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Internat. J. Math. & Math. Scl. VOL. 14 NO. 2 (1991) 283-288 SOME 283 BAZILEVI FUNCTIONS OF ORDER BETA S.A. HALIM Department of Mathematics and Computer Sciences University of Wales University College of Swansea Singleton Park Swansea SA2, 8PP, U.K. (Paper Received April 4, 1988 and in Revised form September 13, 1988) Distortion theorems and coefficient estimates are obtained for a new class ABSTRACT. of Bazilevi functions. Bazilevi KEYWORDS AND PHRASES. Functions, Functions Whose Derivative has Positive Real Part, Coefficient Estimates. 1980 AMS Classification Code. 1. Primary 30C45. INTRODUCTION. Let S be the class of normalized functions regular and univalent in the unlt disc D < z z and S > and Re h(z) showed that the P(B), h(O) P(O). We write P D. 8 for z Bazllevlc [I] Denote by P(), the the subclass of starlike functions. class of functions which are regular in D and such that for h class of normalized regular f functions with representation =f p(t) f(z) > when > S 0. and p , S Re [3], (1.2). (I.1) it follows such that z f’(z) fCz)-gCz) Singh considered Thus f D forms a subclass of S. P for z We denote this class of See also [2]. Then there exists g In dt) = , O, g functions by B(). Let -I g(t) = t BI() if the easily from (I.I) for z > that f B() if, and only if, D (1.2) 0. subclass Bl(e) and only if, for of B(s) obtained > 0 and z D by taking g(z) z in 284 S.A. HALIM z Re I- of (z) > O. (z) We extend this class of functions as follows: DEFINITION. Let f be regular in D with f(z) Then if > + z 0 and 0 z Re n=E2 anzn B < I, f -uf ’(Z)_ > (1.3) BI(,) e If, and only if, for z D (,.4) B. (z) R, the class of functions whose derivative has real part [4]. BI(I,) was considered in [5]. Zamorskl [6] and Thomas [7] solved the coefficient problem for f B(), in the case when N is a positive Integer. In [7], sharp We note that BI(I,O) distortion theorems were obtained for f i5 to extend these results to the class BI() BI(,8). > for 0. The object of this paper The class BI(,B) has also recently been considered in [8]. 2. RESULTS. Distortion Theorems THEOREM I. BI(,). Let f (f) If(z) Q2(r) (r) re D, O r < I, Ql(r) < (ii) if 0 ra-1 Q2 Then for z a (l-r)(l-B)(1+r) + B) (]f’(z) ’ r If’(z)l r Q1 (r) q2 (r) = (+r)(-B) (l-r) + B) and if r where and Q1 (r) (l-r) (l-B) (l+r) + B] (-1 (x (l+r) (l-B) (-r) + ) SOME BAZILEVIC FUNCTIONS OF ORDER BETA f Equality holds in all cases for the function f af (z) t defined by ((l+tei)(l-B) (l-te 10) ’I 285 + B)dt) (2.1) 0 or where PROOF. and it follows from (1.4) that BI(,8) (i) Since f z (l-[)p(z) l-f ’(z) f(z) P. D and p for z l-a Thus f(z)a and since l+r l-r p(z)l $ a t a-I for z (p(t)(1-8) + 8)dt (2.2) D, (see eg. [9]) Q1 (r) To obtain the left-hand inequality in (i), write h(z) ’(z) z f(z) Then (1.4) shows that h (2.3) I-4 p(). Thus from, [5] (Theorem with c=1-28 and n=l), we obtain (l-r) (1-3) + I (1+r) (i+r)(1-1) lh(z)l (l-r) + S (2.4) Hence from (2.3) and (2.4) we have d." Writing to f(zl )a, f(z) (2.5) (1+r) it follows that since f is univalent, the llne segment lles entirely in the image of D. Let be the pre-lmage of from 0 then by (2.5) 286 S.A. HALIM which is the left-hand inequality in From (2.1) we have for z re i8 (2.6) if 0 < a If a ) I, (1) gives I, the follow inequalities at (2.6), from once If(z)ll-e Q2 (2.7) (r) Applying (2.4) and (2.7) to (2.6) gives the required result. f0 and In (II) for f0 when 0 The following shows that as a < and for For 0 r < 1, let Ql(r) f and Q2 > Ql(r) a~ (Ii) Q2(r) (Ill) r (1+r)2(I-) 1. PROOF. We prove (i), since (l i) and (111) are similar. As c*+O re -2(l-8)log(1-r) r (l-r) COROLLARY. Suppose that f(z) for z e D, then are asymptotic to the (r) be defined as in Theorem I. (l-r) 2(I-) Ql(r)~ Qz(r)~ I. 0 (see eg. [9]). as a+O (I) Equallty is attalned in when a 0 the bounds In Theorem distortion theorems for starlike functions of order 8 THEOREM 2. and -a 1-a ql (r) and (i) for (2.4) 2(i-) Then SOME BAZILEVIC FUNCTIONS OF ORDER BETA 287 PROOF. Let a Let L > O, and denote a be the 0 from L and to closest its to the origin. pre-lmage Q2(r) = II (i) gives once, Theorem llne straight f(D) of boundary the on point Thus Theorem 2 (li) gives as o. O. for n O. 3. A COEFFICIENT THEOREM Notation: n=E0 n z n Let f e THEOREM 3. n (( gnZ 0 z + Then (i) f(z) (< Inl I%1 means and be given by (1.3) where N is a positive integer. BI(, D, Suppose also that for z fo(Z) n n 0= YnZ is given by (2.1). fo(Z) where f0(z), and N n PROOF. z Write p(z)= + k=Zl pkzk f(z)N =’" n %znl jm m Then (2.2) gives =: foz t [S(l.-Ig)zN a-1 [[I + kl Pk t + (I-13) k=| k ’:k z (1+8) + ]dt k4 + k+N k z 1 + (I-B) N kl Pk z ) (k+S) Thus f(z) z(:+ (! -8) N gNzN in D. S.A. HALIM 288 (1-) z(l+ f(z) Pkl P, we have and s l,ce p f0(z) k PkZ k=l N (li) Puttlng a 2 [6]. Hence N (k) = in z + nZ2 Tn zn 2z (( z[l+ z[l v +- 2(I-8)N n=El(R) n (n N n (n +z )I 1" n v (v) fo(Z). v 2(I-8) N Dn (V)zn (v (log n) -I as n IN (N) n (log n) N +W) n Thomas [7] proved that D n N (2.1), we have n Let k 0, I, 2, 3 .... ). and so this gives N-1 as n -. REFERENCES I. BAZILEVIC, I.E., On a case of the Loewner- integrabillty in quadratures of 5. Kufarev equation, Mat. Sb. 37(79)(1955), 471-476. THOMAS, D.K., On Bazilevi functions, Ma.th Z. 109(1969), 344-348. SINGH, R., On Bazilevi functions, Proc. Amer. Mat. Soc. 38(1973), 261-271. 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