V25N2-P113-123_OCR

Lemma 1. If Q f; 1 and n and q are positive integers, then (1- Q2) {(~~~)lb!'+ j; [<~~~)lb!'+ 1 ~QQmnRe{b} - m 2n 2 ] . [~! 'f.( Uj] 2 } = { n! ti Uj} 2 , (2.1) 3=0 q ;=O

Theorem 1. If the function f(z) defined by (1.1) is in the class F(b, M, n), M > ! and Q f; 1, and if 2 - 2Q R {b} - ( 1 + Q )Jbl2 > 0 n I-On e 1-r. - ' then (m+I)n L (k - 1)2lakl2 $ (1 + Q)2lbl2, m = 1, 2, · · ·. k=mn+l If, on the other hand, 2$

Theorem 1. The estimates (2.4) and (2.6) are sharp with equality holding in (2.4) for a given m for the fucnction ft: (z)(lcl = 1) defined by !t:(z) = { z(l - £Q::nr(~).:n, Q -:j:. 0, z exp[E%n ), Q = 0, if n2 - .J!LnRe{b} - (!.±Q)\b\2 > 0. 1-Q 1-Q - Also the equality in (2.6) holds for the function ft: (z)(lcl = 1) defined by if n2 - 1:_9QnRe{b} - (~~§)lbl2 < 0.

Corollary 1. If f(z)=z+E~=n+lakzk E F((l-a)e-i>-cos.-\,M,n)=FM(.-\,a,n), I.-\I < i and O :s; a < 1, then form= 1, 2, · · ·. The estimates are sharp for the function ft:(z)(lcl = 1) given by

Corollary 2. If f(z) = z + .L~=n+I akzk E F(b, oo, n) = S(l - b, n), then (m+l)n { m-112b 1} 2 I: (k - 1>21ak1 2 $ <m: 1>, p ~ + j k=mn+l J=O for m = 1, 2, · · ·. The estimates are sharp for the function f~(z)(lcl = 1) given by$

Corollary 3 [14]. If f(z) = z + .L~=n+I akzk E F(e-i>-cos ..\,co~>., n) = (H*)~, then I I ( 2 - cos ..\) cos ..\ ak S k - 1 ' The estimates are sharp for each k 2:: n + 1. k2::n+l.

Corollary 4 [7,8,14]. If f(;) = z+ .L~=n+l akzk E F((l-a)e-i>. cos..\, oo, n) = s>-(a,n),O Sa< 1, I..\I <!,then (m+I)n { m-11 ( )-i>. 1} 2 I: (k - l)21ak12 s m: 1 ' J] 2 1- a: cos,\+ j k=mn+I ( ) ;=O form= 1, 2, · · ·. The estimates are sharp.

Theorem 2. If the function f(z) defined by (1.9) is in the class F*(b, M, n) and M 2:: 1, then (m+i)n-1 I: (k + 1>21ak12 so+ Q)2lb12, k=mn Jorm=l,2,···, and 1 Q = 1- M' (2.14)