NEW ZEALAND JO U RN AL OF MATHEMATICS Volume 28 (1999), 275-284

H. S il v e r m a n a n d E.M. S il v ia

(Received July 1998)

Abstract. Complex-valued harmonic functions that are univalent and sense

preserving in the unit disk A can be written in the form / = h + g, where h

and g are analytic in A. We study subclasses for which h and g have the form

h(z) = 2 - anZn , an > 0 and g(z) = - bnzn , bn > 0, bi < 1 and

/ maps A onto a domain that is either starlike with respect to the origin or

convex. Characterizing conditions involving bounds on the coefficients lead to

various extremal properties. We conclude with some extremal properties for

subclasses having b\ fixed.

SU BC LA SSE S OF H A R M O N IC U N IV A L E N T F U N C T IO N S

1. Introduction

A continuous function / = u + iv is a complex-valued harmonic function in a domain V C C if both u and v are real harmonic in V. In any simply connected domain, we can write

f = h + g, (1)

where h and g are analytic in V. We call h the analytic part and g the co-analytic part of /. A necessary and sufficient condition for / to be locally univalent and

sense-preserving in V is that \h'(z)\ > \g'{z)\ in V. See [2].Denote by Sh the class of functions / of the form (1) that are harmonic univalent

and sense-preserving in the unit disk A = {z : \z\ < 1} with /(0) = f z(0) — 1 = 0.

Thus we may write

OO OO

h(z) = z + ^ 2 anzn and g(z) = ^ & nzn, |&i| < 1. (2)n = 2 n = l

Note that S h reduces to <S, the class of normalized univalent analytic functions whenever the co—analytic part of f is zero. For f = h-\-g € Sjj, ^(0) = 1 >

|6i| implies that the function (/ — b\f)(l — |£>i|2) 1 is also in Sh- Consequently, we may sometimes restrict ourselves to S ^ , the subclass of S h for which b\ = /f(0) = 0. Both S h and 5# are normal families; however, only <S# is compact. See [2].

Let S'h and Kh be the subclasses of Sh consisting of functions / that map A onto starlike and convex domains, respectively. We further denote by Tfi and TJCh

the subclasses of and Kh, respectively, for which the coefficients of / = h + g

1991 AMS Mathematics Subject Classification: Primary 58E20,30C45.

Key words and phrases: Harmonic, sense-preserving, univalent.This work was partially completed while the authors were visiting scholars at the University of

California in San Diego. The authors would like to thank Professor Carl FitzGerald for numerous

stimulating discussions.

276 H. SILVERMAN AND E.M. SILVIA

take the form

h(z) = z — ^ anzn, an > 0 and g(z) = — ^ bnzn, bn > 0, &i < 1. (3)n = 2 71=1

In [5], the families T^° = fl and TKPh = TKh H Sjj were investigated. Proofs for the compact subfamilies <S#, 5# and KQH frequently do not carry over to

the corresponding families Sh , and Kh - The harmonic Koebe function, defined in [2], is conjectured to have extremal coefficient bounds for Sjj. In [3], these

bounds are shown to be extremal for S See also Section 4 of [1] for extremal properties for and an operator that takes functions from Stf to K°H.

In this note, we generalize results for and TK°H to the families and TKh■ There are some fundamental differences in the results, because 7^, unlike Tj!j0, is not a compact family. To see this, note that /n(z) = z — ^ j z € T£, for

n = 1 ,2 ,3 ,..., while limn->oc f n { z ) = z — z is not even univalent in A.We will also obtain surprising results for when b\ is fixed. In the sequel, we

shall take / = h + g, where h and g are in the form of either (2) or (3).

2. M ain Results

We begin with some sufficient coefficient conditions.

Theorem 2.1. For f = h + g with h and g of the form (2), if

OO OO

^ n | a n| + Xn|&n| < 1,71—2 71=1

then f is univalent in A and f eS*H.

