Results & Lemmas (9)
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PROPOSITION 1.
PROPOSITION 1. If φ is holomorphic and univalent in Δ, then and the constant 6 is sharp.
PROPOSITION 1. If φ is holomorphic and univalent in Δ, then and the constant 6 is sharp.
PROPOSITION 2.
PROPOSITION 2. Let Ω be a simply connected domain and let Ω(z) be its Poincare metric. If φ is holomorphic and univalent in Ω, then, zeΩ,…
PROPOSITION 2. Let Ω be a simply connected domain and let \Ω(z) be its Poincare metric. If φ is holomorphic and univalent in Ω, then , zeΩ , and the constant 12 is sharp.
Proposition 1
Proposition 1 is due to Kraus [5] and Proposition 2 is due to Lehto [6]. In this direction Nehari [7] has shown that if φ is holomorphic…
Proposition 1 is due to Kraus [5] and Proposition 2 is due to Lehto [6]. In this direction Nehari [7] has shown that if φ is holomorphic with | Sφ(z) \ ^ 2(1 — | z | 2)" 2 in Δ, then φ is univalent in Δ with the constant 2 being the best possible. 345
LEMMA 1.
LEMMA 1. Let E be a measurable set with a finite Lebesgue measure σ(E) in C. Then
LEMMA 1. Let E be a measurable set with a finite Lebesgue measure σ(E) in C. Then
THEOREM 1.
THEOREM 1. The integral formula x " πiBψ(z:Q-t' holds.
THEOREM 1. The integral formula x " πiBψ(z:Q-t' holds.
COROLLARY 1.
COROLLARY 1. Let ΩgQAD. Then WΦW* ^ CD ζ) with equality holding if and only if Ω is conformally equivalent to the unit disk A less…
COROLLARY 1. Let ΩgQAD. Then WΦW* ^ CD{ζ) with equality holding if and only if Ω is conformally equivalent to the unit disk A less {possibly) a closed null CD-set.
COROLLARY 2.
COROLLARY 2. CD(Z) ^ CB(ζ).
COROLLARY 2. CD(Z) ^ CB(ζ).
THEOREM 2.
THEOREM 2. Let Ω 0AD. If φ is holomorphic and univalent in Ω we have the following sharp string of inequalities φ(z, 01 ^ 6CD(z)CD(ζ)[l
THEOREM 2. Let Ω $ 0AD. If φ is holomorphic and univalent in Ω we have the following sharp string of inequalities \Sφ(z, 01 ^ 6CD(z)CD(ζ)[l$
COROLLARY 3.
COROLLARY 3. Let Ω £ 0^ and let φ be holomorphίc and univalent in Ω. Then I S φ ( z, ζ ) !: £ 6 λ a ( 2 ) λ a ( ζ ) | 1 + ( 1 - £ £ L ) |;…
COROLLARY 3. Let Ω £ 0^ and let φ be holomorphίc and univalent in Ω. Then I S φ ( z , ζ ) ! : £ 6 λ a ( 2 ) λ a ( ζ ) | 1 + ( 1 - £ £ L ) | ; z , ζ e Ω , in particular inequalities are sharp. The inequality \SΦ(z)\ £ 6X%(z) is sharp only when Ω is a disk less (possibly) a closed subset of innear capacity zero. Otherwise, we have the sharp inequality \S,(z)\ ^ 12Xl(z) .