notations and Preliminaries   Throughout this paper, assurne a G (e, oo), let Xm---{(O, a), ta, m} beLebesgue measure space on I iEi (O, a), and denote by .ts' the set of all realvalued measurable functions on X. A function f on aR interval K c ]R imai(-oo, oo) is convex if and only if f satisfies the inequality        f(esx+ (i-a) y) ;$; af (x) + (1-at) fly), x, y, E K, O {$ cr $1.If f is a convex function on K, then K2wr {x: f<x)<2} c K is a convex setfor each 2 E R that is, K2 is an intervai or empty set, and f is continuouson any open intervals contained in K Moreover, g is a concave functionon K lf and only if -g(x) is coRvex on K.   Generalizing the "Stratus"of Takeuchi3) and the "Family S-" of Croweand Zweibel,2) we give the following definition:   DEFINITION 1. Assign a set B(s) E ue to each s ffI [e, a]. If a family" Professor, Department of Applied Science, Faculty of Engineering, Shinshu verslty.t Supported in part by the Grant-in-Aid for Scientific Research (No. O154e175), stry of Education, Science and Cuiture.Uni-Mini-of sets                        y G- {B(s) : Oi!lsiS{a}satisfies the following two eonditions 10 and 20, then we call g a ge7?eraloron X:   10 B(s) c B(s') whenever O;$s<s' E:$ a, and B(a)==L   20 m(B(s)) [== s for every O$sEil a･    Moreover, to each f E ..if' assign a measurable function                 fA(x)ii=sup{s: x E!I B(df(s))},where df(t) ve." m({x: f<x)>t}) is the distribution function of f. We cali f" theequimeasurable rearrangement of f on X with resPect to a generator fY (inshort, Ee7-rearrangement of f>.   It is weli known that df(t> i'sa decreasing and right continuous functionof t E IR and that functionsfand fA satisfy df=dfA. Such a pair of functionsg and h having the same distribution function is said to be equimeasurableto each other (in symbols, g-h). Put B(s)=<O, s), O:Els.<,.,a. Then it follewsthat                 f"(x)=sup{t: df<t)>x}iiif"(x).The above function f* is the well known decreasing rearrangement of f, whichis continuous from the right and decreasing on L (See Chong and Ricei) fordetails. )                 2. Convex gellnetions and EY-Rearrangements   PROpOSITIoN 2. Let EY==' {B(s): O::lls:llla} be a generator on X. Then, {Y-rearrangement fA of f Eil ," satisfiies the fbllowing le and 20.   10 {x: fA(x)>s}=B(df(s)) m-a.e., hence fArs-f.   20 Lf O$s<s'$a, y E B(s) and x E B(s') ft B(s)C, then fA(x);ElfA(y>.   Further, any functions fA E M satis]lying the above le and 20 coincide withthe EY-rearrangement of f m-a. e.   ProofL The proof of the first paragraph is straightforward frorn thedefinition of fA,while that of the second paragraph is not so hard and istherefore omitted.            Convex Functions and 9-Rearrangements on lntervals 3   PRoposllhloN 3. Assum,e f Ei " and Put c==ess. inf{f(x): xe f}. Thenthe following statements are true:    (i) ILf f is convex on I, then df is convex on [c, oo).   <ii) df is convex on [c, oo) if and only ilf f" and f* are convex on I,where f*(x)ii!f"(a-x) (x E I) is the increasing rearrangemenl of fZ   Proof. Proof of <D: Suppose f is convex on I, andcin the propositioR isfinite. It is true, in general, that g(x) is convex on I if and only if g(x)-cis convex on L Further, dg<t)=:dg-c(t-c) on I and (g-c)":= g*-c. Therefore,consideriRg the function {f(x)-c}, if necessary, we may suppose withoutloss of generality that f(x)ille on I and that c in the proposition is e. Supposenow that f(x)lllO aRd c==O. Then, there exist two non-negative convex fun-ctions fi and f2 such that f=fi+f2, where fi is non-decreasing, f2 is non-increasing, and ess. inf{fi(x): x E I} ==ess. inf {f2(x): x E I} =O. But then, itfollows that both df, and df, are convex oR [O, oo) since convex functionsft and f2 are monotone. Therefore, df(t)"df,(t)+df,(t> is convex on [O, oo),which has completed the proof.   On the other hand, the statement (iO is evident. Thus, the proof is com-pleted.   DEFINiTIoN 4. Let g= {B(s): OEEIs;$a} be a generator on X. Define thetransmission function t(x) of the generator fl9 by                  t(x)av inf{s: x E B<s), e;;$snta}.   It is easy to see that the transmission function t(x) satisfies                    O{;;lt(x)giSa for any xEEiL   LEMMA 5. Assume f E" andPut c=:ess. inf{f(x>: x E I}. ILf df is con-tinuous on [c, cx)>, then it is true that          t(x)zadf(f"(x)) fbr any x El such that fA(x) is .17nite.   