Proof. Since X 1 2nla«l + X ^ i n IM < f°r anY 2 with \z\ — r e (0,1), we have that

OO OO

\h'(z)\ > 1 - ^ n | a n|rn_1 > ^ n | 6n|rn_1 > \g'(z)\.71=2 71=1

Thus, / is locally univalent and sense preserving. If g = 0, the univalence of / will follow from its starlikeness; otherwise, for z\ j- z2, it follows that

HARMONIC UNIVALENT FUNCTIONS 277

g { z i ) - g ( z 2)

h(zi) - h(z2)

X > » w - *2)n= l

(zi - Z2) + ^Taniz? - z%)71 = 2

OO

^ 6 n(z” _1 + z” _2z2 + . . . + z2 ~l)71=1

1 + X a n ^ i 1 + z i 2z2 + . . . + z% x)7 1 = 2

OO

X n l6” l< 71=1

i -

71=2

< 1.

Hence,

f { z i ) - f ( z 2)

h(zx) - h(z2)= 1

g(zi) -g{z2)

h(zi) - h(z2)> 0

and we conclude that / is univalent. Finally, starlikeness will follow from showing

that Jj(arg f(re10)) > 0 for 0 < 9 < 2n, 0 < r < 1. Since

d_

d6(arg f{ r e ie)) = % <

1 — bie 126 -I- y ^ ( n a ne^w - nbne *(n+ 1)0)rn 1n=2

1 + bie~i2e -I- X + b n e - ^ + ^ r 71- 1n=2

:= %C,

C will have positive real part if the modulus of

1-C

i + c2b1e~i2e - j P ((n - l K e ^ " 1^ - (n + l)6ne - i(n+1)> n- 1

n=2

2 + E ((" + l)“nei<n-1)e - (n - l)6„e-i(n+1>s)rn-1

n=2

is bounded above by 1. From ^ ^ L 2 n lan|+Z]^=i n IM ^ 1) we see that la«l —

\ and |&i| < 1 — E ^ 2n(M + l&nl)- Hence,

oo oo oo oo

^ ( ( n + 1)|a„| + ( n - 1)|6n|) = nda"l + l6"l) + lflnl ~ H l6nln=2 n=2 71=2 71=2

< - - |6i| < 2-

278 H. SILVERMAN AND E.M. SILVIA

It follows that

1-C

i + c2 N + ( ( n - 1 ) la «l + (n + l )\bn\)

<n = 2

2 - £ ((n + + (n - 1)l&»l) n = 2

/ oo \ oo

2 I 1 “ 5 Z + M H ( ( n “ 1 ) l a n l + (n + 1 ) l ftT»l)

< n = 2 n —2

n —2

= 1.

Therefore, £ > 0 as needed.

Corollary 2.2. For f = h + g with h and g of the form (2), if

□

^ n 2|an| + ^ n 2|6„| < 1,n=2 n=l

then f € /C#.

Proof. A necessary and sufficient condition for / to map |z| = r onto a convex

domain is that

d_d9 arg

d_80

f{reie)

d dde

= % <

1 + bie~i2e + ]T (n2anei(n-1)0 + n2bne-*n+1>>e)rn- 1

n = 2

1 - bxe~™ + Y n ^ ei{n~1)9 ~ bne~i{n+1)e)rn- 1

n = 2

is positive. The remainder of the proof proceeds along the same lines as that of

Theorem 2.1. □

We next look at the subclasses for which these sufficient conditions are also necessary.

Theorem 2.3. Let f = h + g where h and g are of the form (3). Then f if

and only if J2™=2 nan + nbn < 1.

Proof. In view of Theorem 2.1, we only need to show that / ^ if the coefficient

condition is not satisfied. For this case, we will show that / is not even univalent. Setting z = r > 0 , we have /(r) = (1 — &i)r — + bn)rn and f'(r) —

(1 - &i) — Y^=2 n(an + frn)rn-1- Since /'(0) = 1 — 6i > 0 and / '( l) < 0, there must exist an ro, ro < 1, for which f'{ro) = 0. Hence, ro is a local maximum for f(r) and f(r) is not even one-to-one on the real interval (0, 1). □

HARMONIC UNIVALENT FUNCTIONS 279

Corollary 2.4. For f = h + g where h and g satisfy (3), / € T/Ch if and only if E ” X « n + £ “ i n X < l .

