On the definition of a close-to-convex function

$(1978) 125-132 A. W. GOODMAN and E. B. SAFF* Mathematics Dept, University of South Florida Tampa, Florida 33620 Dedicated to Professor S.M. Shah on the occasion of his 70th birthday and in cognition of his outstanding mathematical contributions. (ceived July ii, 1977) ABSTRACT. The standard involves a complex numerical factor e i8 which is on occasion erroneously placed by While it is known to experts in the field that this placement cannot be made without essentially changing the class, explicit asons for this fact seem to be lacking in the literatu. Our purpos to fill this gap, and in so doing we a lead to a new coefficient problem which is solved for n but is open fo n > 2. 2, *search supported in part by the Air Force Office of Scientific search under Grant AFOSR-74-2688. THE . The most common form for this definition is DEFINITION The function f gular in E n-- 2 a z n (i) Izl < i, is said to be close-to-convex in E, if ther a , blZ bnzn, (1.2) that is convex in E and such that in E -, f > 0. (1.3) We denote the set of all such functions by CL. One can begin with mo general expssions but any additive constants disappear on diffentiation and hence may be dropped at the beginning. Further ther no loss of generality in assuming that f (0) it seems natural to set 1 and (0) i, but t a several places in the literatu (0) (for example [3, p. 51]) w this second normalization is expssly forbidden. As far as we a awa, the ason for maintaining b I given. e i is never explicitly In this note we prove that for each is a corsponding function in the open interval (-/2,/2) t F that should be garded as close-to-convex, but is forbidden in Definition would not be in CL if that particular THE EXAMPLE FUNCTION. It is well known and easy to prove that the function F z-e 2ie 1 (- e cosuz iu 2 0 < e < (2.1) z) maps E onto the complex plane minus a vertical slit (see [i] w some intesting properties of this function a obtained). F has no "hairpin" bend that exceeds 180 Hence the boundary curve of Consequently, on geometric grounds (see Kaplan [2]) F is close-to-convex. But we do not need these Indeed geometric facts because we can prove that F satisfies Definition F If we select for our convex function (iSz) 3 is (2.2) -+/-e (2.3) F 3is (l_eiSz) ie -i (lz) 2 F ,. z, -is i -e l_eiSz > 0 (2.4) in E, because this last function carries E onto w > 0. Thus by Definition 1 with e i8 -ie F is close-to-convex. arg(-i) s- /2, (2.5) and since 0 < s < we have -/2 8 < /2. We now prove that if is any convex function diffent from the one given by equation (2.6) the condition F (l_eiSz)3 4 > 0 (2 6) fails to be satisfied. Indeed, if (2.6) holds in E we can write P 3ie z (iz) 3 When P(0) (2.7) i it is well-known that w P has positive al part in E. IP l-r l+r for z 0 < r < i, >_ c(l-r), for z and hence for arbitrary P(0) we have [P[ r < i, w c is a positive constant that depends only on P(0). This last inequality, together with (2.7), and the condition s # 0, as r imply that I (-+/-=) w > 0 is some constant. (-r) (2.8) Now suppose that # is not a half-plane. Since is a convex gion, is contained in a must have two distinct lines of support and hence sector with vertex angle < Consequently, by subordination (see Rogoslnski T, [8]) it follows that for some 0 < T < i, l(iO) l(iS){ and this contradicts (2.8) since O( (l-r) T ). (2.9) But , using Cauchy estimates it is easy to see that (2.9) implies that 0( (l-r) +I) Hence is a half-plane, and + I < 2. has the form in nz/(iSz), lql But q must be the factor selected Consequently, if the associated (2.3), otherwise the inequality (2.4) is false. value of 8 is not permitted in the , the function (2.1) would not be classified as close-to-convex. This completes the justification of the factor e i8 in Definition i, if -/2 < 8 < /2. The maining cases, /2 _< 8 _< 3/2, a trivial. If 8 lies in this interval, {f(0)/$(0)} =< 0 and the condition (3) is not satisfied. We have proved that Definition i is proper for the class we wish to describe, if we add the condition-/2 < 8 < /2. Further no single point of this interval can be dropped without losing at least one function from the class CL. A MARK