ON THE DEFINITION OF A CLOSE-TO-CONVEX FUNCTION

Transcription

1 I terat. J. Mh. & Math. Sci. Vol. (1978) ON THE DEFINITION OF A CLOSE-TO-CONVEX FUNCTION A. W. GOODMAN ad E. B. SAFF* Mathematics Dept, Uiversity of South Florida Tampa, Florida Dedicated to Professor S.M. Shah o the occasio of his 70th birthday ad i recogitio of his outstadig mathematical cotributios. (Received July ii, 1977) ABSTRACT. The stadard defiitio of a close-to-covex fuctio ivolves a complex umerical factor e i8 which is o occasio erroeously replaced by i. While it is kow to experts i the field that this replacemet caot be made without essetially chagig the class, explicit reasos for this fact seem to be lackig i the literature. Our purpose is to fill this gap, ad i so doig we are lead to a ew coefficiet problem which is solved for 2, but is ope fo > 2. *Research supported i part by the Air Force Office of uder Grat AFOSR Scietific Research

2 126 A. W. GOODMAN & E. B. SAFF i. THE DEFINITION OF A CLOSE-TO-CONVEX FUNCTION. The most commo form for this defiitio is DEFINITION i. The fuctio f() + a -- 2 (i.i) regular i E Il < i, is said to be close-to-covex i E, if there is a (), () blz + [ b, b I e =2 (1.2) that is covex i E ad such that i E We deote the set of all such fuctios by CL. f Re () -, () > 0. (1.3) Oe ca begi with more geeral expressios but ay additive costats disappear o differetiatio ad hece may be dropped at the begiig. Further there is o loss of geerality i assumig that f (0) 1 ad (0) e Here it seems atural to set (0) i, but there are several places i the literature (for example [3, p. 51]) where this secod ormaliatio is expressly forbidde. As far as we are aware, the reaso for maitaiig b I e i is ever explicitly give. I this ote we prove that for each i the ope iterval (-/2,/2) there is a correspodig fuctio F() that should be regarded as close-to-covex, but would ot be i CL if that particular is forbidde i Defiitio i. 2. THE EXAMPLE FUNCTION. It is well kow ad easy to prove that the fuctio F() 2ie 2 -e cosu 1 e iu 2 0 < e <, (2.1) (- )

3 DEFINITION OF A CLOSE-TO-CONVEX FUNCTION 127 maps E oto the complex plae mius a vertical slit (see [i] where some iterestig properties of this fuctio are obtaied). Hece the boudary curve of F(E) has o "hairpi" bed that exceeds 180. Cosequetly, o geometric grouds (see Kapla [2]) F() is close-to-covex. But we do ot eed these geometric facts because we ca prove that F() satisfies Defiitio I. Ideed F () If we select for our covex fuctio l-e (l-eis) 3 (2.2) () -+/-e is l-e (2.3) the F () l-e 3is -i 2 Re Re, () ie (l-el) (l_eis) 3 -is F () e -e Re,. Re i > 0 (2.4), l_eis i E, because this last fuctio carries E oto Re w > 0. Thus by Defiitio 1 with e i8 -ie F() is close-to-covex. Here 8 arg(-ieis) s- /2, (2.5) ad sice 0 < s <, we have -/2 < 8 < /2. We ow prove that if () is ay covex fuctio differet from the oe give by equatio (2.6) the the coditio Re F () Re l-e () 1 > 0 (2 6) (l_eis)3 4 ()

4 128 A. W. GOODMAN & E. B. SAFF fails to be satisfied. Ideed, if (2.6) holds i E the we ca write () P() l-e 3ie (l-ei) where P() has positive real part i E. Whe P(0) i it is well-kow that l-r IP() > for re 0 < r < i, l+r 3 (2.7) 18 ad hece for arbitrary P(0) we have [P()[ >_ c(l-r), for re 0 < r < i, where c is a positive costat that depeds oly o P(0). This last iequality, together with (2.7), ad the coditio s # 0,, imply that I (re-+/-=) > where > 0 is some costat. 2 (-r) as r / i (2.8) Now suppose that #(E) is ot a half-plae. Sice (E) is a covex regio, (E) must have two distict lies of support ad hece (E) is cotaied i a sector with vertex agle <. Cosequetly, by subordiatio (see Rogoslski [8]) it follows that for some T, 0 < T < i, l(reio) O( I. ). (2.9) (l-r) T But the, usig Cauchy estimates it is easy to see that (2.9) implies that l (reis){ 0( 1 +I) (l-r) ad this cotradicts (2.8) sice T + I < 2. Hece (E) is a half-plae, ad () has the form /(l-eis), lql i. But the q must be the factor selected i (2.3), otherwise the iequality (2.4) is false. Cosequetly, if the associated value of 8 is ot permitted i the defiitio of a close-to-covex fuctio, the the fuctio (2.1) would ot be classified as close-to-covex. This completes the justificatio of the factor e i8 i Defiitio i, if -/2 < 8 < /2.

