On a Certain Extension of Epstein’s Univalence Criterion

ANNALES UNIVERSITATIS MARIAE CURIE-SKLODOWSKA

LUBLIN-POLONIA

VOL. XIII, 19__________________________ SECTIO A____________________________________ 1988

Instytut Matematyki Uniwersytet Marii Curie-Skłodowskiej

A. WESOŁOWSKI

On a Certain Extension of Epstein’s Univalence Criterion O pewnym uogólnieniu kryterim jednolistnośd Epsteina

Abstract. In this paper a sufficient univalence condition for meromorphic and locally univa­

lent functions in the unit disk is given (Theorem 2).

This condition is an essential generalisation of the Epstein’s univalenos criterion (7). As par­

ticular cases the well-known uni valence criteria of A hl for s [l] and Nehari [8] are obtained.

Moreover, a sufficient univalence criterion for meromorphic and locally univalent functions in the upper half plane is given (Theorem 3).

1. CL Pommerenke has recently given a simplified proof of a univalence criterion obtained earlier by Epstein in another way. In his proof an additional assumption made by Epstein was dropped (see e.g. [7, p.143]).

Let D = {» : |x| < 1} and let Sj denote the Schwaizian derivative

Theorem 1. (CL. Epstein, see e.g. (7]) Let f be meromorphic and g analytic in D. If both function» arc locally univalent in D and if

(1.1) ||(1 - |s|’)’(5/(s) - S,(*)) + (1 - U|’)I < 1

for * eD then f i» univalent in D.

If <(i) s s then (1.1) gives

(L2) |(l-l«l’)’Sz(x)|^2

and this is the well-known univalenoe criterion of Nehari [6].

If g = f then (1.1) implies

172

and this is also a well-known univalence criterion (see e.g. [9, p.172]).

2. The following theorem is a generalization of the univalence criterion given by Epatein.

Theorem 2. Let f be meromorphic and g analytic in D. If both functions are locally univalent in D and if there exists a function h, analytic in D, satisfying Re h(z) > | and suck that

(2-1)

zh'(t) */'(»)

AU) i'U)

)-

-i(l-W ’ ) ’ |A(x)(S z W-Ss(z))|< l, z e D then f is univalent in D.

Proof. Let /(*) = <

0 + <h* + • •• » uU) = A o + Aix+ ••• , «i / 0, At 0. Then it is possible to consider the functions :

v _ ______ Z — <»o_______ .. « 9 ~ Ao

* ” + -£)(/- «o) ’ 9 ' “ A, instead of f and g.

These normalizations don’ t impair the generality of our considerations and we may also assume that

f(z) = g(z)+O(z 9 ) and — fi = l + O(xJ ) as z — » 0 . v

Let us introduce now

(2-2) (2-3)

®U) = = l + 3*’ + O(x’),

«(«) = ZU) • »(«) = » + + O(x3).

Both functions are analytic in D because f cannot have multiple poles and be ­ cause /' and g' do not vanish in D.

For I € I =< 0, oo ) we consider [3, p.38]

(2.4) ^t , _ ”U<~ ‘) + (<* ~ g-> )xA(xe- < )«, (xe-<) M ’ 7 “ v(xe-<) + («< -

The function f(z,t) is for each fixed t 6 I meromorphic in D. FYonr(2.2) and from the assumption on the function h given in Theorem 1 it follows that the denominator in (2.4) has the form 1 + O(z*) as z -* 0, uniformly with respect to t.

It is easy to show, that = z + •■• , t € I, is a normal family in D, where

°iv)

(2.5) «I (t) = «-< + (e ‘— e _,)A(0) and |«i (t)j —» oo as t — • oo .

Ona Certain BxteruionofEpxitan’« UnivalenceCriterion

173 FYom (2.3) and (2.4) we have

(2.0) /(*,<) = ai(t)z + O(z l) as z -» oo , where ai(f) is defined by (2.5).

Let us denote /'(»,<) = , /(z,f) = ~ ■ • After some calculations

OZ Ot

we obtain from (2.4)

(2-7)

- */'(*,f) h - 1 -at .„ re- 1 A' /(z,t) + z/'(z,t) h ' > k

(1 - <-»)*<-<(«% - vv") + (1 - e-^pz^fttV - t'v") n'v —

where t»,e, A,A',«",»" are evaluated at the point ze~*

FYom (2.3) and the assumption (2.2) we have

«'o - uv' = f1 o2

*"v - ««" = /%* + 2f'v'v =

«%' - n'v" = f'v'v - f'v"o + 2/'(n') ’ = \f't>2(Sj - S,) .

Thldng this and (2.7) into account we have for z € D

(2-8) W- kite-') ‘ ( ’ V A(ze-‘ ) + ,'ze-n - |(1 - e~2t) 2 z2h(ze~ t }(S/(ze~ t) — St (ze~t}) .

, *(«)“ ! The right-hand side is equal —r-r-r—

"1*1 if t > 0. Then putting ze~‘ = f , « “ * (2.8) and the assumption (2.1) that

for t = 0 and is analytic in T5 = {z : |z| < 1}

= |f | and replacing f through z we have from

1)1-

80/(*>*) = */*(*» *)•?(*> 0 » R«p(*»0 > 0 f°r * € P, tel.

Since tel , is a normal family it follows from (2.5) that /(z,t, is a

«»(<)

LSwner chain and /(z, f) is univalent in D [8, Corollary 3J.

<(x) • In particular we conclude from (2.3) and (2.4) that f(z) = = p^z) ’*

univalent in D and this ends the proof.