ON THE COEFFICIENTS. The class CL is naturally divided into subclasses CL(8) in accordance with the value of 8 that may be uaed in f > 0, (-.) The subclasses z in E, (3.1) w now .... CL(8) a exhaustive, but they a certainly not mutually exclusive. Thus if f is itself convex we may take # f in (3.1). Hence a convex function is in CL(8) for every 8 in (-/2, /2). intersection In fact, it is easy to show using a normal families argument that the CL() is pcisely the collection of all normalized convex functions. We can ask for extme properties of functions in the subclasses we pause only to discuss the magnitude of the coefficients. CL(8). THEOM If f given by (I) is in CL(S), for ,3,... lanl If , the sult _< I + (n-l)cos 6- (3.2) la21 _< i cos 6 is sharp. PROOF. part and If p =i Pn z is a normalized function with positive al =2 bn z is the associated convex function, (3 i) yields coSI isf @(z,) sln8] n=l [ pn z (3.3) or na z n n-i (i + nb zn-l) (i + eiScosS n=l Pn z )" Hence [ nan nbn + eiScsS(Pn-i + k=2 kbkPn-k )" The known bounds, n-I (3.4) [bk[ i (Loewner), and [pk[ _< 2 (Caratheodory) yield the inequality (3.2). If we define G to be the solution of G l+eiSz -iSz =i+ nAz n--2 (3.5) (l-z) for n > i n-i A =I+ If G(0) 0 and cos8 # I, 2 cos S ke -i(n-k-l) (3.6) k--1 G If cosS I-cosS {cosS n i-z .-iSz) i sins }. (3.7) i, G z/(l-z) In either case, for n 2, equation (3.6) gives A2 cos 8, the sharp upper bound. To see that G is in CL(S) we put (3.5) in the form G (l-z) 2 l+eiSz - iS z + cosS (3.8) =-i sins l+e-iSz -iS z Since the convex function @ z/(l-z) yields i/@ (l-z) the function G satisfies the condition (3.1). The problem of finding the maximum for difficult for n la n in the class CL(8# seems to be __> Although G, given by (3.7) furnishes the maximum when 2, ther no ason to believe that it continues to play this role when Indeed if we use equation (3.7), we can obtain an alternate form for the n > 2. coefficient A (the sum indicated in (3.6)): n Thus with 8 fixed, A n (-inS)cos l-cos8 a constant as n + isin (3.9) I BnZ (3.10) In contrast, if F, given by equation (2.1), has the expansion B w 8 + (n-l)cos 28 + i(n-l)slnScos8 a nonzero constant as n + /2, and Bn /n approaches G maps E onto We observe that for 8 # 0, the extmal function (for a slit half-plane. la21) Both the boundary and the slit make an angle 8 with the al axis. Thus the complement of G contains a half-plane. It is very unusual for the extmal solution of a coefficient problem to omit an open set when t a competing functions (such as F) in the same class that do not omit any open set. As 8 0, the function G z/(l-z) < i the Koebe function given in Theom Since every We turn to the bound lan + (n-l)cos8 convex function belongs to CL(8) for every 8 in (-n/2,z/2) this inequality includes the Loewner bound lanl 1 for convex functions as a special case. It also a sult that includes the inequality lanl =< n for all close-to-convex functions; was obtained much earlier by M. ade [5]. Work on the coefficients of subclasses of CL has been done by nyi [7], Pommenke [4], and ade [6], but their subclasses a diffent from the ones consided . ade [6] proves that if the condition (3) is placed by f larg " 0 < i, (3.11) lanl =< i + and the sult is sharp when n (n-l)s (3.12) FENCES Goodman, A. W. and E. B. Saff. Functions that a Convex in one Diction, to appear, Proc. Amer. Math. Soc. Kplan, W. Close-to-Convex Schlicht Functions, Mich. Math. J. i (1952) 169-185. Pommenke, C. 1973. Univalent Functions, Vandenhoeck and Rupcht, Gttlngen, Pommenke, C. On the Coefficients of Close-to-Convex Functions, Mich. Math. J. 9 (1962) 259-269. ade, M. On Close-to-Convex Univalent Functions, Mich. Math. J. 3 (1955) 59-62. Pommenke, C. The Coefficients of Close-to-Convex Functions, Duke Math. J. 23 (1956) 459-462. nyl, A. Some marks on Univalent Functions, Bulgar. Akad. Math. Inst. 3 (1959) 111-121. Nauk Izv. Rogosinski, W. ber Bildschranken bel Potenzihen und ihn Abschnitten, Math. Zeit 17 (1923) 260-276. KEY WORDS AND PHRASES. Convex functions, univalent funetions. AMS(MOS) SUBJECT CLASSIFICATIONS (1970). 30A52, 30A66.$

On the definition of a close-to-convex function

Hindawi Publishing Corporation – Aug 23, 2007