5 DEFINITION OF A CLOSE-TO-CONVEX FUNCTION 129 The remaiig cases, /2 _< 8 _< 3/2, are trivial. If 8 lies i this iterval, the Re{f (0)/$ (0)} =< 0 ad the coditio (3) is ot satisfied. We have proved that Defiitio i is proper for the class we wish to describe, if we add the coditio-/2 < 8 < /2. Further o sigle poit of this iterval ca be dropped without losig at least oe fuctio from the class CL. 3. A REMARK ON THE COEFFICIENTS. The class CL is aturally divided ito subclasses CL(8) i accordace with the value of 8 that may be uaed i Re f () > 0, i E, (3.1) e (-.) where ow () +... The subclasses CL(8) are exhaustive, but they are certaily ot mutually exclusive. Thus if f() is itself covex the we may take #() f() i (3.1). Hece a covex fuctio is i CL(8) for every 8 i (-/2, /2). I fact, it is easy to show usig a ormal families argumet that the itersectio CL() is precisely the collectio of all ormalied covex fuctios. We ca ask for extreme properties of fuctios i the subclasses CL(8). Here we pause oly to discuss the magitude of the coefficiets. THEOREM i. If f() give by (i.i) is i CL(S), the for =2,3,... lal _< I + (-l)cos 6- (3.2) If =2, the result la21 _< i + cos 6 is sharp. PROOF. If p() i + =i P part ad () + =2 b is a ormalied fuctio with positive real is the associated covex fuctio, the (3 i) yields

6 130 A. W. GOODMAN & E. B. SAFF cosi isf (,) + i sl8] 1 + [ p () =l (3.3) or Hece -i i + a (i + b -l) (i + eiscoss P )" =2 =2 =l -I a b + eiscss(p-i + [ kbkp-k )" k=2 The kow bouds, [bk[ < i (Loewer), ad [pk[ _< 2 (Caratheodory) yield the iequality (3.2). If we defie G() to be the solutio of (3.4) G () l+eis i l-e-is (l-) =i+ --2 A (3.5) the for > i A =I+ 2 cos S -i k--1 -i(-k-l) ke (3.6) If G(0) 0 ad cos8 # I, the G() e is.l-e-is) I-cosS {coss i- i sis }. (3.7) 2 If coss i, the G() /(l-) I either case, for 2, equatio (3.6) gives A cos 8, the sharp upper boud. To see that G() is i CL(S) we put (3.5) i the form G () (l-) 2 1 l+eis (3.8) e is e is l-eis l+e-is =-i sis + coss l-e-is 2 Sice the covex /(l-) yields i/@ () (l-) the fuctio

7 DEFINITION OF A CLOSE-TO-CONVEX FUNCTION 131 G() satisfies the coditio (3.1). The problem of fidig the maximum for la i the class CL(8# seems to be difficult for > 3. Although G(), give by (3.7) furishes the maximum whe 2, there is o reaso to believe that it cotiues to play this role whe > 2. Ideed if we use equatio (3.7), we ca obtai a alterate form for the coefficiet A (the sum idicated i (3.6)): A (l-e-is)cos 8 isi (3.9) l-cos8 Thus with 8 fixed, A / a costat as +. I cotrast, if F(), give by equatio (2.1), has the expasio I BZ the B i + (-l)cos 28 + i(-l)slscos8 where 8 /2, ad B / approaches a oero costat as +. We observe that for 8 # 0, the extremal fuctio (for la21) (3.10) G() maps E oto a slit half-plae. Both the boudary ad the slit make a agle 8 with the real axis. Thus the complemet of G(E) cotais a half-plae. It is very uusual for the extremal solutio of a coefficiet problem to omit a ope set whe there are - competig fuctios (such as F()) i the same class that do ot omit ay ope set. As 8 0, the fuctio G() / /(l-) 2 the Koebe fuctio We retur to the boud la < i + (-l)cos8 give i Theorem i. Sice every covex fuctio belogs to CL(8) for every 8 i (-/2,/2) this iequality icludes the Loewer boud lal < 1 for covex fuctios as a special case. It also icludes the iequality lal =< for all close-to-covex fuctios; a result that was obtaied much earlier by M. Reade [5]. Work o the coefficiets of subclasses of CL has bee doe by Reyi [7], Pommereke [4], ad Reade [6], but their subclasses are differet from the oes

8 132 A. W. GOODMAN & E. B. SAFF cosidered here. Reade [6] proves that if the coditio (3) is replaced by the larg f () "() < 0 < < i, lal =< i + (-l)s (3.11) (3.12) ad the result is sharp whe 2. REFERENCES i. Goodma, A. W. ad E. B. Saff. Fuctios that are Covex i oe Directio, to appear, Proc. Amer. Math. Soc. 2. Kpla, W. Close-to-Covex Schlicht Fuctios, Mich. Math. J. i (1952) Pommereke, C. Uivalet Fuctios, Vadehoeck ad Ruprecht, Gttlge, Pommereke, C. O the Coefficiets of Close-to-Covex Fuctios, Mich. Math. J. 9 (1962) Reade, M. O Close-to-Covex Uivalet Fuctios, Mich. Math. J. 3 (1955) Pommereke, C. The Coefficiets of Close-to-Covex Fuctios, Duke Math. J. 23 (1956) Reyl, A. Some Remarks o Uivalet Fuctios, Bulgar. Akad. Nauk Iv. Math. Ist. 3 (1959) Rogosiski, W. ber Bildschrake bel Potereihe ud ihre Abschitte, Math. Zeit 17 (1923) KEY WORDS AND PHRASES. Covex fuctios, uivalet fuetios. AMS(MOS) SUBJECT CLASSIFICATIONS (1970). 30A52, 30A66.

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