Meromorphic univalent functions with positive coefficients

THEOREM 1. Let f(z) = - + \ a zn be regular in D . If n=l 00 (.2.1) I C(.l+6)w + (2a-l)B + 1] \a \ < 2B(l-a) , n=l 0 < a < l , 0 < B < l , then f e Z*Ca,6) .

THEOREM 2. Let f(z) = - + Y a zn , a > 0 , be regular in D . z n=\ n n Then f e E*(a,8) if and only if (2.1) is satisfied.

THEOREM 3. If f(z) e Ip(a,B) , then for 0 < \z\ = r < 1 , (3.1) p - where equality holds for the function ,_ ~» n . . 1 p(l~CX) • (3.2) / (s) = — + s at 3 = ^r , r

THEOREM 4. If f(z) is in E*(a,B) , then f(z) is meromorphically convex of order 6(0 < 6 < 1) in \z\ < r = r(a,B,6) , where 1 , „ _ 1,2 __ 1 27ze bound for \z\ is sharp for each n , with the extremal function being of the form (2.5).

THEOREM 5. Let f (s) = - and f (z) = 1 + Jn z 2P(l-g) z",n= 1,2,... https://doi.org/10.1017/S0004972700009874 Published online by Cambridge University Press

THEOREM 6. The class E£(a,6) is closed under convex linear combinations.

THEOREM 7. If f{z) is in E*(a,6) , then the integral transforms 1 Fc(z) = c I u f(uz)du , 0 < c < » , are in Z*(6) , where (5.2) 6 = 6(a,B,e) = Bc(l-a) + (1+aB) (<3+2) ' result is best possible for the function f(z) = — + S(1~") s . 3 1+aB https://doi.org/10.1017/S0004972700009874 Published online by Cambridge University Press

THEOREM 8. If f(.z) = i + £ anzn and g(z) = i + £ Z^s" are in CO E*(a,B) £?ien f{z)*g(z) = \ + Y abzn is in E£(Y,B) wfcere (6.1) Y = (1+aB)2 - 8(l-a)2 (1+ag)2 + 6 2U-a) 2 The result is best possible for the functions « « • » < • > • $ • ? & • •$

Theorem 2, we have (6.2) and C2a-3.)B+1 n=l 28(l-a) an ~ 2Btl-a) In view of Since /(s) and M=l are regular in D , so is ftz)*g(.z] . Further, Cl-a) + 1 2B 1- (1+aB)2 - BU-a) 2

THEOREM 9. If f(.z) e E*(a,B) and g{z) e Z*tY,B) , then f{z)*g(z) e Z*(.T,B> where T = (1+aB) (1+By) - B(l-aUl-v) (l+aB)(l+BY) + B2Cl-a)(l-Y) The result is best possible for -hr Z . https://doi.org/10.1017/S0004972700009874 Published online by Cambridge University Press

THEOREM 10. If f(z) =\ I*,(a,B) and en =^+ I b bzn with \bn\ < 1 , n = 1,2,... , then n=l " f(z)*g(z) e E*(a,B)