MacGregor, T.H

Tepper, D.E

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JOURNAL OF APPROXIMATION THEORY 52, 315-321 (1988) 
Finite Boundary Interpolation by 
Univalent Functions 
T. H. MACGREGOR 
Depurtment q/ Mutlzemarics and Statistics, 
State Uniwr.rity of NPM. York, Alhnny, New York 12222, (/..%A. 
D. E. TEPPER 
Depurtment of Mcrthematirs, Bartrch College, 
City Unirrrsity of‘ New, York, Ntw York, New York 10010, U.S.A. 
Communicated by T. J. Riulin 
Received April 22, 1985; revised Novemher 22, 1985 
1. INTRODUCTION 
Let D = {z E C: 1: 1 < 1 }, and suppose that z,, zz, . . . . z,, and IV,, w2, . . . . w, 
are two collections of distinct points on 8o arranged in counterclockwise 
order. Let Z~ = r’“’ and uak = &Ok, where a,<cc,<...<a,<cc,+2rt and 
/I, < p2 < ... </?,, </I, + 27~. We are interested in functions f which are 
analytic and univalent in D and satisfy the boundary interpolation f(zp) = 
~~,fork=l,2 ,..., n. 
In particular we prove the following theorem. 
THEOREM 1. There is a function f which is analytic and univalent in the 
union of D and a neighborhood of (z,, z2, . . . . z,} and continuous on d such 
thatf(z,)=w,fork=1,2,...,n. Furthermore, If(z iflzl=l andzis 
sufficiently near any of the points 7 ‘k. 
Theorem 1 is related to considerations in the recent paper [3], where the 
following theorem about simultaneous peaking and interpolation is proved. 
THOEREM A. (Clunie, Hallenbeck, and MacGregor). There is a 
function f that is analytic and univalent in D and satisfies 1 f (z)l < 1 for 
/z I < 1 and z # zk (k = 1, 2, . . . . n) and f (zk) = wk for k = 1, 2, . . . . n. 
315 
X121-9045/88 $3.00 
316 MACGREGOR AND TEPPER 
The proof of Theorem A is rather long and in several places non-con- 
structive. The main steps in the argument rely on the following ideas: a 
peaking result for polynomials [ 1, p. IO1 1; an interpolation result for finite 
Blaschke products [2]; a starlike mapping having suitable properties [3, 
Lemma 11; and an application of the Riemann mapping theorem for a 
domain formed from a disk by adding “channels.” 
Theorem 1 can be used to give a somewhat simpler and more construc- 
tive proof of Therem A. The argument relies on the following reslt, which is 
cntained in [3, Sect. 31 and is a weakened version of Theorem A. The 
proof of this result is elementary and provides a step by step procedure for 
obtaining the function from the given points. This function is a com- 
position of a finite number of functions which are power functions, 
exponentials, or Mobius transformations. The argument for Theorem 1 
also relies of properties of explicit functions which map D onto the com- 
plement of spirals. 
THEOREM B. There is a function f that is analytic and univalent in D and 
satisfies 1 f(z)1 < 1 for IzI 6 1 and z #zk (k = 1, 2, . . . . n) and 1 f(zk)l = 1 for 
k = 1, 2, . . ., n. 
Our proof of Theorem A is as follows. Let g be a function given by 
Theorem B and let ck = g(zk) for k = 1, 2, . . . . n. Let h be a function given by 
Theorem 1 for the two collections of points, [, , c2, . . . . [, and w,, w2, . . . . w,. 
Then f = h 0 g satisfies Theorem A. 
We also prove the following similar results. 
THEOREM 2. Suppose a<x,<x,<-..<x,,<b and y,<y,<...<y,,. 
There is a real-valued polynomial p which is univalent in a domain containing 
[a, b] such that p(xk) =yk for k = 1,2, . . . . n. 
THEOREM 3. Let R = {c E @: Re [ > 0} and let S be a subset of R such 
that dS n aR = {c,, iz, . . . . [,,>. Given E > 0, there is a function f that is 
analytic and univalent in a neighborhood of Su {[, , t2, . . . . [,}, and con- 
tinuous in S such that ,f (ck) = ck for k = 1,2, . . . . n and f(S) c { [ : 0 < 
Re[<e}. 
