Results & Lemmas (8)
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THEOREM 2.1.
THEOREM 2.1. Let f z) = Σ^U α^ %, ^fcere αΛ are real, and ϊϊm I an 1/n =
THEOREM 2.1. Let f{z) = Σ^U α^ %, ^fcere αΛ are real, and ϊϊm I an \ 1/n =
THEOREM 2.2.
THEOREM 2.2. Let f(z) = Σ"=o αΛs Λ, wfeere an are real and ^oo 1/n = l//e, JB > 1. Tfeβ^ limL*(/) = Im(/);ί-»0 uniformly on ^ 1.
THEOREM 2.2. Let f(z) = Σ"=o αΛs Λ, wfeere an are real and ^oo \an\ 1/n = l//e, JB > 1. Tfeβ^ limL*(/) = Im(/) ;ί-»0 uniformly on \z\ ^ 1.
THEOREM 3.
THEOREM 3. If U θ) = ^1± nl c, ( c o s θ) satisfies a Lipschίtz condition of order a, 0 < μ < a < 1, on [0, π], then its conjugate V θ) = 2…
THEOREM 3. If U{θ) = ^1± nl c , ( c o s θ) satisfies a Lipschίtz condition of order a, 0 < μ < a < 1, on [0, π], then its conjugate V{θ) = 2 2<" Σ jη—cτ 7Γ a* C^ Ό*
THEOREM 4.1.
THEOREM 4.1. If U is a generalized axisymmetric potential which is regular on an axiconvex region Ω, then its conjugate V is also regular…
THEOREM 4.1. If U is a generalized axisymmetric potential which is regular on an axiconvex region Ω, then its conjugate V is also regular on Ω. The following result is also immediate.
THEOREM 4.2.
THEOREM 4.2. F is μ-pseudo analytic on an axiconvex region
THEOREM 4.2. F is μ-pseudo analytic on an axiconvex region
THEOREM 5.1.
THEOREM 5.1. Let Ω be a bounded axίconvex region. If F is μ-pseudo analytic on C1(J2), then there exist μ-pseudo analytic polynomials WM=άo…
THEOREM 5.1. Let Ω be a bounded axίconvex region. If F is μ-pseudo analytic on C1(J2), then there exist μ-pseudo analytic polynomials WM=άo such that ,ι ^ M \F - where || || denotes the sup norm over Cl (Ω), M is a constant inde- pendent of n9 and R > 1.
THEOREM 5.2.
THEOREM 5.2. Let Ω be a bounded axiconvex region, w — φ z)~ z + α0 + axz~~ ι + map the compliment of Ω onto the circular region > p, and…
THEOREM 5.2. Let Ω be a bounded axiconvex region, w — φ{z)~ z + α0 + axz~~ ι + map the compliment of Ω onto the circular region \w\ > p, and let ψ — φ' 1. If F(x, y) is μ-pseudo analytic on the region BB bounded by ΓR = {z: \Φ(z)\ = R}, then Fcan be expanded into a series of μ-pseudo analytic Faber polynomials (5.3) F(x, y)= ± anFn(x, y), where (5.4) an = μ Γ ^ ) \
THEOREM 5.3.
THEOREM 5.3. Let Ω be a bounded axiconvex region and C* maximize |F(CJ| over Cl(i2). lim LΛ(z;C*;F) = F(z) uniformly on Cl (42) for every…
THEOREM 5.3. Let Ω be a bounded axiconvex region and C* maximize |F(CJ| over Cl(i2). lim LΛ(z;C*;F) = F(z) uniformly on Cl (42) for every function F which is μ-pseudo analytic on Cl (42). Further, the convergence is at a geometric rate, i.e., ii π _ j . ii < fa + 2)M where |] |] denotes the sup norm over Ω and the constants M, R>1 are as given in Theorem 5.1.