CONVEX FAMILIES OF STARLIKE FUNCTIONS. H. Silverman 1 and E. Silvia Introduction. Let S denote the class of functions of the form f(z) = z +

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1 HOUSTON JOURNAL OF MATHEMATICS, Volume 4, No. 2, CONVEX FAMILIES OF STARLIKE FUNCTIONS H. Silverma 1 ad E. Silvia 2 1. Itroductio. Let S deote the class of fuctios of the form f(z) = z + 2 =2aZ that are aalytic ad uivalet i the uit disc U, with S* ad K desigatig the subclasses of S that are starlike ad covex. A fuctio f(z) is said to be starlike of order oz (0 < oz < 1 ), deoted by S*(o0, if R zf'(z) e f - > o (zg U), ad is said to be close-to-covex of order a (0 < a < 1), deoted by C[a], if there exists a fuctio O(z) C S*(O) = S* such that R >a (zc U). It is well kow [2] that C[0] C S ad that S* C C[0]. Sice fg K if ad oly if0 = zf' G S*, it follows that C[ 1 ] = K. I short, we have ad K=C[1] CS*CC[0] CS K=C[1] CC[a] CC[0] CS (0 <a<l), with the cotaimet beig proper i each case. There is o ice relatioship betwee S* ad C[a], 0 % a % 1. I this paper we lk at subclasses of C[a] that are starlike, ad fid coefficiet bouds, distortio theorems, ad extreme poits. Let T be the fuctios i S of the form f(z) = z - E=21alz, ad set T*(a) = S*(a) fh T ad L = K T. It is kow [4] that T = T*(0) = T* ad that a ecessary ad sufficiet coditio for f i T to be i T*(a) is that -a < 1 (1) 2= 2 l _ _a lal. This work was completed while both authors were visitig at the Uiversity of Ketucky. 1The research of the first author was partially supported by a College of Charlesto Summer Research Grat. 2The secod author was o sabbatical leave from the Uiversity of Califoria, Davis. 263

2 264 H. SILVERMAN ad E. SILVIA Most of the classes we cosider will be cotaied i T. 2. The Class B a. Deote by B[a] fuctios of the form which ther exists a fuctio 0(z) = z + Z=2b z G S* such that (2) Z=2(Ia - bl + (1 - a)lbl ) < 1 - a (0 < a < 1). THEOREM 1. B[a] C C[a]. f(z) = z + Z=2a z for PROOF. Note firsthat if either a = 1 or Z=21bl = 1, the b = a for every. Thus 0(z) = zf'(z), so that f G K = C[1]. We may therefore assume that Z=21bl < 1. The Sice the iequality zf' 0 ' 11 = }Z =2(a - b)zl < Z=21a. - bl ' z + Z=2bz Z= 21a - b I < 1 - a 1 - =21bl is equivale to (2), the prf is complete. a<l. iequality 1 - Z=2[bl COROLLARY 1. BIff] C B[a] for 0 < a <fi <1, but BIll = K PROOF. If / % 1, the Z=2lb[ < 1. Hece the first part follows from the Z=21a - bl <(1-/ )(1- Z=2lb[) <(1- a)(1 - Z=21bl). The fuctio f(z) = Z=lz G BIll with 0(z) = Z=lz. But fora%l, fg would imply the existece of a fuctio ad 13 - b3l < 1, which is clearly impossible. COROLLARY 2. B[a] C S'for 0 < a < 1. 0(z) = z + Z=2b z satisfyig both lb31 < 1 PROOF. I view of Corollary 1, it suffices to cosider oly the case a = 0. Sice Z=2la I < 1 is a sufficiet coditio for the starlikeess of f (see[ 1 ]). The result follows from the iequality 2;=2la[ < Z= 2 (}a - bl + Ibi) < 1. I the sequel, families deoted by subscripts will cosist of fuctios with power series represetatios of the form z - Z= 2[A Iz. Afuctio f(z)=z- Z= 2la [z is said to be i Ca, 0 <a <l, ifthere existsa

