Results & Lemmas (12)
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LEMMA 1.
LEMMA 1. If p z) e P£(p), then (1.1) ««"*») = where ψ(θ) is a function with bounded variation in [0, 2π] satisfy- ing (1.2) 2πdf(θ) = 2π…
LEMMA 1. If p{z) e P£(p), then (1.1) ««"*») = where ψ(θ) is a function with bounded variation in [0, 2π] satisfy- ing (1.2) \ 2πdf(θ) = 2π and ΓW(#)I ^ kπ . J Jo
LEMMA 2.
LEMMA 2. f(z) is in V k(ρ) if and only if there is a function fQ(z) in V k(p) such that The function fo(z) in V k(p) has associated with it…
LEMMA 2. f(z) is in V k(ρ) if and only if there is a function fQ(z) in V k(p) such that The function fo(z) in V k(p) has associated with it a function go(z) in Vί c(0). ([9], Lemma 2.)
LEMMA 3.
LEMMA 3. f(z) is in V£(p) if and only if there is a function gQ(z) in V0 k(0) such that
LEMMA 3. f(z) is in V£(p) if and only if there is a function gQ(z) in V0 k(0) such that
LEMMA 4.
LEMMA 4. f(z) is in V£(p) if and only if there exists a func- tion g(z) in Va(0) such that /'(*) = to W ' •
LEMMA 4. f(z) is in V£(p) if and only if there exists a func- tion g(z) in Va(0) such that /'(*) = to W ' } •
THEOREM 1.
THEOREM 1. f(z) is in V£(p) if and only if there exists a func- tion ψ(θ) with bounded variation on [0, 2π] satisfying condition (1.2) and…
THEOREM 1. f(z) is in V£(p) if and only if there exists a func- tion ψ(θ) with bounded variation on [0, 2π] satisfying condition (1.2) and f\z) - exp — (1 — p)e ίacosa [ 2π π log (1 - ze iθ)dψ(θ)} .
THEOREM 2.
THEOREM 2. f(z) is in V£(p) if and only if (A) there exist starlike functions Slt S2 such that f'(z) = rrJ z J (1—p)e i a cosα (B) there…
THEOREM 2. f(z) is in V£(p) if and only if (A) there exist starlike functions Slt S2 such that f'(z) = rrJ z J (1—p)e i a cosα (B) there exist a-spiral functions Tlt Γ2 such that ΓΓtCg) T + 2 1 / 4 ' L 2J J VT2(z) Ύ k- 2)/4
COROLLARY 1.
COROLLARY 1. Suppose f(z) = z + a2z 2 + is in V£(p). Then 2 ^ k(l — p) cos a/2, and this bound is sharp.
COROLLARY 1. Suppose f(z) = z + a2z 2 + is in V£(p). Then \a2\ ^ k(l — p) cos a/2, and this bound is sharp.
LEMMA 5.
LEMMA 5. If f(z) is in V£(p), then F(z) defined by F'(z) = - a z, J F ( O ) - 0, z <l, is also in V£(p).
LEMMA 5. If f(z) is in V£(p), then F(z) defined by F'(z) = - a z , J F ( O ) - 0 , \a\ z\<l, is also in V£(p).
THEOREM 3.
THEOREM 3. // f z) is in VJί(ρ) and 0 < k(l - p) cos a ^ 1, then f(z) is univalent in < 1.
THEOREM 3. // f{z) is in VJί(ρ) and 0 < k(l - p) cos a ^ 1, then f(z) is univalent in \z\ < 1.
COROLLARY 1.
COROLLARY 1. If f(z) is in F« (0), / is univalent in E when- ever (2.1) 0 < cos a ^ 1/fc. ήs simplifies and improves bounds previously…
COROLLARY 1. If f(z) is in F« (0), / is univalent in E when- ever (2.1) 0 < cos a ^ 1/fc . ήs simplifies and improves bounds previously published for this class [7].
COROLLARY 2.
COROLLARY 2. // f(z) is in VQ k(p), then f is univalent in E for (2.2) p:> A z i l.
COROLLARY 2. // f(z) is in VQ k(p), then f is univalent in E for (2.2) p :> A z i l .
COROLLARY 3.
COROLLARY 3. // f(z) is in VZ(ρ), then f(z) is univalent in E when 0 < cos a <£ 1/2(1 — p). f need not be univalent if cos a > 1/ [2(1 -…
COROLLARY 3. // f(z) is in VZ(ρ), then f(z) is univalent in E when 0 < cos a <£ 1/2(1 — p). f need not be univalent if cos a > 1/ [2(1 - p)]. Chichra [4] has shown that for each a, 1/[2(1 — p)] < cos a < 1, there exists a function f{z) in F£ = Fα 2(^) such that /(z) is not univalent in E. Hence the problem of univalence in V%(p) is solved. 3* We may use the same function / as in [4] to study condi- tions on k, a, and p which will allow functions in V£(p) to be non- univalent. Let (3.1)