NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 … · Proof. It suffices to show that 2/(|) G TJCh•...
Transcript of NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 … · Proof. It suffices to show that 2/(|) G TJCh•...
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
E.M. Silvia
Department of Mathematics
University of California
Davis CA 95616-8633
U.S.A
emsilvia@ucdavis. edu