Stated briefly, Theorem 3 provides mappings which keep ck fixed while 
“squashing” the set S toward the imaginary axis. We ask whether 
Theorem 3 can be improved to include the conclusion f(S) c { [ : m - E < 
Im [ < M + E}, where m = min, Im ik and M= maxk Im ck. Such a result 
may have implications when combined with Theorem 2 or similar facts. 
FINITE BOUNDARY INTERPOLATION 317 
2. PROOF OF THEOREM 1 
The proof of Theorem 1 depends on the following construction of a map 
of D onto a set consisting of D less a finite number of slits. 
Suppose that m is an integer, m > 2, and let ik = eiYr for k = 1, 2, . . . . m, 
where Y,<~~<...<~~<Y,+~Tc. Assume that [‘=e’7’, where y,,-,<y’< 
;‘, + 271. We will obtain a function h which in particular is analytic in D and 
at 5,, iz, . . . . i, ,, i’ such that h(ck)=ck for k = 1, 2, . . . . m- 1 and 
A(<‘) = [, 
If [’ = [,,, then the identity function serves for h. Otherwise, define y” by 
y” = i 
tb,, I + r,,,L if y’>y, 
g-y,,, + y 1 + 27c), if y’ < y,,. (‘1 
This gives a point [” = eiYw so that on the counterclockwise arc on 8D from 
i,, ~ , to [, , [,,, is between [’ and [‘I. For I z I < 1, let 
“I - ’ 1 + [, z 
p(z)=-& c 
,=, =cp’ 
and define CY by 1 c( 1 < 7112 and 
tan a = ’ ycot (“1”‘). 
m-1 ,=, 
Also, let the function g be defined by the differential equation 
d(z) 
-=e’“[(cosU)P(z)+isincc] 
g(z) 
(2) 
(3) 
and the conditions g(0) = 0 and g’(0) = 1. Since Rep(z) > 0 for 1 z 1 < 1 and 
p(O) = 1, it follows from [S, p. 521 that g is E-spiral-like. The function g 
maps D one-to-one onto the plane slit along m - 1 Jordan arcs (spirals) 
connected only at infinity. 
Let C, , CZ, . . . . C,. , denote the circular arcs on 8D which correspond to 
the individual slits comprising @\g(D). On each arc Ci there is a unique 
point 5, mapping to the tip of the corresponding slit. Equations (2), (3), 
and (4) imply that g’(e”“) = 0, and therefore t,,.. , = r”. 
Let aj = g( t,) for j = 1, 2, . . . . m - 1, and for t > 0 let G, denote the subset 
of @ defined by w#ai exp[-e’“s] for O<s<t and j=l, 2,..., m-l. 
Properties of a-spiral-like functions imply that G, c g(D). Thus the function 
g, defined by g,(z) = g -- ’ [ { exp( - e’?) } g(z)] for t > 0 is analytic in D. Also 
g, maps D one-to-one onto a subset of D formed by removing m - 1 
Jordan slits joined to aD at the points t,, t2, . . . . t,+, . Each point z with 
318 MAC GREGOR AND TEPPER 
lzl=l and z#[, (j=1,2 ,..., m - 1) is mapped by (exp[ -e”t]} g onto 
another (finite) point on the spiral containing g(z) or on the extension of 
that spiral toward the origin. Therefore, g,([,) = ii forj= 1, 2, . . . . m - 1. The 
function g, is continuous in t for each z in D, and if 1 z 1 6 1 and z # [, 
(j= 1, 2, . . . . m - 1) then g,(z) + 0 as t -+ m. This implies that g,,([‘) = [,,, 
for some t, > 0. The function g,, is analytic in D and is continuous in 6. 
Also, ) g,,(z)] = 1 if I z I = 1 and z is sufficiently near any of the points 
Y ” 
4 I2 i2, . ..> 4, , , i’. The reflection principle implies that g,, is analytic at 
[, , cz, . . . . [,,, , , c’ and the reflection also shows that g,, is univalent in the 
union of D and a neighborhood of { [, , c2, . . . . i,,- , , [’ }. 
This obtains h =g,“. Geometrically stated, for each t, g, fixes c,, 12, . . . . 
[,,, ~, as g,([‘) moves along CJD (which monotone argument) until it reaches 
i,,, for the value t = t,. The slit at [” effectively pulls [’ toward c,,,. 
