Results & Lemmas (7)
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THEOREM 1.
THEOREM 1. A function /(a) = is in P*(a, 8) if and only if rt=2 £ n(l+B)|a I s 2B(l-a). n=2 n This result is sharp.
THEOREM 1. A function /(a) = is in P*(a, 8) if and only if rt=2 \a \zn £ n(l+B)|a I s 2B(l-a) . n=2 n This result is sharp.
THEOREM 2.
THEOREM 2. If f € P*(a, g), then (3.1) r and (3.2) i
THEOREM 2. If f € P*(a, g) , then (3.1) r and (3.2) i
THEOREM 3.
THEOREM 3. Let f Z P*(a, 3). Then the disc < 1 is mapped onto a domain that contains the disc < P. The result is sharp with extremal •…
THEOREM 3. Let f Z P*(a, 3) . Then the disc \z\ < 1 is mapped onto a domain that contains the disc \w\ < P . The result is sharp with extremal • function z ~ ~ / z . -L+p
THEOREM 4.
THEOREM 4. If f € C*(a, B), then f i ^ [ 3 ^, 3 ^ ) • This result is sharp with the extremal function f(z) = z -
THEOREM 4. If f € C*(a, B) , then f i ^ [ 3 ^ , 3 ^ ) • This result is sharp with the extremal function f(z) = z -
THEOREM 5.
THEOREM 5. If f t. P*(a, &), then f is convex in the disc < r = r(a, 3), where ( » - 2. 3. --. This result is sharp, the extremal function…
THEOREM 5. If f t. P*(a, &) , then f is convex in the disc \z\ < r = r(a, 3) , where ( » - 2 . 3. --. This result is sharp, the extremal function being of the form https://doi.org/10.1017/S0004972700022917 Published online by Cambridge University Press
THEOREM 6.
THEOREM 6. If n=2 and https://doi.org/10.1017/S0004972700022917 Published online by Cambridge University Press
THEOREM 6. If n=2 \ay and https://doi.org/10.1017/S0004972700022917 Published online by Cambridge University Press
THEOREM 7. · coeff
THEOREM 7. iet " (n = 2, 3,... ). Then f € P*(ot, 3) £/ and onlj/ if it can be expressed in the form where X 2 0., n=l n n Z X = 1 • n=l n…
THEOREM 7. iet " (n = 2 , 3 , . . . ) . Then f € P*(ot, 3) £/ and onlj/ if it can be expressed in the form where X 2 0 ., n=l n n Z X = 1 • n=l n Proofs of Theorems 6 and 7 follow on the lines of the proofs of Theorems 8 and 9 in [/]. The details are omitted. References [J] V.P. Gupta and P.K. Jain, "Certain classes of univalent functions with., negative coefficients", Bull. Austral. Math. Soa. 14 (1976),
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