On a theorem of Kobori

NORTH- HOILAND

On a Theorem

of Kobori

Li 3ian Lin

Department of Applied Mathematics Northwestern Polytechnical University Xian, Shaan Xi 710072 People's Republic of China and H. M. S r i v a s t a v a

Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8 W 3P4 Canada

ABSTRACT Let ~9~ and ~ denote the well-known classes of normalized analytic functions which are, respectively, univalent and starlike in ] z[ < 1. A theorem of Kobori states that, for a function ov

g ( z ) = z + ~ ak z k ~ 5 ~*, k=2

the sequence of the nth sections Sn(z)=z+

~. 1

k~2 -~ akz

k

1 must be starlike in ]zl < ~. Recently, Silverman [1], Gruenberg et al. [2], and R~nning [3] extended the above result. In this note, we first improve the results of Ilieff [4], Ruscheweyh [5], and Silverman [1]. We then give a remarkable simple proof of a result of Gruenberg et al. [2] and R0nning {3] that $4(z) is starlike in i zi < ~1 for g ~ ~ . © Elsevier Science Inc., 1997

APPLIED MA THEMA TICS AND COMPUTATION 85:287-296 (1997) © Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010

0096-3003/97/$17.00 PII S0096-3003(96)00143-9

288

L . J . LIN AND H. M. SRIVASTAVA

1. INTRODUCTION Let 5~' denote the set of analytic functions f in the open unit disk D normalized by f(0) = 0

and

f ' ( 0 ) = 1.

Also let ~9~ be the set of univalent functions in ~4 (cf., e.g., [6, 7]). Denote by Dr the set {z: z • C and ]z] < r}. A function f • ~ ¢ is said to be convex (starlike) in Dr if it is univalent in Dr with f ( D r) convex (starlike with respect to the origin). By 5Y and ~ * we denote the subclasses of functions in .9~ which are, respectively, convex and starlike in D. For ¢¢

f( z) ~- E ak zk • •

( a 1:= 1),

k=l

let

S~(z,f) =

~ akzk

(ne~;al:=l),

k=l

be the nth section of f(z). In 1934, Kobori [8] proved that, for f • ~ , all sections Sn(z, f ) are starlike in DW2. Since then, many papers (see, e.g., [1-5, 9, 10]) have appeared concerning the section S~(z, f). Ilieff [4], Ruscheweyh [5], Bernardi [9], and Silverman [1] determined the disks Dr, in which Sn(z, f ) is univalent, close-to-convex, and starlike, respectively. On the other hand, the well-known relationship

g • 5 f* ¢~ h(z) = fo z g(tt) dt • .Yf furnishes us with an alternative statement of Kobori's theorem: If g • . ~ * ,

then Sn( z, h) is starlike in D1/2 for all n. Recently, Gruenberg et aL [2] and RCnning [3] extended Kobori's theorem by showing that, if g • .9~, then ~{S'n( z, h)} > 0 in D1/2 for all n, and Sn(z, h) is starlike in D1/2 for positive integers n < 4 and n >t 6. In this note, we first improve the aforementioned result of Ilieff [4], Ruscheweyh [5], and Silverman [1]. We then give a simple proof of the assertion that $4( z, h) is starlike in D1/2.

289

On a Theorem of Kobori

2.

RADIUS OF STARLIKENESS DEPENDING ON n

Let f ~ . Ilieff [4] gave bounds depending on n for the radius of univalence of S,( z, f). Let r~ denote the positive root of the equation: 1-(l+n)rn-nr~+l=O

(n~

N).

(1)

Ruscheweyh [5] improved the result of Ilieff [4] to the following form.

THEOREM 1. If f ~ ~ , then Sn( z, f ) is univalent in Drn, and maps this circle onto a close-to-convex domaiu Furthermor6 for an even positive integer n, r~ cannot be replaced by any larger number with respect to ~ . Subsequently, Bernardi [9] proved the following theorem.

THEOREM 2. If f ~ , then S~( z, f ) is starlike in D r . This result is sharp for each even positive integer n for the function f ( z ) ~ z/(1 - z). Recently, Silverman [1] proved that, if f ~ ~ , then S~( z, f ) is starlike in I zl < {1/(2n)} 1/~. In view of Theorem 2, the above-mentioned results of Kobori [8], Ruscheweyh [5], and Silverman [1] can be extended to the following form.

THEOREM 3. If f ~ ~ , then S,( z, f ) is starlike in Dr, where r~ denotes the positive root of (1) constrained further by 1 -~ ~ ( 2 n ) - 1 I n ~ r n ~ n - l / ( n - 1 )

(2)

and 1

rn = 1 - --log(2n) + ~ n

log( n) log(4 en) + o

.

(3)

Furthermor6 for an even positive integer n, % cannot be replaced by any larger number.

In fact, it follows from Theorem 2 that S~( z, f ) is starlike in D~. For the number r n defined above, Ruscheweyh [5, Theorem 2] showed that (2n) -1/~ <. r n ~ n -1/(~-1) and (3) hold true. Since the sequence (2n) -1/n increases with n, the assertion of Theorem 3 follows easily.

290 3.