We now prove Theorem 1. Let z,, z2, . . . . z,, and w,, wZ, . . . . ,v, be as 
described in the Introduction. If n = 1, the function f is obtained by a 
rotation. For n = 2, first rotate D mapping z, to u’, . Let z; be the image of 
z2 under this rotation. The constructon above with m = 2, (, = w, , iz = wz, 
and c’=z; yields a function g such that the composition of the rotation 
with’g gives a suitable function ,J 
Suppose that the theorem holds for n = N. We will show that it holds for 
n=N+ 1. Let z,,z ,,..., zN+, and w,, wz ,..., We+, be the given sets of 
points. There is a function ,fN satisfying the theorem for the sets of points 
-?I,-?2 )...) ZN and u’, , M’~, . . . . ivy. In particular this provides a suitable 
neighborhood A of {z,, z2, . . . . z,}. Also A contains a disk {z: 
lz-zNI <E}, for some e>O, which does not contain z,,z2, ,zNP,,zN+, 
and such that wk$fN[(-7: lz-zNI <E}] for 1 <k<N+ 1 and k#N. Let 
?.’ 
‘N+Ize “’ be a point in {z: Iz-zNI <E} with c(~< a’<~,~+, (where 
? -N=e ‘x.% and 7 ‘Nfl =r “A + I). Let wh + , = fN( zh + , ). 
The earlier argument gives a function h, which is analytic and univalent 
in the union of D and a neighborhood of {MI,, ,,I~, . . . . wN, wl,, , } such that 
h,(w,) = ,,I~ for k = 1, 2, . . . . N and h,(w~+ ,) = wN + , Since fN is analytic 
and univalent in D u A, it also has these properties in D u B, where B is a 
smaller neighborhood consisting of open disks centered at z, , z2, . . . . 
zN, z;+ , so that fN maps these disks into the neighborhood of {w,, w2, . . . . 
w  N, wh + , } above. 
The earlier argument also gives a function h, which is analytic and 
univalent in the union of D and a neighborhood of {z,, z2, . . . . zN+, ) such 
that h,(z,) = zk for k = 1, 2, . . . . N and h,(z,+ ,) = zl,, ,. Some smaller 
neighborhood of {z,, z2, . . . . zN + , } is mapped by h, into B. 
The properties of h, and h, imply that f= h, ofNo h, satisfies the con- 
clusions of the theorem associated with the points z,, z2, . . . . zN+, and 
u’I > w2, . . . . wN+ I 
FINITE BOUNDARY INTERPOLATION 319 
3. PROOF OF THEOREM 2 
The proof of Theorem 2 depends on interpolation and approximation 
reslts about polynomials and is set up by the following lemmas. 
LEMMA 1. Suppose a<h, c<d, and O<&<(d-c)/(h-a). There is a 
cubic polynomial f such that ,f(a) = c, f(b) = d, f’(a) =f’(b) =E and 
min { f’(x) : a 6 x ,< h} = E. 
Proof: A translation of variables implies that there is no loss of 
generality to assume that a = c=O. Let t >E and let g be the quadratic 
polynomial such that g(0) = E, g(h/2) = t, and g(b) = E. Define ,f by f(x) = 
s; g(s) ds. Thenf(0) = O,f’(O) =f’(b) = E, and,f’(x) = g(x) 3 E for 0 6 x < h. 
Since 1; (g(s) - E) d. 7-0 as t +E, it follows that f(b)-+~h as t -+E. Also, 
f(h) + cc as t + x. The condition 0 < E < d/h and the continuity off assure 
that there is a value of t for which J‘(h) = d. 
LEMMA 2. Suppose a<x,<x,<...<x,,<h and y,<y2<...<yn. 
There is a function f defined and continuously differentiable on [a, b] such 
that f (xk) = y, ,for k = 1, 2, . . . . n and min { f ‘(x): a <x < b} > 0. (Here and 
later, derivatives at end points are one-sided limits.) 
Proqf: Choose y0 and y, + , such that y, < y, and y, + , > y, and let 
xO=a and x,+, =b. Choose E such that O<c<min{(y,+,-yk)/ 
t-x k+, -xk): k=O, l,..., n}. Lemma 1 implies that there is a cubic 
polynomial in each of the intervals [x,, xk+ ,] for k = 0, 1, . . . . n which 
piecewise defines a functionf on [a, b] which is continuously differentiable. 
Also, min{f’(x): a<x<b} =E>O. 
LEMMA 3. Suppose a<x,<x,<...<x,,<b and y,<y,<...<y,. 
There is a polynomial p such that p(xk) = y, .for k = 1, 2, . . . . n and 
min{p’(x): adxdb} >O. 
