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Results & Lemmas (5)
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LEMMA 1.
LEMMA 1. Let Re c 0 > 0 and for z in Δ, (2.1) />(z) = (Rec0) ί I m c o = c z" y-i where for j = l,2,...,m, t and λj are real with λj ^ 0…
LEMMA 1. Let Re c 0 > 0 and for z in Δ, (2.1) />(z) = (Rec0) ί I m c o = c z" y-i where for j = l,2,...,m, t} and λj are real with λj ^ 0 and Σ™=1 λ y = 1. Define for a fixed n the s by s determinants Q ( k s\ s > 1, k = 1,2,... ,m by Qί s) = -2)lk
THEOREM 1.
THEOREM 1. Let P(z) = c0 + cxz + ReP(z) > 0 for z in Δ. For s = 1,2,3,... analytic in Δ iέ? ^Λe (^ + 1) 6y (s + 1) n + s/ CO cχ/co s-l/ CO
THEOREM 1. Let P(z) = c0 + cxz + ReP(z) > 0 for z in Δ. For s = 1,2,3,... analytic in Δ iέ? ^Λe (^ + 1) 6y (s + 1) n + s/ CO cχ/co s-l/ CO
COROLLARY 1.
COROLLARY 1. Let P(z) = c0 + cxz + be analytic in Δ and satisfy Re P(z) > 0 for z in Δ αn<i let 1/P(z) — d0 + dλz +, then for p = 1,2,...,…
COROLLARY 1. Let P(z) = c0 + cxz + be analytic in Δ and satisfy Re P(z) > 0 for z in Δ αn<i let 1/P(z) — d0 + dλz + , then for p = 1,2,..., p Σ
LEMMA 3.
LEMMA 3. Let F z) = axz + a2z 2 + be in K(p) and be analytic for < 1 with an real for each w, then there exists f(z) = bλz + b2z 2 + • in…
LEMMA 3. Let F{z) = axz + a2z 2 + be in K(p) and be analytic for \z\ < 1 with an real for each w, then there exists f(z) = bλz + b2z 2 + • in S(p) and analytic for \z\ < 1 with bn real for each n, such that Re(zF'(z)/f(z))>Ofor\z\<l.
THEOREM 2.
THEOREM 2. Le/ JP(Z) = αxz + a2z 2 + 6e ι/ι ΛΓ(/?) w/7Λ ΛΠ real for each «, (3.4) - t) - p - l) (n 2 - t
THEOREM 2. Le/ JP(Z) = αxz + a2z 2 + 6e ι/ι ΛΓ(/?) w/7Λ ΛΠ real for each «, (3.4) - t)\\n - p - l)\(n 2 - t
Function classes studied:
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