On univalent polynomials and related classes of functions. David Alexander Brannan. ii. Chapter Index. 1. Preliminaries 1 2. Univalent polynomials of any degree 28 3. Univalent polynomials of small degree 63 4. A conjecture of Ilieff 82 5. Composition of coefficients 90 6. The theorem of Bernstein 94 References 99 Abstract. The behaviour of the coefficients of poly- n nomials pn(z) = z + a2z2 + + anz and µn(z) = + biz + + bnzn univalent in Izi < 1, 0 < Izi < 1 respectively, has received surprisingly little attention. After a survey of those significant facts known about pn and µn, bounds are established for an, bn, bn_i; several interesting results an-1' are obtained for special types of univalent polynomials when a and b are maximal. The correct order of n n growth with n of alp (for fixed k), where the Bieberbach conjecture is assumed to hold for k, is established. The coefficient regions for p3, µ2, µ3 p4, are then studied, with complete results for p3. We conclude with the proof of a special case of a conjecture of L. Ilieff, some results from the theory of apolar polynomials, and several examples connected with a theorem of S. Bernstein. All published papers with significant results on univalent polynomials appear in the list of references, markee. iv. Acknowledgements. I would like to thank Professor J.G. Clunie for introducing me to the analytic theory of polynomials and univalent functions. This thesis could never have reached completion without his unfailing help and encouragement over a long period. It is impossible to pick out anything in which his hand cannot be seen clearly, but results marked are due in large measure to Professor Clunie. Finally I wish to thank Professor W.K. Hayman, A.W. Norton, M.B. Zaturska, and D. Hornblower, all of whom have assisted in the birth of the concepts which I have used. 1. Chapter 1. Preliminaries. 'A day's work - getting started'. - Gaelic proverb 1. Introduction. The class of polynomials univalent in lz I < 1 has been studied relatively little, and surprisingly few significant results are known concerning them. Let us, first of all, introduce some of the notation which we shall use. Definition 1.1.1. A function f z is univalent in a domain D if it is regular, sin le-valued and does not take any value more than once in D. Definition 1.1.2. A function f(z) is said to belong to the class S if it is of the form: co f(z) = z + l a zn n n . 2 and is regular and univalent in zt < 1. Definition 1.1.3. A function f(z) is said to belong to the class ) if it is of the form: f(z) = 2. and is regular and univalent in 0 < lz I< 1. Definition 1.1.4. A polynomial pn(z) of the form: n z + a z2 a z ion(z) = 2 + n which is univalent in Id < 1 is said to belong to the class P. The P contains P , since pn(z) is not n+1 n necessaIlLY91MMIE....2YQL1111- Definition 1.1.5. A function µn(z) of the form: n µ (z) = 1— + a1z + + a z n which is univalent in 0 < 1z < 1 is said to be a meromorphic univalent polynomial of degree n, belonging to the class N. . The following two theorems give important results concerning infinite sequences of polynomials in P and n Mn respectively. orp Theorem 1.1.1[21] Let (in(z)) 1 be a uniformly convergent sequence of functions regular and univalent in a domain DJ and let f(z) be the limit function of the sequence. Then f(z) is either constant or univalent in D. Proof. By the Weierstrass Limit Theorem, f(z) is regular in D. If f(z) is not univalent in D, there are two points z and z at which w = f(z) takes the 1 2 3. same value w0. Describe, with z1 and z2 as centres, two circles which lie in D, do not overlap, and such that f(z) - w does not vanish on either circumference 0 (this is possible unless f(z) is a constant). Let m be the lower bound of I on the f(z) w two circumferences. Then we can choose n so large that f(z) - f (z) # < m on the two circumferences. Hence, n i by Rouche's theorem, the function: f (z) - w0 = (f(z) - wo ) + (fn(z) - f(z) has as many zeros in the circles as f(z) w i.e. at o1 least two. Hence f (z) is not univalent, contrary to n hypothesis. This proves the theorem. In a similar way, we may prove: OD Theorem 1.1.2. Let (in(z)) be a sequence of 1 functions of the form: CD 1 n fn(z) = anz ÷ O regular and univalent in 0 < 1 z 1 < 1, and uniformly convergent to a function f(z) in any compact subset of 0 < 1z 1 < 1. Then f(z) is regular and univalent in 0 < lz 1 < 1. In view of Theorem 1.1.1, any function univalent 4. in lz i< 1 may be approximated arbitrarily closely by a sequence of polynomials univalent in Izi < 1 (for example, renormalisations of partial sums of the original function). Consequently, it might be expected that many important results for the class S could be obtained as the limiting cases of the corresponding results for Pn. By Theorem 1.1.2, a similar relation holds between I and N. Unfortunately, few of the usual techniques for dealing with univalent functions are of any value when we consider polynomials in Pn or Mn. For example, the application of the bilinear transformation to a poly- nomial in Pn or Mn does not generally yield another such polynomial. In addition, if pn(z) Pn and 1-1.1,1(z)E Mn, then pn(z2)2 and µn(z2)2 do not generally belong to the classes Pn and Mn. However, if pn(z) and µn(z) are odd polynomials in Pn and Mn, then pn(z-)-e: Pn and µ (z) 1- ] µ (22)2 - 2 [ - E M . No other n n z z2 z=0 really useful variation for P or M is known at n n present. Furthermore, in dealing with S it is often help- ful to guess that the Koebe function --I-- may be (1-z)2 5. extremal for whatever property we are investigating. Unfortunately, there are no such convenient 'possible extremals' known in Pn. Consequently, it is necessary to develop new techniques for dealing with the classes Pn and Mn in order to obtain other than the simplest results. Thus the consideration of the special subclasses Pn of S and Mn of does not appear to simplify the task of establishing such things as, for example, coefficients and maximum modulus estimates, but only makes it more difficult. This means that, in general, we do not expect to solve problems for S or ) by using the solutions of the corresponding problems for Pn or Mn. As a result, we will study the classes P n and mainly for their own independent interest. M11 2. The Dieudonne Criterion. A fundamental result concerning univalent functions may be expressed in the following form: Theorem 1.2.1.[21] A function w= f z regular on a domain containing a simple closed rectifiable curve C and its interior Dy is univalent on D if it is univalent on C. Proof. The curve C corresponds to a curve C° in the w-plane. C° is closed, since f(z) is single-valued; and it has no double points, since f(z) does not take any value twice on C. Let DI be the region enclosed by at . Clearly f(z) takes in D values other than those on C, say at zo. Then if 6,0 denotes the variation round C, 1 c arg f(z) f(zo) ] 27c is equal to the number of zeros of f(z) f(zo) in D, by the Argument Principle. It is therefore a positive integer, since there is at least one such zero. But it is also equal to: w0) tic C2 arg (w where w = f(z0); and this is either 0, if w is outside o o 02, or +1, if wo is inside C2, the sign depending on the direction in which C2 is described. Hence it is equal to 1. Hence wo lies inside C', C' is described in the positive direction, and f(z) takes the value w just once in D. Thus D is mapped univalently onto DI. o Let us consider the radius of univalency R of polynomials: