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31Труды Петрозаводского государственного университета
Серия “Математика” Выпуск 7, 2000
YJIK 517
UNIFORMLY STARLIKE AND UNIFORMLY CONVEX MAPPINGS IN SEVERAL COMPLEX VARIABLES
Stanislawa Kanas, Gabriel a Kohr, Mirela Kohr
In this paper we introduce some subclasses of biholomorphic mappings on the unit ball of Cn. These mappings called uniformly starlike and uniformly convex are introduced by geometric interpretation, like in the case of one variable.
§ 1. Introduction
Let Cn be the space of n complex variables z — (zi,...,zny with the Euclidean inner product (z,w) = zj an(^ the norm INI =
(z, z)1/2. The symbol ' means the transpose of vectors and matrices. For z_= (zi,... ,zny E Cn let z = (zi, • • • ,zny and for A = (a^-) i<ij<n let A be the conjugate of matrix A. We denote by 0 the origin of Cn and by L(Cn, Cn) we denote the space of all continuous and linear operators from Cn into Cn with the standard operator norm. Further, let I denote the identity in L(Cn, Cn) and let Br denote the open ball {z E Cn : ||z|| < r}. The open unit ball is abbreviated by B = B\. In the case of one variable the ball Br is denoted by Ur and the unit disk U\ by U.
Also, let H(G) be the set of holomorphic mappings from a domain G C Cn into Cn. We say that / E H(G) is locally biholomorphic on G if its Frechet derivative
Df(z) =
as an element of L(Cn,Cn) is nonsingular at each point z E G. For a mapping / E H(G), let D2f(z)(u2) = D2f(z)(u, u), for z E G and
© Stanislawa Kanas, Gabriela Kohr, Mirela Kohr, 2000
u E Cn, where D2f(z) denotes the second order Frechet derivative of / at
z. Also we say that / E #(G) is biholomorphic on G if the inverse /-1 exists and is holomorphic on a domain 0 such that /_1(0) = G.
The mapping / E H(B) is said to be starlike if /(0) = 0, / is biholo-morhic on B and /(5) is a starlike domain in Cn with respect to zero. Also, if / E H{B), we say that / is convex if / is biholomorphic on B and f{B) is a convex domain in Cn.
Matsuno [15] and Suffridge [20] proved that a locally biholomorphic mapping / E H(B) with /(0) = 0 is starlike if and only if
On the other hand, Kikuchi [8], Gong, Wang and Yu [1] showed that a locally biholomorphic mapping / E H{B) with /(0) =0 is convex if and only if
for all z E B \ {0} and v E Cn \ {0} with Re(z, v) = 0.
Very recently Goodman [2, 3] introduced and studied the notion of uniform starlikeness and uniform convexity on the unit disk U. Several authors have continued the study of these mappings and deduced important results in this direction (see for details [17, 18], [4, 5, 6], [11], [13, 14], etc.).
Definition 1. An univalent function f is called uniformly starlike in U if f maps every circular arc 7 contained in U with center ( also in U onto the arc /(7) starlike with respect to /(C)-
In [2] is showed that if
The class of all functions uniformly starlike in U was denoted by UST.
Re ([Df(z)] 1f(z),z) > 0,z G B \ {0}.
(1.1)
\\v\\2-Re(Df(z)]-1D2f(z)(v,v),z)>0,
(1.2)
then / is uniformly starlike in U if and only if
>0, (z,()&UxU.
(1.3)
Definition 2. An univalent function / is called uniformly convex in U if f maps every circular arc 7, contained in U with center ( also in U, onto the convex arc /(7).
In [3] is showed that an univalent function /, normalized by /(0) = /'(0) — 1 = 0, is uniformly convex in U if and only if
Re
1 + (z - o
/"(*)
/'(*)
> 0, (z, OeUxU.
(1.4)
The class of all univalent functions, uniformly convex in U was denoted by UCV.
As a matter of fact the assumption C E U in the case of uniform convexity can be dropped. The analytic condition which characterizes such class of functions coincides with (1.4) but with the assumption C E C (for details we refer to [4, 5, 6]).
In the present paper we will extend the above definitions to the case of locally biholomorphic mappings on the unit ball of Cn and we will investigate some properties of such mappings.
§ 2. Main results
We begin this section with the following definitions which are natural extensions in Cn of Definitions 1.1 and 1.2.