Proof. Let f satisfy Lemma 2 and let E = min { f ‘(x): a < x 6 b}. Given 
6 > 0, then by [4, p. 1131 there is a polynomial q such that 
and 
max{If(x)-q(x)l:a<xdb)<6 (5) 
max { ) ,f’(x) - q’(x)1 : a < x < b } < 6. (6) 
Let r be the polynomial which interpolates the values f(xk) -q(xk) for 
k = 1, 2, . . . . n. Then r can be expressed 
r(x)= f Cf (xJ - dxk)l Pk(xL 
k=l 
(7) 
320 MAC GREGOR AND TEPPER 
where P, is the polynomial of degree at most n such that P,(x~) = 0 for 
j#k and P,(x,)=l. Let M, = max { 1 P;(x)1 : a d x < 6) and let 
M = C; =, M,. Equations (5) and (7) imply that T’(X) 2 -6M for 
a 6 x 6 6. 
The polynomial p = r + q satisfies p(xk) =f(x,) = y, for k = 1,2, . . . . n. If 
a 6 x < b then (6) and the lower bound on r’ imply p’(x) = r’(x) + q’(x) > 
,f’(x)-6-6M>,s-6-6M. Therefore min{p’(x): a<x<b}>O for 
sufficient small 6. 
Remark. No claim is made in Lemma 3 about the degree of p. In 
general, the Lagrange solution of the interpolation p(xk)= y, with the 
conditions of Lemma 3 is not necessarily increasing on [x,, x,]. This 
suggests the problem of determining whether there are upper bounds on 
the degree of p which depend on n and/or the “spread” of the points xk, y,. 
Proof of Theorem 2. Let p be a polynomial given by Lemma 3. We will 
show that there is a neighborhood (in the plane) of [a, b] in which p is 
univalent. On the contrary, assume there is no such neighborhood. This 
implies there are two sequences {zk} and {z;} with zk #zK and p(zk) = 
p(z;) for k = 1, 2, . . . . and each sequence has an accumulation point in 
[a, b]. Consideration of subsequences implies that we may assume that 
7 + x0 and z; + xb with x0 and xb in [a, b]. Thus p(xO) =p(xb), and since 
yis strictly increasing on [a, b], this requires x0 = xb. However, p’(x,) # 0, 
and therefore p is univalent in some neighborhood (in the plane) of x0. 
This contradicts p(zk) =p(z;) for sufficiently large k. 
4. PROOF OF THEOREM 3 
We first note that Theorem 1 has an equivalent formulation for suitable 
domains which are conformally equivalent to D. For R, this is obtained by 
the introduction of a Mobius transformation and applies to two sets of n 
complex numbers on { [ : Re [ = 0} in the same conformal order. 
In the case S is unbounded, first consider a mapping c + l/([ - [‘) which 
sends S to a bounded set T in R any complex number [’ with Re [’ = 0 and 
[’ # ck for k = 1, 2, . . . . n. In particular, Tc { [ : 0 < Re c < M} for some M. 
Let [; = l/(ck - [‘). Theorem 1 implies that for each p with 0 < p < 1 there 
is a function gp which maps { [ : 0 < Re [ < M} into itself and is analytic at 
pi;. Also, g,(p[;) = ch for k = 1, 2, . . . . n and g,(O) = 0. The function g, 
with pM < E satisfies the conditions on f in the theorem. In the case S is 
bounded, the auxiliary mapping [ H l/([ - [‘) is not needed. 
FINITE BOUNDARY INTERPOLATION 321 
REFERENCES 
1. L. BRICKMAN, T. H. MACGREGOR, AND D. R. WILKEN, Convex hulls of some classical 
families of univalent functions, Trans. Amer. Math. Sot. 156 (1971), 91-107. 
2. D. G. CANTOR AND R. R. PHELPS, An elementary interpolation theorem, Proc. Amer. Math. 
Sot. 16 (1965), 523-525. 
3. J. G. CLIJNIE, D. J. HALLENBECK, AND T. H. MACGREGOR, A peaking and interpolation 
problem for univalent functions, J. Math. Anal. Appl. 111 (1985), 5599570. 
4. P. J. DAVIS, “Interpolation and Approximation,” Ginn (Blaisdell), Waltham, MA, 1963. 
5. P. L. DUREN, “Univalent Functions,” Springer-Verlag, New York, 1983. 


Finite boundary interpolation by univalent functions 

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Finite boundary interpolation by univalent functions

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