The proof given here is analogous to the proof given bv Pommerenke in [7].

3.

Corollary 1. I]h(z) si, z € D , then (2A) ¡ivez (1.1).

174

Corollary 2. 1J we put h(z) = --- r-r, where |w(r)| < 1 , w(x) 1 for z € D, 1 - w(x)

then we obtain from (2.1)

(3.1)

- J<* - (s'w * s,w) l s 1

The univalence criterion obtained by Z.Lewandowski and J.Stankiewicz [4] shows to be a particular case of (3.1) as ÿ(x).= x.

Corollary 3. On putting w(x) = e = const. , |c| < 1, e # 1 and g = f, or g[z)

z, respectively, in (2.1) we obtain

|‘ W ’ -(i-M’)i^ÿ|<i, *en

or

|(1 - |x| ’)’S,(x) ~ *0 - <)*| < 2|1 - e| , zeD

These are the well-known univalence criteria given by Ahlfors [1], which for e = 0 give (1.3) and (1.2), respectively.

Remark. The univalenoe criterion given here is a generalization of the criterion obtained by Epstein as the following example shows.

Let /(*) = (1 + *)’ . 9(*) * » + I»3- For z e D we have

I id - |x|’ ) ’(s z - S,) + (1 - |x| ’)T £| = (1 - w’)|r ^| < W +’W 3,

thus the inequality (1.1) isn’ t satisfied in D.

The same pair of functions /and g and A(x) = | gives

- (1 - |*| ’)(~ + ^) - 1(1 - lap)’| h(Sf - S,)| = |x|’ < 1 .

Also the criteria obtained by Ahlfors [1], Lewandowski and St ankiewicz [4]

are not satisfied by the function f(z) = (1 + x)1.

The criteria ci univalenoe in the unit disk D can be transferred on the upper half

plane U = {z € G , Im z > 0}. The mapping w = I * + 2 » , t > 0 , z € D, together

with Theorem 2, gives

On aCertain Extensionof Epstein’sUnivalence Criterion 175

Theorem 3. Let f be meromorphie and g analytic in U. If both functions are locally univalent in U and if there exists a function h, Re h > j, analytic in U such that

(3.2) h(.

hjz) - 1 1 x - il I» z-it (h'(z) g"(z) 2 \ , z) Iz + it I *z—it\h{z) g'(z] z + it)

+ 2y2^ h(z)(Sf(z) - S,(x))j < 1 , zeU , f>0, y = Imx,

then f is univalent in U.

Putting in (3.2) y(x) = -— r- , I > 0 , z 6 U , h(z) = , Re p(z) > 0

2 i” 1» «

in U we obtain a theorem due to Lewandowski and Stankiewicz [5].

Putting in turn h(z) = where e is a constant satisfuing |e — 1| < 1, g(z) = , t > 0, z € U we obtain the inequality of A hl for s [1] for t -» oo from (3.2) :

|2y’S/(x) + e(l - e)| < |e| , z € U , Imx = y.

f

If in the last inequality the right hand side is replaced by &|e|, 0 < k < 1, then the inequality implies the possibility of a K-quasiconformal extension of f on G, K = (1 + k)/(l — k). This was proved in [2] and was a positive answer to the conjecture put forward by A hl fors [1]. A similiar question arises in the case of the inequality (3.1) for I — • oo. This problem is in the course of study.

REFERENCES

[1] Ah lfors , L. V. , Sufficient conditions forquasiconformal extension ,Princeton Annals o{

Math 79 (1974),23-29.

[2] Anderson , J. M , Hinkkanen , A. , A univalencecntenon , Michigan Math. J. 32 (1985),33-40.

[3] Becker , J. , LàwnerteheDifferentialgleichvng und quasikonform forisetxbareschltchle RiiJt- tionen, J. reine angew. Math. 255(1972), 23-43.

[4] Lewando wski , Z. , Stankiewicz ,J. , Some sufficient conditions for univalence, Fblia Sdentarum Universitatis Technicae Resoviensis14 (1984), 11-18.

[5] Lewandowski , Z. , Stankiewicz , J. , Sufficient conditions for univalence and gua- sionformalextension tn ahalfplane , FoliaSa. t’niv. Tech. Resoviensis 48 (1988),67-76.

[6] Nehari , Z. , TheSchwaman denvative and schluhi functions ,Bull. Amer. Math. Soc.

55 (1949), 545-551.

[7] Pommerenke , Ch. , On theEpstem univalence cntenon ,Résulta inMathematics vol 10 (1986), 143-146.

[8] Pommerenke , Ch. , (Ther die Subordination analytischer fonhtionsn , J. rane angew.

Math. 218 (1965),159-173.

[9] Pommerenke,Ch. , Univalent fonctions ,GCttingen1975

176

W pracy podano warunek dostateczny jednolislnoid funkcji meromorficznej i lokalnie jedno- hstnej w kole jednostkowym (tw.2). Warunek ten jest istotnym uogólnieniem warunku podanego przez Epsteina [7]. Przy odpowiednich założeniach otrzymuje sit znane kryteria jednolistnofó Ahlforsa [l] lub Nehariego [6].

W twierdzeniu 3 podano dostateczny warunek jednolistncdci funkcji meromorficznej i lokalnie iednolistnej w górnej półplaszczyżnie.

I

X'Jic s-wy aa*. et /y/so B4«.aS«»?»cj.

ANNALES UNI VERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN-POLONIA

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