Proof. In view of the corollary to Theorem 2.1, it suffices to show that / ^ T/C#

if the coefficient inequality does not hold. For f = h + g where h and g satisfy (3), a necessary and sufficient condition for / to map \z\ = r onto a convex domain is

that

l ( arg{ l /(re‘9)})

=n=2 n = l

-i(n+ l)6r n - l

n= 2 n = lis positive. When 0 = 0, the last expression is

1 - n2anrn 1 - ^ n2bnrn 1

n = 2 n = l1 - ^ nanrn 1 + nbnrn 1

n=2 n = lIf n2fln + X ^ L i77,2 > 1, the numerator is negative for r sufficiently close to1. Thus, there exists r € (0, 1) for which the quotient is negative. Hence / ^ TJCh and the proof is complete. □

Corollary 2.5. If f € Tfi, then f maps \z\ < \ onto a convex domain. The result_ 2 _ —2

is sharp, with extremal functions z — bz — ^ and z — bz — , 0 < b < 1.

Proof. It suffices to show that 2/(|) G TJCh• From

_ —nr

n = 2 — n = lit follows that

oo / , \ oo

+ £ n2 + - 6l + 5Z n(fln + M < 1

as needed. □

For any compact family T, the maximum or minimum of the real part of any continuous linear functional occurs at one of the extreme points of the closed con­vex hull, clco^. We will use the necessary and sufficient coefficient conditions of Theorem 2.3 and its Corollary to determine the extreme points of the closed convex

hulls of and TJCh .

280 H. SILVERMAN AND E.M. SILVIA

Theorem 2.6.(a) Set hi(z) = 2, hn(z) = z - £ (n = 2 ,3 ,...); and gn(z) = z - £

(n = 1 ,2 ,3 ,...). Then f e clco 7# */ and only if

OO

f(z) — ^ (An^n ~i~ 7nffn);n= l

w/iere Ai = 1 — X^*L2 An " Z !^L i7n> Aj > 0 and 7? > 0. In particular, the

extreme points of are {hn} and {<7n}-

(b) Set fci(z) = z, hn{z) = z - £ (n = 2 ,3 ,...), and gn(z) = z - £

(n = 1 ,2 ,3 ,...). Then f £ clcoTK,h if and only if

OO

f{%) ~ ^ ^(Xnhn -|- r)n9n)t n—1

where Ax = 1 - An - E ^L i 7»> A, > 0 and 7,■ > 0 .

Proof. We prove (a). Because the proof of (b) is similar, it is omitted. Suppose

OO OO * OO

f (z) = Ys(Xnhn+7n5n) = z ~ Y l ~ znn—l n=2 n= l

Then

oo / \ \ 00

£ > ( ! ) + £ » ( 5 )n—2 ' ' n = l

and / G clco 7^.Conversely, if / 6 c l c o w e set An = nan (n = 2 ,3 ,...), 7n = nbn

(n = 1 ,2 ,3 ,...), and 71 = 1 - J2n=2 An - E ^= i 7n- Then /(z) = £~ =1(An/in + 7ngn) as needed. □

Remark 2.7. Most families investigated for univalent functions with negative co­efficients are both convex and compact, including T^° and TK?H, so that the closed convex hulls are the families themselves. See [4] and [5]. This is not the case for

the families in Theorem 2.6, since the extreme point <71 (z) = z — z e clco7^ — T£.

In [5], it is shown for / £ and \z\ = r < 1 that

r ~ y < \ f{z)\ < r+ T— .

We next show that the corresponding bounds for Tfi are quite different.

Theorem 2.8. If f € Tfi, then

HARMONIC UNIVALENT FUNCTIONS 281

Proof. In view of Theorem 2.3,

OO

\f(z)\ < {l + b1)r + ' 2 (an + bn)rnn= 2

oo

< (l + bi)r + r2 ^2 (an + bn)n=2

< (1 + b\)r + r2-

The upper bound follows upon noting the right side is an increasing function of b\.

The lower bound is a consequence of the inequality

|/(*)| > (1 - b i)r- (^—^ ) r 2.

To see that the result is sharp, we note that, for f(z) = z — bz, setting z = ir and z = r yields the upper bound and lower bound, respectively, upon letting

b — >1". □

Remark 2.9. Since z — bze TK-h , the same bounds remain valid for this subclass

°f T&.