L . J . LIN AND H. M. SRIVASTAVA T H E STARLIKENESS OF S,(z, h) FOR g ~

Let ov

n=2

and

h(z) =

z g(tt) dr.

It was shown by Li [10] that $2( z, h) and $3( z, h) are starlike in D,/2. More recently, Gruenberg et al. [2] proved that ~{S',(z, h)} > 0 in D1/2, and hence, that S,( z, h) is univalent in D1/2 for all n. R0nning [3] has further proved the following result.

THEOREM 4. Let g ~ ~ . Then Sn( z, h) is starlike in D1/2 for positive integers n <~ 4 and n >~ 6. It should be pointed out that Theorem 4 provides a verification of a certain multiplier conjecture for univalent functions in a special case (cf. [2, 11]). R0nning [3] obtained this result for a little larger set. However, his proof concerning the special case n = 4 is difficult and rather lengthy. Here we shall give a shorter (and simpler) proof of Theorem 4 in the special case n=4.

Proof of Theorem 4 in the Special Case n -- 4. Let ch = x +

iy,

a3 = u + iv,

and

a4= s + it.

(4)

We need to prove that

=

>0

( z ~ Dll2).

(5)

On a Theorem of Kobori

291

Since the classes ~ and .9 ~* are preserved under rotations, it is sufficient to prove (5) for the point z = r = 1 / 2 . Now = G(x,y,u,v,s,t) -- 768 + 576x + 9 6 ( x 2 + y2) + 256u +80(xu+

yv) + 16(u 2 + v 2)

+ 3( s ~ + t ~) + 120 s + 36( ~s + yt) + 14(us + vt). It is well known t h a t 1

g ~

=z+

z-~._ i

=1

~:~

with

bl = - ~ ,

1

3

2

b~ = ~a~ -

3 5 1 b5 = ~ a 2 oa - ]-~ a~ - ~ a a.

1

~, (cf. [6, p. 132]).

(6)

Applying the area theorem to { g(1/z2)}-1/2, we obtain 15112 + 315312 + 51 b512 ~< 1.

(7)

Substituting from (4) and (6) into (7), we conclude that G I ( x , y, u, v, s, t)

= 64( x 2 + y2) + 108( x 2 + y~): + 125(x 2 + y~)3 + 192(~2 + v:) - [ 2 8 8 + 6oo(x2 + y~)l(~,x~ - ,,y2 + 2~yv) + 320( s ~ + t ~) + 720( ~ + y ~ ) ( u ~ + ,2) + 4 0 0 4 :~3 - 3xy :) - 960~( ~ -

y~)

+400t(3x2y_

(8)

y3) _ 9 6 0 t ( x v + yu) - 256 ~< 0.

292

L . J . LIN AND H. M. SRIVASTAVA

In order to prove (5), it is sufficient to prove t h a t G + G 1 >t 0. We write G+

GI = { 5 1 2 + 5 7 6 x +

160(x 2 + y2) + 108(x 2 + y2) 2

+ 1 2 5 ( x 2 + y2) 3 + 256u + 208(u 2 + v 2)

+ 7 2 o ( x 2 + y2)(u ~ + v ~) + so( xu + yv)

- [ 2 s s + ~oo(x~ + y~)](u~ - uy~ + 2 ~ v ) } + 323(~ + t~) + 2s{200(x 3 - 3 x y ~) - 4 8 0 ( x u + 2t{200(3x2y_

yv) + 18x + 7u + 60}

ya) _ 4 8 0 ( x v + yu) + 18y + 7v}

= G:( x, y, ~, ~) + 323( ~ + t ~) + 2 ~C~( ~, y, u, ~) + 2 tG4( x, y, u, v).

Since

323(~: + t :) + 2~G~(~, y, ~, v) + 2tC~(~, y, u, v) 1

>~

_

- -2ffi ( a~ + a: ) 1 {40000(x 2 + y2) a 323

192000(x 2 + y 2 ) ( u x 2 -

uy 2 + 2 x y v )

+ 7200( ,~ - y') + 2ao4oo(x ~ + y~)(u s + ,~) + a 2 4 ( ~ + y~) + 4 9 ( ~ ~ + v ~) + ~ s o o [ ~ ( x 3

a~y2) +

v(ax2y_ y3)]

- 17280u( x 2 + y2) _ 6720x( u 2 + v 2) + 24000( x a - 3 xy 2) - 57600( x u -

yv)

+ 2 5 2 ( xu + yv) + 2160x + 840u + 3600} = Gs( x, y, u, v),

(9)

293

On a Theorem of Kobori

we have G + G~ ~ G 2 + G5 183888

_ { -161776323 - ÷

323

24000 323 ( xa - 3xy2)

51356 x ÷ - 323 (x2 + y2)

27684

42084 x 4 ÷ 216x2y 2 . ÷ 323 323

÷ - -

375

y4 ~_ ~_~_~( X 2 ÷ y 2 ) 3

}

67135 6720 2160 )} +(u2 + v2) 32--'-~ + 32--'--3-x+ 32-----~(x 2 + y2 900

4

1400

- 2~ ~-yS(~ - y4) ÷ -5Y2( x3 - 3~Y~) ÷

37872 x 2

55152 y2

41594

40924 }

- -

- -

- - X

- -

323

323

v/ 1800

323

323

1400

- 2 t "-3ff2 xy('x2 ÷ y2 ) + ~

y2 (3 x2y -

)

