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Results & Lemmas (5)
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THEOREM 1.
THEOREM 1. Let. f(z) = z + a2z 2 + + anz n + • be regular and univalent in z | < 1 and such that (1 — te iΛ)f(z) is sub- ordinate to f(z)…
THEOREM 1. Let. f(z) = z + a2z 2 + + anz n + • be regular and univalent in \ z | < 1 and such that (1 — te iΛ)f(z) is sub- ordinate to f(z) in I z I < 1 for an interval 0 ^ t ^ t0, a a real constant I a I < π/2, then (2.4) ^ > o , |*| < 1 . For the proof of Theorem 1 we take (2.5) F(z, t) = (1 - te")f(z)
THEOREM 2.
THEOREM 2. Let f(z) of (1.1) be regular and univalent in z < 1. For an interval 0 ^ t ^ t0 let the function (2.6) ^r[f(e uz) + f(e~»z)] be…
THEOREM 2. Let f(z) of (1.1) be regular and univalent in \ z \ < 1. For an interval 0 ^ t ^ t0 let the function (2.6) ^r[f(e uz) + f(e~»z)] be subordinate to f(z) in \ z | < 1. Then. (2.7) (#) is convex in 12 | < 1.
THEOREM 3.
THEOREM 3. Let (2.10) /(*) - z + ta^z^-i, 2 be an odd function, regular and univalent in < 1. For all real a and for an interval 0 ^ t ^ ί0…
THEOREM 3. Let (2.10) /(*) - z + ta^z^-i , 2 be an odd function, regular and univalent in \z\ < 1. For all real a and for an interval 0 ^ t ^ ί0 Zeί £fce function (2.11)
THEOREM 4.
THEOREM 4. Let the function (2.17) (1 - t)f(z) + tf(-z) be subordinate to the univalent, regular function
THEOREM 4. Let the function (2.17) (1 - t)f(z) + tf(-z) be subordinate to the univalent, regular function
THEOREM 5.
THEOREM 5. A necessary and sufficient condition that the function f(z) = z + a2z 2 + + anz n +, regular and univalent in < 1, be convex in…
THEOREM 5. A necessary and sufficient condition that the function f(z) = z + a2z 2 + + anz n + , regular and univalent in \z\ < 1, be convex in \z\ < 1 is that the de la Vallee Poussin means
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