3. Fixed Coefficient Results

Denote by 7^(6), 0 < b < 1, the subfamily of consisting of functions for

which bi = b is fixed. It is easy to see that Tfi(b) is a compact and convex family for which the extreme points are

h\(z) = z — bz, hn{z) = z — bz — zn, and

gn{z) = z- b z- zn, n = 2,3,4 ,.. ..

It is of interest to determine bounds on the modulus as well as the real and imagi­nary parts of functions that are in T^(b). The results are unexpected.

Theorem 3.1. If f G T^(b) and \z\ = r, then

\f(z)\ > r(l - b) ( l - 0 , 0 < b < 1,

and, at least for 0.029 < b < 1

(4)where

(r(h rW2 3 + 3b + 262 2 (1-6)2 [G{b,r)) = -------- r + -- ---

—8b3 + yj(462 + 3(1 — 6)2(3 -I- b)r2)

+ 27(1 - b)2

282 H. SILVERMAN AND E.M. SILVIA

Proof. It suffices to consider the extreme points of Tfi(b). Since (1—b)/n decreases

with n, the lower bound clearly holds for h2(r) = g2(r).

For n > 3,

\hn(z)\ < r( 1 -I- 6) -I- -— - rn < r(l + b) + r3 = \h3(ir)\. n o

Similarly, \gn(z)\ < l^3(*r)l when n > 3. Thus, it suffices to compare \hs(ir)\ with max# \g2 (z) | and max \h2(z)\.

Straightforward, though messy, calculations reveal, for \z\ = r that

il t m2 2 -+■ 36 + 362 2 (1 — 6)2 4 max \h2(z)\ = -------- r + — ~ r

-8b2 + J 6(46 + 3(1 — 6)2(1 + 36)r2)3

+ ------ 276(1 - 6)2----------

for all r when < 6 < 1 and for 0 < r < when 0 < 6 < ;

max|02(z)|2 = (G(6,r ) )2U

as defined above. Basic algebraic arguments lead to the conclusion that, at least

for 0.02853 < 6 < 1,

max\h2(z)\2 < max |<72(-z)|2- (5)6 6

Thus, for 0.029 < 6 < 1, we conclude that the maximum of |/(z)|2, / G 6)

and \z\ = r, is given by either |/i3(ir)|2 or (G(b, r))2. To see that both functions given on the right side of (4) can yield the maximum, note that when 6 = 0.4 we

have that max# 1 2(0-4) |2 « 0.33061 > \hs(0Ai)\2 « 0.3281 while max# |2(0-8)|2 « 1.4804 < |/i3(0.8*)|2 « 1.4943. □

/ 2 \Remark 3.2. For 6 = 0, we know that max# \h2(z)\ = max# \g2(z)\ = (r +

while, for 6 G (0,0.029], numerical considerations involving Maple, support the claim that inequality (5) holds for all r G (0,1). However, the bounding arguments used did not carry over to the whole range of 6.

Remark 3.3. Applying basic algebraic arguments, it is easy to prove that, for fixed6 > 0.48, \hs(ir)\ > G2(b,r) for all r G (0,1). Numerical considerations involving Maple indicate that, at least for each fixed 6 with 0 < 6 < .3, \hs(ir)\ < G2(b,r)

for 0 < r < 1.

The bounds on %ef(z) must occur at an extreme point. Note that 'Re h\(z) — z — bz = r(l — 6) cos 9 and, for n > 2,

!Rehn(z) = %£gn(z) = r(l - 6)r

cos 9 -----cos nO

We have %&hn(z) > h2(-r) = - r(l - 6)(l + ^).

The upper bound is much more surprising. Observe that $£ h\(z) < h\(r) — r( 1 — 6). Thus for H(e hn(z) to be maximal for n > 2 and some r, there must be a

rn-l9 for which tn (r, 9) = cos 9 -----cos n9 > 1.

n

HARMONIC UNIVALENT FUNCTIONS 283

We first look at the boundary r = 1. Then tn{ 1,9) = tn(9) = cos9 — cosJ l6- . If

tn(9) > 1, we have cos# > 0 and cosn9 < 0. Now t'n(9) = — sin9 + sinnO = 0 when sin# = sinn9, and hence cos# = — cosn# = cos(7r — n9) > 0. Thus, for each n,

tn{9) < cos ( ^ j ) . Numerical considerations lead to the conclusion

that ^ n ( ^ i ) increases for n < 8 and then decreases with n. Hence,

tn{9) < t8 | COS Q ) « 1.057.