Definition 3. Let f : B —>• Cn be a biholomorphic mapping on B, normalized by /(0) = 0 and Df{0) = /. We say that f is uniformly starlike in the unit ball B if for every part T of the sphere dB((,r) = {z e Cn : \\z — (\\ = r} contained in B, with center ( also in B, the hypersurface f(T) is starlike with respect to /(C)-
We remark that if we denote by Cr(() = f(T) then Cr(() C {w E Cn : VKw) = 0}, where w) = w E f(B) and ip(z) = \\z — C||2 —
r2,z E B. Also, the hypersurface Cr(() is starlike with respect to f(() if and only if
Re (NWJw — /(C)) > 0, weCr(C), (2.1)
where Nw denotes the outward normal vector to Cr(() at w.
Definition 4. Let f : B Cn be a biholomorphic mapping on B, normalized by /(0) = 0 and Df(0) = I. We say that f is uniformly convex in the unit ball B if for every part T of the sphere dB((,r) = {z E Cn : \\z — Cl I = r} contained in B, with center ( E B, the hypersurface f(T) is
convex.
Note that the hypersurface Cr(() = /(T) is convex if and only if
S(u,u)>0, ueTw(Cr(0), ^GCr((), (2.2)
where S(u,u) and Tiy(Cfr(C)) mean the second fundamental form of Cr(() at w, and the real tangent space to Cr at w, respectively.
Taking into account the above definitions, we can prove the following results, which characterize analyticaly the notions of uniform starlikeness and uniform convexity in B.
Theorem 1. Let f : B —>• Cn be a locally biholomorphic mapping on B, normalized by /(0) = 0 and Df{0) = I. Then f is uniformly starlike in B if and only if
Re([JD/(z)]-1(/(z) - f(C)),z - 0 > 0, (2.3)
for all z / C-
Proof. First, assume that / is uniformly starlike on B. Then / is starlike, hence biholomorphic on B. Next, let ( E B and r > 0. Also, let V = dB((,r) H B and Cr(C) = /(r). Then CV(() C {w e J{B) : ^(w) = 0} is a starlike hypersurface on Cn, with respect to /(C), where ij){w) = (f o f~1(w), w G f(B) and ip(z) = \\z — C||2 — r2,z e B. Therefore
Re(Nw,w-f(()} >0,
for all w E Cr(().
Now, let w G Cr((), then the outward normal vector to Cr(() at w is Nw = ^Lip(w) and a short computation yields that Nw = (\Df (z)]-1)'(z— C), where z = f~1(w). Hence, it is obvious to see that the above condition is equivalent to the following
Re{[Df(z)]~1(f(z) - f(Q),z - C) > 0,
that completes the proof of the first part of our result.
Conversely if the relation (2.3) holds for all z, C £ B, with z / then if we let C = 0 into (2.3), then we deduce that
Re{[Df{z)]~1f{z),z) >0, z^0,
that means / is starlike, hence biholomorphic on B. Further, the relation
(2.3) implies that
Re(Nw,w- /(C)) > 0,
for all w E CV(C) and C £ so, taking into account Definition 2.1, we conclude that / is uniformly starlike on B, as desired conclusion. □
Theorem 2. Let / : B Cn be a locally biholomorphic mapping on B, normalized by /(0) =0 and Df(0) = I. Then / is uniformly convex in B if and only if
IMI2 - Re([JD/(z)]-1£>2/(z)(t;, v),z - Q > 0, (2.4)
for a 11 z,( G B, z / £ and v E Cn \ {0}, Re (z — (,v) = 0.
Proof. First, assume that / is uniformly convex on B, therefore / is biholomorphic on B and each sphere dB((,r) contained in B is mapped by / onto a convex hypersurface in Cn.
Now, let C E B and r > 0. Also, let T = dB((,r) D B and Cr(() = /(r). Then it is well known that the second fundamental form S(u,u) of CV(C) = f(dB((,r)) at w E CV(C) can be written as follows
Re [tt'0(»)«]
S(u,u) =------^---------y1---------------- (2.5)
|f*H
for all u E Tw(Cr), where — V?0/-1^) and — \\z ~ (\\2 ~r2, z £ B.
After short calculations we deduce the following relations
^ = ll^^)]“luH2 > (2-6)
and
= ~^ ~ (\.Df(z)]~lu’ iDf(z)]~lu^ >
(2.7)
where w = f{z).
Indeed, using the formulas of matrix derivatives described in [7] we have
= §~z ((§;«)’ wwr1) (Wi*)]-1 x /)
= 0(z) ([Df(z)}-' x [Df(z)}-')
- [Df{z)]~'D2f{z) ([.Df(zJ]-1 x [Df(z)]~1) ,
where the sign x designates the Kronecker product.
Hence
“'Sw“ =
- (§j(*)Y [o/M]-‘o2/W ([C/W]-1», [c/M]-1»).
Since
d^p d2 (p
—— = z — C and -7—7(2:) = 0 (the null matrix),
dz dzz
then, in view of the above relations, we obtain
= -(*-([£/(*)]-YlA/WrM •
In a similar manner we can show that
*iL(w)u=i|№irl*>'
o 2
and since = /, then
Now if we set 1? = [2}/(z)]_1u and use the fact that u E T^((7r(C)) we deduce that
0 = Re (^j^j2{w),w - = Re |u' i\Df{z)]~1)' (z - £)J
= Re(z-<,t;), where f{z) = w.
Hence, Re(z — (,v) =0 that means v E Tz(dB(£,r)). Finally, combining (2.5), (2.6), (2.7) and the relation (2.2), we conclude that if / is
uniformly convex, then the condition (2.4) holds for all z, C £ B with z / C an(l v E Cn \ {0}, with Re(z — (,v) = 0.
Conversely, if we assume that / satisfies the relation (2.4), then, using the condition (1.2), we conclude that / is convex, hence biholomorphic on the unit ball. Further, using similar kind of arguments as in the first step, we conclude that each hypersurface Cr(() = f(T) is convex, where T = dB((,r) fl B, for all r > 0 and ( E B, that means / is uniformly convex on B.
This completes the proof. □
Remark. Notice that in the case of one variable, we obtain from (2.3) and (2.4) the usual notions of uniform starlikeness and uniform convexity of functions in the unit disk U. For this aim, it suffices to see that if z, ( E U, z / ( and v E C \ {0} such that RefiJ^ — ()] = 0, then
v(z - ()+v(z-() = 0, hence \v\2(z — C) = —v2(z — £). Therefore, the relation
i.e.
for z,( eU, z / C
Further on, let US*(B) and UK(B) denote, respectively, the classes of uniformly starlike and uniformly convex mappings on B, normalized by /(0) = 0 and Df(0) = I.
We next give some examples of mappings from US*(B).
Example 1. Let /i,..., fn be uniformly starlike functions on the unit disk U and f(z) = (/1(21), —,fn(zn))',z = (z1,...,zny. Then / is uniformly starlike on B.
Indeed, it is clear that / is normalized and locally biholomorphic on B and also
Re {[Df (z)]-1 (f (z) - f(0),z- C) = ERe|^)~J((^| \Zj - 01
(2.8)
1 Re
Re
1 +
/'(*)
(z-or(z)
/'(*)
>0,
>0,
for ^,( G 5, with z / C- Since the functions /j, j G satisfy
condition (1.3) then the right hand side of (2.8) is positive and so the left hand side.
Therefore, using results due to Goodman [2], the following mappings belong to US* (B)
(i) —’-"’1—~—) ’ z = (z1,...,zny G B,
\ 1 — a\Z\ 1 — anzn J
where ^
aj £ C, \a,j\ < -^=, for j £ {l,...,n}.
(ii) f(z) = (z! + aiz[\ ...,zn + anzlJ ) , z = (zi, ...,zn)' £ B, where
/2
aj ^ ^5 IajI ^ 5 ^j^N\{l}, j E {1,..., n}.
(iii) f(z) = (z! - aizl, ...,z„ - anzl)', z = {z\,..., zn)' £ B,
where
\/3
aj £ C, l^l < —, j£{l,...,n}.
Example 2. Let n = 2 and /(z) = (zi + for 2 = (21,22)' G
where a G C, |a| < Then / G US*(B).
Clearly, f is normalized and locally biholomorphic on the unit ball of C2. After simple calculations we deduce that
[Dm]-i(m-m) =
= (zi - Cl + a(z22 - C22), -2az2(z! - Ci + a(z22 - C22))' ■ Therefore, for |a| < y/^4~1, we obtain
Re<[Df(z)]-1(f(z)-f(0),z-0 > P-CII2 -\a\-\z2 + C2\^^-\a\-\z2\-\\z-C\\2
—2|a|2 ■ \z2 \ ■ \z2 — C2I • \z2 + C2I >||z-C||2[l-2|a|-4|a|2] > 0.