3 CONVEX FAMILIES OF STARLIKE FUNCTIONS 265 fuctio 0(z) = z - Z=2lblz i T such that If f G C a so that R z >a (z U). R_zf'(z) 1 - Z '=2lalz- 1 c 7.,) = Re( 1-Z '=2lblz-1 } we may chse values of z o the real axis ad let z--> 1- to obtai the coefficiet iequality Z=2(lal - albl) < 1 - a. This ad aother iequality give us the subclass that is our mai cocer. A fuctio fuctio ad 0(z) = z - Z= f(z) = z - Z=21alz 2 lb [z i T such that (i) Z=2(lal-albl) < 1 - a (ii) lal - lbl > 0 for every. is said to be i Ba, 0 < a < 1, if ther exists a REMARK. I [4] it is show that Z '=2lbl < g2, so that (3) Z=2la[ < 1 -a/2 is a ecessary coditio for f to be i B a. If coditio (ii) were dropped from the defiitio of B a, the we could always take 0(z) = z- z2/2, ad (3) would also be sufficiet. As we shall see, the apparetly artificial coditio (ii) will lead to a more iterestig subclass. THEOREM 2. B a C C a (3 B[a]. PROOF. Takig (ii) ito accout, z[½-[ [ZOO 2( la I _ ibl)z_ 1-11 = 1 _ Z=2lblz-1 I < Z=2(la[- 1 - Zøø=21bl I(bl) which is bouded above by 1 - a if (i) holds. Hece B a C C a. The cotaimet of B a i B[al follows from (ii) ad the idetity lal-albl = Ilal-lbl I+ (1 REMARKS. 1. T*(a)C Ba, which ca be see by settig O(z)= f(z) ad applyig ( 1 ).

4 266 H. SILVERMAN ad E. SILVIA 2. B C Ba for0 <a< < Sice B 0-- T* ad B 1 = L, the families B a represet a cotiuous passage from T* to L. Compare this to the families of a-covex fuctios, 0 < a < 1, defied i [3], which represet a cotiuous passage from S* to K. We ext determie some extremal properties of B a- 3. Coefficiet bouds ad distortio properties. < (1- )* THEOREM 3. If ffz) = z - Z=21alz GB a the la[ 2, with equality oly for fuctios of the form f(z) = (1-a)-t-az PROOF. From (i) we have that z- 2. lal < 1 - a+ albl. Sice [b[ < 1/ for 0 G T, the coefficiet iequality follows. The fuctio f(z) = z - (1-a)+az is i B a with respect to O(Z) = z -l- G T. 2 THEOREM 4. If f B a, the with equality i all cases for r-(2- r2 < If(z)[ < r + (%r 2 ([zi < r), 1 -(%r < If'(z)l < 1 + (%r, f(z) = z -( -- - )z 2 z = +r. (Izl < r), PROOF. For f(z) = z - E=21alz, we have from (3) that 2Z=2la ] < Z=2la ] < 2-a 2' Hece Similarly, Iffz)l r + Z=21alr < r + r 2 l;=21al < r + ( ) r 2. If(z)l > r- r2z =21a I > r-(-2- -)r2. The iequalities for f' follow from (3) ad 1 - rz=2la I < If'(z)l < 1 + rz=2la I. COROLLARY. The disk U is mapped oto a domai that cotais the disk tw[ <-2 for ay f G Ba. PROOF. Let r-* 1 i the first iequality.

