Results & Lemmas (7)
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THEOREM 1.
THEOREM 1. Suppose p(z)£ F, h is defined by (1), and a is defined as in Theorem A. Then Re where rβ is the unique root of the equation (1 -…
THEOREM 1. Suppose p(z)£ F, h is defined by (1), and a is defined as in Theorem A. Then Re where rβ is the unique root of the equation (1 - 2β)r 3 - 3(1 - 2β)r 2 + 3r - 1 = 0 in the interval (0,1]. This result is sharp.
Theorem 1
Theorem 1 to prove
Theorem 1 to prove
THEOREM 2. · radius
THEOREM 2. Suppose rβ is the unique root of ί(r) = ( l - 2 β ) r 3 - 3 ( l - 2 β ) r 2 + 3 r - l in the interval (0, 1]. Set, „, 1 2— Q, ~…
THEOREM 2. Suppose rβ is the unique root of ί(r) = ( l - 2 β ) r 3 - 3 ( l - 2 β ) r 2 + 3 r - l in the interval (0, 1]. Set , „, 1 2— Q, ~ Σβ -\- \ <x la. + 4 p op •+• 3 Then the radius of convexity of C(α, β) is r(α, β) when 0 < r(α, β)^ rβ, and is otherwise the smallest root greater than rβ of the polynomial equation
THEOREM 3. · radius
THEOREM 3. If f(z)G S with Ref(z)> β, then f(z) is convex in a disk of radius 1 1 +,, V - This result is sharp.
THEOREM 3. If f(z)G S with Ref(z)> β, then f(z) is convex in a disk of radius 1 1 + ,, V -\ This result is sharp.
THEOREM 4.
THEOREM 4. // f(z) E C(α, β), then This result is sharp.
THEOREM 4. // f(z) E C(α, β), then This result is sharp.
THEOREM 5.
THEOREM 5. Suppose /(z), g(z)E C(α,/3). 77ιen λ/(z) + (l-λ)g(z) ( O ^ λ ^ l ) is univalent in a disk < r, w/iere r is ί/ie smallest…
THEOREM 5. Suppose /(z), g(z)E C(α,/3). 77ιen λ/(z) + (l-λ)g(z) ( O ^ λ ^ l ) is univalent in a disk \z \ < r, w/iere r is ί/ie smallest positive root of the equation /5 s/iαrp.
Theorem 4.
Theorem 4.
Theorem 4.
Function classes studied:
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