We note that if /1,..., fn are uniformly convex functions on the unit disk, then / is not necessarily uniformly convex on B, where f(z) = (fi(zi),fn(zn)y since this mapping is not always convex (see, for example [1] and [19]).
We shall now give a distortion result for uniformly starlike mappings on B. To this end, we will prove the following lemma.
Lemma. Let P : B x B —>• Cn be a holomorphic mapping, such that P(z, C) = z- C + \d2P( 0,0 )(z2, C2) + • • •,
for (z, C) € B x B.
If
Re (P(z,C),z- 0 > 0, z, C e B, Zj£(,
then
<2, (2.9)
±(D2P(0,0)(z2,(2),z-()
for all \\z — Cll = 1.
Proof. Let z, ( E B, \\z — £|| = 1 and let p : U C, given by
f \(P(zt,Ct),z-Q, teU\ {0}
Pit) = {
[1, t = 0.
Then p G H(U) and Re p(t) >0, t £ U, then it is well known that
|p'(0)|<2. (2.10)
in\ (tDP(zt, Ct)(z, C) - P(zt, Ct),z- C)
p w =-------------------$------------------’
for t e U \ {0}, then
p'(°) = limp'W = \ (D2P(0,0)(z2,(2),z ~ C) •
Taking into account the relation as desired. □
Considering this result we can formulate the following distortion theorem.
Theorem 3. If f e US*(B) then
- (D2f(0)(z, z) + D2f( 0)(C, C) - 2D2f(0)(z, (),z-()
for all Zj C G B, \\z — C|| = 1- In consequence, 1
(D2f(0)(z,z),z)
<2, N| = l.
< 2, (2.11)
(2.12)
Proof. First we will show the inequality (2.11). To this aim, let P(z,() = [Df(z)]-1(f(z)-f(0) for all z,(eB, \\z - Cll = 1. Since / £ US*(B) then P £ H(B x B),
1
P(z, C) = z-C+ -D P(0,0 )(z2, C2) + • • •,
and
Re {P(z, Q,z - C) > 0.
Hence, considering the result of Lemma 2.5 we conclude that
- (D2P(0, 0)(z2, (2),z - ()
< 2.
(2.13)
On the other hand, since f(z) — /(C) = Df(z)P(zX)? then using the Taylor expansion of / and P and equating the coefficients of the second order in the last equality, we easily deduce that
~\ (D2f(0)(z, z) - ±D2f(0)(C, C) + D2f(0)(z, 0) = \d2P{0,0)(z2, (2),
for zX £ B.
Thus, considering the relation (2.13) and the above equality, we get inequality (2.11), as desired.
In order to obtain relation (2.12) it suffices to let C = 0 into (2.11).
Note that, some similar distortion results for starlike mappings and other biholomorphic mappings have been recently obtained by the second author (cf. [9, 10]).
We shall finish this paper with the following remarks and questions. Let US*(B) be the class of those uniformly starlike mappings on B of the form
f(z) = (fl(zi), ...fn(Zn))', z = (z1,...,zny e B,
where /1,..., fn are uniformly starlike functions on the unit disk. Clearly, in the case of one variable US*(B) = UST, but in Cn, n > 1, US*(B) is a
proper subclass of US*(B), since f(z) = ^zi + ^2) is in US*(B),
but not in US*(B)
In this case, we have the following result.
Theorem 4. If f e US*(B), then
<1, *g<C", ||*|| = 1.
\D2fmz,z)
Proof. This inequality is a simple consequence of the relation (4.2) [2].
Indeed, since fj is uniformly starlike in the unit disk, for all j E {1,..., n}, then
<1, j e {1
2/, (0)
Thus,
;D2fmz,z)
\
n 1
3=1
j=i
for all 2; G Cn, ||z|| = 1. This completes the proof. □ Conjecture. If / g US*(B), then
1
-D2f(0)(z,z)
Open problems. Find the best estimates for coefficients of second order for mappings from UK(B) (This problem has been solved by the second author in the case of convex and normalized mappings on B (see [9])).
Find the growth and covering results for uniformly starlike and uniformly convex mappings on the unit ball of Cn.
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Department of Mathematics,
Rzeszow University of Technology,
ul. W. Pola 2, 35-959 Rzeszow, Poland
Faculty of Mathematics,
Babes-Bolyai University,
1. M. Kogalniceanu, 3400 Cluj-Napoca, Romania