16006 ÷288

+

= a s ( x, y) + ( u ~ + v ~) aT( ~, y) - 2 ~ a s ( ~, y) - 2 r a g ( x, y)

1

where we have used the fact that G 7 > 0. But a~a~ - ( a~) ~ - ( a~) ~

= /45955

t%~-(

318960 ~

x2 + y2) +

1330344 ÷__x2y2 323 19822400 ÷

323

x2

2

1746292x 4

~

x/( ~ + y~)2 +

21892 __y4 17

7418432 ÷ _ _ x 3 323

3012544 __y2 323

+

31046656 323

32---57224768 __xy2 323

1496832 x+ ~ 17

(11)

294

L. J. LIN AND H. M. SRIVASTAVA

In view of the inequality: 45955 318960 323 (12 + y2) + 3 - ~ x / ( 12 + y2)2

454100 323 ( x ' + 212y 2 + y'),

(12)

(11) reduces to the following form:

G6G7 - ( a s ) ~ - ( a g ) ~

45792 y4 + 17

3012544 323

7224768 x + 24832 X2 ) y2 323 19

1496832 31046656 19822400 - + x+ x2 17 323 323 7418432 1292192 A- - x 3+ - - x 4 323 323

= Glo(X, y~).

(13)

Setting y2 = w and writing

G10( x, w)

45792 - 17 w 2 + Gll ( x ) w ~ - G12 ( x ) ,

we find that, for each x ~ [ - 2, 2], the graph of G10( x, W) as a function of w is a parabola opening downward. Thus, the inequality G10 > 0 needs to be verified only at the endpoints of the interval 0 ~< w < 4 - x 2, that is,

Glo(X,W) >i min{G,o(Z, 0), G,o(X, 4 - x~)}.

(14)

295

On a Theorem of Kobori

But

a,o(

o) =

1292192 ( x2

x +

115913]2 288 1286465209 T

323 +

40381 ]

31046656

x~

323

40381

1492192 X - l - - -

323

17

(15)

6736 and G,o ( x, 4 - x 2) = x2 ( 14643200 x + 323

31483904 ] + 2147584 - - X 323 ] 323

2468864 + - 323 ~

2197504 2147584 ~ x 2+ ~ x 323 323 2197504

-

- - x - I 323

8389 ]2 17168 ]

2468864 ~ 323

+

2468864 +

-

-

323

2 04 / 8389)2 323 6019.

[ 17168 (16)

Hence, (14) shows t h a t G10 > 0 and the desired result follows at once from (10) and (13). The proof of Theorem 4 in the special case n = 4 is thus completed. In conclusion, we remark that the elementary area theorem provides sufficient information to Prove Theorem 4 in the case n = 4. Unfortunately, along similar lines, we are unable to prove Theorem 4 in the case n -- 5. We do indeed conjecture that Theorem 4 is also true in the case n = 5. [See also a recent work of Li and Srivastava [12] for various starlikeness criteria associated with a family of integral operators which include, as a very special case, the one used here in the definition of h( z).]

296

L. J. LIN AND H. M. SRIVASTAVA

The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant 0GP0007353. REFERENCES 1 H. Silverman, Radii problems for sections of convex functions, Proc. Amer. Math. Soc. 104:1191-1196 (1988). 2 V. Gruenberg, F. ROnning, and S. Ruscheweyh, On a multiplier conjecture for univalent functions, Trans. Amer. Matb~ Soc. 322:377-393 (1990). 3 F. Rcnning, Integrated partial sums of convolutions of univalent functions, J. Math. Anal Appt 175:186-198(1993). 4 L. Ilieff, On the partial sums of univalent functions which are convex in I zl < 1, Ann. Uniu Sofia Fac. Sci~ Livre I46:153-157 (1950). 5 S. Ruscheweyh, On the radius of univalence of the partial sums of convex functions, Bull London Mat]~ Soc. 4:367-369 (1972). 6 P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften 259, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo, 1983. 7 H. M. Srivastava and S. Owa (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London, and Hong Kong, 1992. 8 A. Kobori, Zwei S~itze iiber die Abschnitte schlichter Potenzreihen, Mew. College Kyoto 17:172-186 (1934). 9 S.D. Bernardi, New distortion theorems for functions of positive real part and applications to the partial sums of univalent functions, Proc. Amer. Matl~ Soc. 45:113-118 (1974). 10 Li Jian Lin, On the radius of starlikeness for some classes of analytic functions, J. Jiangxi Normal Univ. Natur. Sci. 4:17-21 (1986). 11 R. Fournier and S. Ruscheweyh, Remarks on a multiplier conjecture for univalent functions, Proc. Amer. Matl~ Soc. 116:35-43 (1992). 12 Li Jian Lin and H. M. Srivastava, Starlikeness of certain integral operators, C.R. Math. Rep. Acad Sci. Canada18:93-98 (1996).