v27,no01,1996,p015_026_OCR

Theorem 2.1. Let f(z) =~-~lanlz" be regular in U'and belongs in n=l TM(A, B) if and only if 00 芷{n(l - B) + 1- A}lanl::; (B - A) n=l (2.1)

Theorem 2.2. Let f(z) =钅- L lanlzn (where a~l). If f is regular in U n=l and satisfies f(zo) = -fa, then f E Tu(A, B, zo) if and only if f [{n(l - B) + 1- A} - (B -A)z;;十l] 屈I~B - A. (2.4) The result is sharp.

Theorem 2.3. / E 呤(A, B, zo), then f is meromorphically convex of order 8(0~8 < 1) in the disc lzl < R, where R = inf (1 - 8){n(l - B) + 1 - A} 1/(n+l) n>l[ n(n+8)(B-A) ]·

Theorem 2.4. If f E 呤(A, B, z0), then the integral transform F(z) = c/1 刃(uz)du, for O < c < oo 。 (2.8)

Theorem 2.5. Let'Y be a real number such that'Y > 1. If f E 呤(A, B, zo), then the function F defined by F(z) = ('Yz~1)「 P打(t)dt 。 also belongs to TM(A, B, z0). CX)

Theorem 2.6. Let fi(z) = 竺- I: lanilzn, j = 1, 2, ... , m. If Ji E Tu(A, B, zo) n=l for each j = 1, 2, ... , m, then the function b 00 h(z) = - - z 2 阯l 玕 n=l also belongs to TM(A, B, zo) where b =f Ajaj, lbnl =立ilanjl (n= 1,2, ... ,m), J=l J=l m

Theorem 2.7. Let f(z) =~and {n(l - B) + 1 - AH - (B - A)zn fn(z) = {n(l - B) + 1 - A} - (B - A)z尸' n = 1, 2, 3, .... Then h E TM(A, B, zo) if and only if it can be expressed in the form 00 h(z) = A J(z) + L Anfn(z), n=l where A;:: 0 and A+ E:=1 An= 1.

Theorem 2.9 If J(z) = 钅－立~1 Ian 丨zn E TM(A,B,zo) and g(z) = 钅－ I:::"=1 lbnlzn with lbnl :S 1 for n = 1, 2, ..., then f * g E TM(A, B, zo).