On the curvature oî level-lines in certain classes oï starlike meromorphic functions

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A N N A L ES S O C IE T A T IS M A T H E M A T IC A E P O LO N A E Series I : C O M M E N TA TIO N E S M A T H E M A T IC A E X X I I I (1983) R O C Z N IK I P O L S E IE G O T O W A R Z Y S T W A M A TEM A TY C ZN EG O

Sé ria I : P E A C E M A T E M A T Y C Z N E X X I I I (1983)

E . M^{a zu k} (Kielce)

On the curvature oî level-lines in certain classes oï starlike meromorphic functions

1. Introduction. In this paper, we continue the study initiated in [2], [3], and [4] of the classes of starlike meromorphic functions.

Suppose, that

<1-1) P(£) = l + p yZ-\-$2Z* + ...

is holomorphic in the unit disc К and further, let ft e ( 0 ,1 ) and Ж > 1.

By 3?(ft) and 3P(M) we denote the classes of all functions P of the form (1.1) satisfying

(1.2)

P(*) + 1 < P ,

(1.3) \P{z)-M\ < M,

for z e ^J^l^,respectively. Let 2 * {ft) and E* (M) denote the classes of functions of the form

(1.4) F(z) = 1

0 + a0 + a xz + a 2z2 + ...

that are meromorphic in К and satisfy

(1.5) -z F '(z )

F{z) ' = P(z), z e K , where P belongs to 0>(ft) or 0>(M), respectively.

2. Let z = rei<p, 0 < r < l , 0 < 9 ? < 2 7 t and К г(ф) denote the curvature of the image of the circle \z\ — r under the mapping w — F (rel<p), where F is in 2 * (ft) or 2*(M ), respectively. I t is easy to notice that the functions F are univalent in the unit disc and thus

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60 R. M azur

If F belongs to 27*(/?) or P*{M ), then

(2.2)

and (2.3) where (2.4)

/ zF"(z)\ j

- r e ( 1 + ^ r ) = r e H

zP'(z)

~p W z e K ,

\z2F ’(z)\

r TeJJ(petf) — l , exp --- dg,

J о

U(z) = P(z) zP' (z) P{z)

and P is in ^(/?) or 0>{M), respectively. From (2.1)-(2.3) we have r re P ( Qei4>) — 1

(2.5) K r(qo) = r r e P ( 0)exp J dg,

) о "

where ü (geie) is defined by (2.4) for z = gei<p. We shall prove the following theorem.

Th e o r e m 1. I f F e P * ( f i) , then

r ( l — 2fir2 — r2) r ( l + 2 f r 2 — r2) (1 - r 1)1- '

and these inequalities are sharp.

P ro o f. From (2.2) we have

(2.6),

— re| l + zF"(z)

re {P(z) zP'(z)

P e & {P ).

F'(z) J ~~\ P(z) Г

By the well-known theorem of Zmorowicz [5] we obtain (see [2 ]) 1 — (2/?-f l ) r 2

1 — r2 — re I1 l еЯ" ( * ) \ < l + ( 2 ^ - l ) r 2 .

F'(z) 1 — r2

hence from (2.3) and (2.5) we have (2.6).

The functions

F ^ z) = — exp 0 (z ,p ),

z P , 1

2ехрФ(г, - £ ) , where

&

P) = / y 2(}e~i<p(r — e~i,p%)

( l - f ) r e - ivd- e ~ 2iç,f d%

at z = гег> indicate the sharpness of the result.

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Curvature of level-lines 61

B e m ar k . When = 1, we obtain the known result (see [1]):

r ( l - 3 r 2) < F r(<p) ^ r ( l -fr2) (1 - r 2)2' Theorem 2. I f F e Z * ( M ) , then

(2.7) (2 Vab + bc + l — 2b — e)A (r, 31) < K r(<p)

3Ir2 — r + M

< --- ---

(1 — 3 I)r2- r + 3 I B(r, M), where

A(r, if ) = r ( l — r2)

B(r, if ) =

I_____ 2a \2a l

\ aVl — r2+Va2 — v2r2 I \

vVl — r2+Va2 — v2r2 ra-f Va2 — v2r2

M — l YlW -M ) .[a _ 4 + - = L , ) ]

with

v = Va2 — 2b,

1 — r

a — ^{2 i f}

231 — 1 ’ 2 3 I - 1 1

a + v

31 ф 1, i f = 1 ,

)

^2t>

’

c = ¹ ^{_|_r 2} 1 - Г 2 Inequalities (2.7) are sharp.

Proof. For every function F of S * ( if ) we have (see [4])

2Vab + bc + l — 2b — c —re (14 zF"(z)\ Air2- r +31 F'(z) f ^ ~ ( l - 3 I ) r 2- r + 3 f ' Hence from (2.3) and (2.5) we obtain (2.7).

Equality is attained for z = retq> by the functions F * and F*, respect

ively, where

F*(s) =

(i+fcXi-ь)

K)

(1 — 3 -14~1/2)5/9 exp(141/23_1s)

F*(z)

— ez 1 z

M — \ \ (2 3 f-l)/ (M -l)

“ i f ez)

for M > 1, for i f = 1 ,

for i f > 1 , for i f = 1

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62 R. M azur

w ith

- Ъ а -V A

%л —■

1 - 6 7 A — 6 2a 2 + l — 6 2,

■baArV A 1 - 6 ^’

2 a - l

h Va

И = 1 . a(4a —1 ) ’

E e m a rk . In the limit case M = oo, we obtain the result of [1].

References

[1] G-. W . K o r i c k ii , On the curvature of level lines for schUcht conformal mappings, Dokl. AN SSSR 4 (1957), 653-654 (in Russian).

[2] R. M azu r, A. T re s k a , On some classes of f -starlike and of quasi-f-starlilce meromor- phic Tc-symetric functions, Demonstratio Math. 11.3 (1978), 735-750.

[3] K. S. P a d m a n a b h a n , On certain classes of starlike functions in the unit disc, Indian Math. Soc. 1-2 (1968), 89-103.

[4] P. W ia tro w s k i, On the radius of convexity of some family of functions regular in the ring 0 <■ \z\ < 1, Ann. Polon. Math. 25 (1971), 85-98.

[5] W. A. Z m o ro w icz , On a class of extremal problems connected with the regular functions which have positive real part in the circle \z\ < 1, Ukr. Mat. Zurn. 17, No 4,

12-20 (in Russian).

IN S T I T U T E O F M A T H E M A T IC S P E D A G O G IC A L U N I V E R S I T Y , K ielce