5 CONVEX FAMILIES OF STARLIKE FUNCTIONS 267 Usig the covexity of the family T, we ow prove THEOREM 5. The family B a is covex. PROOF. Suppose fl(z) = z- E '=2[a[z ad o0 f2(z) = z- E=2[ClZ are i Ba o0 o0 with respect to Ol(Z) = z - E=2[blz ad 02(z) = z - E=2ldlZ i T. For 0 < X < 1, we will show that Xf 1 (z) + (1- X)f2(z) = z- E =27(X)z C Ba with respect to X01(z) + (1- X)02(z) c = z - E=26(X)z C T. Sice Eo0=2(- (X ) - ab(x)) = XZ =2(lal - albl) +(1 - X)Z =2(lCl-aldl) < 1 -a, the result follows. Before examiig additioal properties of Ba, we shall lk at a proper subclass that cotais the extremal fuctios of Theorem 3. a 1, 4. The class Da. The fuctio ffz)= z- ;=21alz is said to be i Da, 0 < if 1' Z '= 2 -( l_ 2 }a½ - <. Observe that D O = B 0 = T*, D 1 = B1 = L, ad that Da is a covex family for all o0 THEOREM 6. The extreme poits of Da are give by fl(z)=zadf(z)=z-(l'a)+az (=2,3... ). 2 PROOF. We must show that f Da if ad oly if it ca be expressed i the form f(z) = Z=lXf(Z), where X > 0 ad Z '=lx = 1. Sice D is a covex family, f(z) = Z=jXf(Z)is i Da. Coversely, if f(z)= z- Z=2lalz is i Da, we set X= - (l )+ala[ ( = 2,3,...), ad X 1 = 1- Z=2X. The f(z)= Z =lxf(z), ad the prf is complete. COROLLARY. Da C Ba. PROOF. This follows from Theorems 6 ad 5. We-ext determie sharp bouds for the order of starlikeess of fuctios i Da. 2-2 THEOREM 7. Da C T*( -, ), with extremal fuctio f(z!2 z-( )z. z a o0 a, ß I l < ' ' S PROOFß Accordig to (1), we must prove that Z=2(l )+a- 1 mphe that a =21_2a/(2+a ) I < 1. It suffices to show that I; ø0.-2a/(2+a)

6 268 H. SILVERMAN ad E. SILVIA (4) i-a)+oe 2-2oe/(2+oe) 1-2oe/(2+oe) = (2+oe)-2oe 2-a ' ( = 2,3... ). But (4) is equivalet to > 0 ( = 2,3... ), ad the theorem is proved. We ow show that for 0 oe 1 the cotaimet of Doe i Boe is proper. THEOREM 8. Fuctios of the form are i Boe - Doe for 0 oe 1. f(z)=z z 2,1-oe, - - t - -fi- z ( = 3,4,...) PROOF. The fuctios f(z) are easily see to satisfy the coditios for cotaimet B w with 0(z) = z - z2/2. Sice the fuctios f(z) are ot i Doe. We coclude with two ope problems. 1 + (l<r) >1 for0<oe<l, 2-oe 2((1- )+oe) OPEN PROBLEM #1. Fid the extreme poits of Boe for 0 ( oe ( 1. It follows from Theorem 3 that each extreme poit of Doe is a extreme poit of Boe. However, by Theorem 8 we kow that there must be additioal extreme poits. OPEN PROBLEM #2. Fid the largest/3 for which Boe C T*(/3). As a cosequece of (3), Boe C T*(oe/2). But this caot be sharp because B 1 = L C T*(2/3). See [4]. O the other had, the fuctios f(z) i Theorem 8 have order 2oe ad of starlikeess 3-2+2oe 2oe. Lettig -, we see that a upper boud for/ is, cojecture that this is best possible. REFERENCES 1. A. W. Gdma, Uivalet fuctios ad oaalytic curves, Proc. Amer. Math. Soc., 8(1957), W. Kapla, Close-to-covex schlicht fuctios, Michiga Math. J., 1(1952), P. T. Mocau, Ue propridtd de covexitd ggdralisge das la reprdsetatio coforme, Mathema fica (Cluj), 11 (1969), H. Silverma, Uivalet fuctios w th egative coefficiets, Proc. Amer. Math. Soc., 51(1975), College of Charlesto Charlesto, South Carolia Uiversity of Califoria, Davis Davis, Califoria Received October 30, 1977