ON THE DEFINITION OF A CLOSE-TO-CONVEX FUNCTION

The standard definition of a close-to-convex function involves a complex numerical factor eiβ which is on occasion erroneously replaced by 1. While it is known to experts in the field that this replacement cannot be made without essentially changing the class, explicit reasons for this fact seem to be lacking in the literature. Our purpose is to fill this gap, and in so doing we are lead to a new coefficient problem which is solved for n=2, but is open for n>2.

i. THE DEFINITION OF A CLOSE-TO-CONVEX FUNCTION.The most common form for this definition is DEFINITION i.The function Izl < i, is said to be close-to-convex in E, if there is a (z), that is convex in E and such that in E f' (z) > 0.
Re -, (z) We denote the set of all such functions by CL.
One can begin with more general expressions but any additive constants disappear on differentiation and hence may be dropped at the beginning.Further there is no loss of generality in assuming that f' (0)   1 and ' (0) e Here it seems natural to set '(0) i, but there are several places in the literature (for example [3, p. 51]) where this second normalization is expressly forbidden.
As far as we are aware, the reason for maintaining b I e i is never explicitly given.
In this note we prove that for each in the open interval (-/2,/2) there is a corresponding function F(z) that should be regarded as close-to-convex, but would not be in CL if that particular is forbidden in Definition i.

THE EXAMPLE FUNCTION.
It is well known and easy to prove that the function 2ie 2 z-e cosuz F(z) 1 e iu 2 0 < e < , (2.1) (-z) maps E onto the complex plane minus a vertical slit (see [i] where some inter- esting properties of this function are obtained).Hence the boundary curve of F(E) has no "hairpin" bend that exceeds 180 .Consequently, on geometric grounds (see Kaplan [2]) F(z) is close-to-convex.But we do not need these geometric facts because we can prove that F(z) satisfies Definition I. Indeed l-e z F'(z) and since 0 < s < , we have -/2 < 8 < /2.
We now prove that if (z) is any convex function different from the one given by equation (2.6) then the condition Re F'(z) ' (z) (l_eiSz)3 4' (z) fails to be satisfied.Indeed, if (2.6) holds in E then we can write 3ie l-e z '(z) P(z) 3 (2.7) (l-eiz) where P(z) has positive real part in E. When P(0 for z re 0 < r < i, l+r 18 and hence for arbitrary P(0) we have [P(z)[ >_ c(l-r), for z re 0 __< r < i, where c is a positive constant that depends only on P(0).This last inequality, together with (2.7), and the condition s # 0, , imply that I' (re-+/-=) > 2 as r / i (2.8) (-r) where > 0 is some constant.Now suppose that #(E) is not a half-plane.Since (E) is a convex region, (E) must have two distinct lines of support and hence (E) is contained in a sector with vertex angle < .Consequently, by subordination (see Rogoslnski [8]) it follows that for some T, 0 < T < i, l(reiO) O( I. ).
(2.9) (l-r) T But then, using Cauchy estimates it is easy to see that (2.9) implies that l'(reiS){ 0(' 1 (l-r) +I)' and this contradicts (2.8) since T + I < 2. Hence (E) is a half-plane, and (z) has the form nz/(l-eiSz), lql i.But then q must be the factor selected in (2.3), otherwise the inequality (2.4) is false.Consequently, if the associated value of 8 is not permitted in the definition of a close-to-convex function, then 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.
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 REMARK 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 Re f' (z) > 0, where now (z) z + ....The subclasses CL(8) are exhaustive, but they are certainly not mutually exclusive.Thus if f(z) is itself convex then we may take #(z) f(z) in (3.1).Hence a convex function is in CL(8) for every 8 in (-/2, /2).In fact, it is easy to show using a normal families argument that the intersection CL() is precisely the collection of all normalized convex functions.
We can ask for extreme properties of functions in the subclasses CL( 8).Here we pause only to discuss the magnitude of the coefficients.
If n=2, the result la21 _< i + cos 6 is sharp.n PROOF.If p(z) i + =i Pn z is a normalized function with positive real n z n is the associated convex function, then (3 i) The known bounds, [bk[ __< i (Loewner), and [pk[ _< 2 (Caratheodory) yield the inequality (3.2).
If we define G(z) to be the solution of G'( In either case, for n 2, equation (3.6) gives A 2 1 + cos 8, the sharp upper bound.To see that G(z) is in CL(S) we put The problem of finding the maximum for la in the class CL(8# seems to be n difficult for n __> 3.Although G(z), given by (3.7) furnishes the maximum when n 2, there is no reason to believe that it continues to play this role when n > 2. Indeed if we use equation (3.7), we can obtain an alternate form for the coefficient A (the sum indicated in (3.6)):In contrast, if F(z), given by equation (2.1), has the expansion I BnZ then B i + (n-l)cos28 + i(n-l)slnScos8 n (3.10)where 8 /2, and B /n approaches a nonzero constant as n + . n We observe that for 8 # 0, the extremal function (for la21) G(z) maps E onto a slit half-plane.Both the boundary and the slit make an angle 8 with the real axis.Thus the complement of G(E) contains a half-plane.It is very unusual for the extremal solution of a coefficient problem to omit an open set when there are competing functions (such as F(z)) in the same class that do not omit any open set.
As 8 -0, the function G(z) / z/(l-z) 2 the Koebe function We return to the bound la < i + (n-l)cos8 given in Theorem i.Since every n 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 includes the inequality lanl =< n for all close-to-convex functions; a result that was obtained much earlier by M. Reade [5].
fixed, A / a constant as n + .n