Proo.L It follows from the definition of f" that there exists a to E iRsuch that fA<x)- s <to and x e B(df(te)) for any s >O. Then, it foilows fromthe definition of t(x> that                                '                     t(x)Sdf(te))Edf(f"(x)-s),since df(t) is non-increasing. Suppose now the continuity of df on [c, oo)･Then, it follows from the above inequality that                      t(x>i:;ldf(fA(x>), x E LHere, note that the assertion is clear if df(f"(x)>=e, as seen from the aboveinequality. Therefore, it sufllces to prove the assertion when df(f"(x)>>O,which condition yields t(x)>e.    Suppose now that O<t(x)<df<fA(x)), and fA(x)< f"(t(x)>, with contradiction.Then, x G B(dfCf"(t(x))) by the definition of fA. Hence it follows that  (1) df<f'(x))<t(x>,by the definition of t(x). But, since df is continuous on [c, oo), it is true thatt<x)==dfif'(t<x)), which is contradictory to (i). Thus, the assertion is true,which has completed the proo£   THEoREM 6. Let g== <B(s>: O::SsEEa} be a grtenerator on X such that B(s)E 9 is concave for any O$s$a and that the transmission function t(x) is concaveon L Then,          fA(x) is convex on I whenever f(x) is convex on L   Moreover, it is true that          fA(x>=f'(t(x)), x Ei! I, with a convention f*<a)==f*(a-O)･   Proof: First we claim that fA(xKoo (x E I). Suppose fA(x)== oo for some xE I, with contradiction. Then, there exists a sequence {sn} such that sntoo and x EmaB(df(s.)) for any natural nurnbers n. But, then m(B<df(s.)))==df<sn)--.e, on our letting n-oo, which is contradictory to the assumption thatB(s) is concave for any O;;{s::la.   Suppose now that f is convex. Then it follows that df is convex andhence continuous on (ess.inf{f(x): x E I}, oo), by Proposition 3. Therefore,it follows from Lemma 5 that (2) t(x)=df<f"(x)) for any xEwaLHence f*(t(x>) =f*(df(fA(x>)==fA(x), that is, (3) fA(x)=f"(t<x)) for any xELThen, (3) yieids the assertion that fA(x> is a convex function on I, since t(x)is a concave function on I and f*(x) is a non-increasing convex function onI, by Proposition 3. Thus the proof is completed.   THEoREM Z Let fg7=={B(s>: OjiSsESa} be a generator on X such that B(s)is concave for every Oi:$siE;la, and denote by t(x) the transmission function of fY.            Convex Functions and g-Rearrangements on lntervals 5Then, l(x) is coneave on I if and only iLf B(s) =(e, Ps] U (a-(1-P)s, a) m--a.e..for some P E [O, 1]. Moreover, if t(x) is concave, then we can write down t(x)as follows:    (i) ILIC`P:=O, then t(x)=a-x.    (ii) Ilf` P=:1, then t(x)=x.   (iii) ILfpykO, 1, then                   t(x)-rmIi./2.)/(,-i) lpO;g2Z').   Proof. The proof is so easy to be omitted.   TIffEoREM 8. Let [Y== {B(s):O$s$a) be a generator on X. Then the foll-owing statements (i) and (ii) are equivalent to each other.    (i) f"(x) is convex on I whenever f(x> is convex on L    (iO B(s>=(O, Ps] U (a-<l-P>s, a> m-a.e. fbr some OiEIP$1.   Proof. Put f(x)==x. Then f<x) is a convex function on i such that f"(x)Nx. Suppose <i), and then it results in only the following three cases:                                      (O<x$pa)c-1 f"(x)- ((a.ZXp/.P)/a-p)                                      (Pa<x<a)for some O$P{fll.     C-2 f"(x) =x on L     C-3 f"(x)xa-x on LThen, it follows that B(s) =(O, Ps] U (a-(1-P)s, a) m-a.e. for some O$P:i{1,by Proposition 2.   Further, if B(s) is defined as above, then t(x) is concave, and then, byTheorem 7, it follows that f"(x) is convex on I whenever f(x) is convex onL Thus the proof is completed.                              References(1) K.M. CHoNG and N.M. RicE, "Equimeasurabie Rearrangements of Functions,"  Queen's Papers in Pure and Applied Mathematics, No. 28, (1971).(2) j.A. CRowE and J.A. ZwEiBEL, "Rearrangements o£ Functions," Journal of Fun-  ctional Analysis, 66, 432-438 (1986).(3> ¥uldo TAi<Eucm, "A New Rearrangement and the Theorem of Hardy-Littlewoed-  P6Iya," Journal of the Facu!ty of Engineering, Shinshu University, 41, 1-21 (1976).

Convex Functions and g-Rearrangements on Intervals

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Convex Functions and g-Rearrangements on Intervals

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