V22N2-P133-143_OCR

Theorem 1. For f(z) E C[A, B; C, D,p, {J, a], izl ~ r < 1, C-D r11-1P - [pB + (A - B)(p - /3)]r (1- Dr)( D )(p - a) < I f'(z) I · 1 - Br C-D s; ,.,,-,P + [pB + :: ~:)(p - /1)]r (1 + Dr/ D )(p- a), D # 0, r11-1 P - [pB + (A - B)(p - /3)]r e-C(p-a)r ~ I /'(z) I . 1 - Br < rp-1 P + [pB + (A - B)(p - /3)]r eC(p-a)r, D == 0. -

Theorem 1] C-D C-D prP - 1 (1 - Dr) ( D ) (p - a) ~ I g' ( z) I ~ prP - 1 (1 + Dr/ D )(p - a) , D f:. 0, and prP-le-C(p-a)r ~ lg'(z) I~ prP-leC(p-a)r, D = 0. (2.2) Also for p(z) E P[A,B,p,/3], we have for lzl ~ r < 1 [1, Corollary 1] p - [pB + (A - B)(p - f3)]r < I (z) I < p + [pB + (A - B)(p - /3)]r. 1 - Br · - p -

Theorem 2. For /(z) E C[A, B; C, P,P, /3, a], lzl ~ r < 1, I arg f'(z) I< zP-1 - {( C-D . _1 • _1 (A-B)(p-/3)r D )(p- o)sm (Dr)+ sm p- [pB +(A_ B)(p- /3)]Br2, D #- 0, ( . _1 (A - B)(p - f3)r

Lemma 1 (Ruscheweyh and Sheil-Small, [17]). Jf 'f/J(z) is regular in U, </>(z) and h(z) are convex univalent in U such that 'f/J(z)-< </>(z), then 'f/J(z) * h(z)-< ¢,(z) * h(z), z EU.

Theorem 3. If f(z) E C[A, B; C, D, p, /3, a], then there exists p(z) E P[A, B, p, /3] such that for alls and t with Isl ::::; 1, Jtl ::::; 1 (s f. t), f'(sz)p(tz)tP-l f'(tz)p(sz)sP-1 \ C-D (l+Dsz)( D )(p-a) ,D f. 0, (3.1) -< 1 + Dtz eC(p-a)(.,-t)z ,D = 0. (3.2)

Corollary 1. If f(z) E C[A,B;C,D,p,/3,a], then there exists ap(z) E P[A,B,p,/3] and a Schwarz function w( z) E n such that { C-D ~:~; = p(z)(l + Dw(z))( D )(p- a)' D 'F 0, p(z)eC(p-a)w(z)' D = 0.

Corollary 2. If f(z) == zP + L anzn E C[A,B;C,D,p,/3,a], then n=p+l I a I< p(C - D)(p- a) + (A- B)(p- /3) p+l - p + 1

Lemma 2. For g(z) = zP + L bnzn E I<[C, D, p, a] andµ complex n=p+l p lb,+i I~ (p+l)(C-D)(p-o), (4.1) and I bp+2 - µb;+1 I< "'_P, o\ (C - D)(p- a)· 2p(p + 2) -max{l,I , . n? µ(C-D)(p-a)-[(C-D)(p-a)p-D] I}. (4.2) The result is sharp.

Theorem 5. For f(z) = zP + L an Zn E C[A, B; C, D, p, ,B, a] n=p+l l a I< p(C-D)(p-a)+(A-B)(p-/3) p+l - p + 1 (4.9) and I ap+2 I~ E___,..(C - D)(p- a)+ (A - B?(~ - /3) [(C - D)(p - a) + 1] + P:2(C-D)2(p-a)2+ P:2B~, l(C-D)(p-a)p-DI ~ p -(C - D)(p- a)[(C- D)(p- a)p - D] (A- B)(p - f3)[(C - D)(p- a) + 1]