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Results & Lemmas (5)
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LEMMA 1.
LEMMA 1. For p > 1, q > 0 and τ e [-1, 1] let f L B(τ) = / (cosh(pqs) + τ sinh(p qs)) ι/p ds. J-L
LEMMA 1. For p > 1, q > 0 and τ e [-1, 1] let f L B(τ) = / (cosh(pqs) + τ sinh(p qs)) ι/p ds. J-L
THEOREM 1.
THEOREM 1. If g(z) = z+a2Z 2+a3z 3- — is a normalized univalent function on Ό, then This inequality is sharp for all p > 0. For p > 3/2,…
THEOREM 1. If g(z) = z+a2Z 2+a3z 3-\— is a normalized univalent function on Ό, then This inequality is sharp for all p > 0. For p > 3/2, equality holds if and only if g is a rotation of the Koebe function.
THEOREM 2.
THEOREM 2. Suppose f is univalent in D. There is a constant P G ( 1, 3/2] swcΛ that for any p>P and all a, beΌ,
THEOREM 2. Suppose f is univalent in D. There is a constant P G ( 1 , 3/2] swcΛ that for any p>P and all a, beΌ,
Theorem 2.
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Theorem 2. COROLLARY. Let Ω be a simply connected hyperbolic region in C. Then for any p>P and all A,B eΩ, -B > Equality holds if and only…
Theorem 2. COROLLARY. Let Ω be a simply connected hyperbolic region in C. Then for any p>P and all A,B eΩ, \A-B\> Equality holds if and only if Ω is a slit plane A and B lie on the extension of the slit into Ω.
THEOREM 3.
THEOREM 3. Suppose Ω is a convex hyperbolic region. Then for any p> 1 and all A,B eΩ, ύήh dΩ(A9B)) 1 ' - Equality holds if and only if Ω is…
THEOREM 3. Suppose Ω is a convex hyperbolic region. Then for any p> 1 and all A,B eΩ, ύήh{dΩ(A9B)) 1 ' - Equality holds if and only if Ω is a half plane and A and B lie on a line perpendicular to the edge of the half plane. Conversely\ if Ω is a hyperbolic region in C and the preceding inequality holds for some p > 1 and all A, B eΩ, then Ω is convex.
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