In particular, for / G T£(b), $gf(z) attains its maximum when \z\ = r is close

to 1 for hg(z). Moreover, max tn(r,9) < maxin(l,#) < ^ ( X f ) w 1.057. This appears to indicate that tn(r,9) < 1 for r outside of some neighborhood of 1, in

which case ^ f ( z ) < h\{r) = r(l — b).These observations are summarized in the following theorem.

Theorem 3.4. If f G T^(b), then for \z\ = r we have f(z) > — r ( l—b)(l-F|) =

h2{—r). In addition, %gf (z) < r( 1 — b) = hi(r) for 0 < r < ro « 0.900; %e f(z) attains its upper bound with hg(z) for ro < r < r\ « 0.915 and with hs(z) for

ri < r < 1.

For the bounds on Im f over / G Tj {b), again restricting ourselves to the ex­

treme points of T£(b), we note that Imh\{re%e) = (1 4- b)rsin9, Imhn(reie) =

(1 -1-b)r sin 9 — rn sin n9, and

sn(r,9) := Imgn{re%e) = (1 + b)rsin# + -— -rnsinn#.

We have that Imhn(reld) and Imgn(rel9) assume the same set of values for —7r < 9 < 7T. Furthermore,

max{Img3(reie)} = s3 (r, = (1 + b)r+ - y - r3 > Imhi(ir)

and

mm{Img3(re’ie)} = s3 (r, = -(1 + 6)r - ^-y^ r3 < Iw i/ii(-2r).

It follows that finding bounds for Im f, f G Tfi(b), is equivalent to comparing

s3(r, |) and ^ ( r , -|) with mdLXo{Img2{reie)} and mine{Img2(rete)}, respec­

tively. The results of the comparison are summarized in the following.

Theorem 3.5. For fixed b, 0.2 < b < 1, and f G T^{b),

- ^(1 + b)r + r3 < ifft f[reld) < (1 + b)r + “ y-"r3> for \z\ = r• (6)

The result is sharp.

Proof. For fixed b and r, the extrema for Img2(rel6) = (1 -I-b)r sin 9 + r 2 sin 29

occur when cos#* = where m(b,r) = a/(1 + 6)2 + 8(1 — b)2r2. Then

the minimum and maximum are s2(r,—9*) = —H(r,b) and s2(r,0*) — H(r,b),

respectively, where

j-y /, v (3(1 + 6) + m(6, r)) /8(1 - b)2r2 - 2(1 + b)2 + 2(1 -I- b)m(b, r)

H{b’r) = ------------------ ------------------------

284 H. SILVERMAN AND E.M. SILVIA

Finally, we note that

(3(1 + b) + m(b, r)) \/8(1 - b)2r2 - 2(1 + b)2 + 2(1 + b)m(b, r)

at least for b > 0.2. Consequently, for b > 0.2, the sharp bounds for Im f are given

by 03- D

Remark 3.6. Note that, for b € [0,0.2), the maximum and minimum of Im f ,

/ 6 are given by g2 for some r. For example, #(0.1,0.25) « 0.28048 >53(0.1,0.25) « 0.27969 while #(0.1,0.75) « 0.94179 < p3(0.1,0.75) « 0.95156.

References

1. J.A. Cima and A.E. Livingston, Integral smoothness properties of some harmonic mappings, Complex Variables, 11 (1989) 95-110.

2. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Annales Academici Scientiarien Fennicae, Series A I Mathematica 9 (1984) 2-25.

3. T. Sheil-Small, Constants for planar harmonic mappings, J. London Math Soc.

(2) 42 (1990) 237-248.4. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math.

Soc. 51 (1975) 109-116.5. H. Silverman, Harmonic univalent functions with negative coefficients, J. Math.

Anal, and Appl. 220 (1998), 283-289.

< 16(1 — b) M l + b)rsin# +

H. Silverman

Department of Mathematics

University of Charleston

Charleston SC 29424

U.S.A

silvermanh@cofc.edu

E.M. Silvia

Department of Mathematics

University of California

Davis CA 95616-8633

U.S.A

emsilvia@ucdavis. edu