Results & Lemmas (11)
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LEMMA 1.
LEMMA 1. Iff(z) — z + Σ?=2 6»«* ί iS> *Λβw- there exists an ε > 0 ίfeαί geFε(f) implies that g
LEMMA 1. Iff(z) — z + Σ?=2 6»«* ί iS> *Λβw- there exists an ε > 0 ίfeαί geFε(f) implies that g
THEOREM 1.
THEOREM 1. 1/ f z) = z- Σ?=2 K|« n wiίfe Σ»=2?Φ»I > exists an e > 0 sucfe ί/iαί gr e -P£(/) implies that g £ LS.
THEOREM 1. 1/ f{z) = z- Σ?=2 K|« n wiίfe Σ»=2?Φ»I > exists an e > 0 sucfe ί/iαί gr e -P£(/) implies that g £ LS.
THEOREM 2.
THEOREM 2. // Σ»-2^|αΛ( > 1, then there exist real numbers a2,, aN (—π < a3- <; π) such that
THEOREM 2. // Σ»-2^|αΛ( > 1, then there exist real numbers a2, , aN (—π < a3- <; π) such that
COROLLARY 1.
COROLLARY 1. 1/ Σ ^ ^ Ί β J > 1> ίAβw ίfeβrβ exist real numbers azi '' Ί aN ( — π < w3- <> π) and an ε > 0 such that for each λ, — ε ^ λ ^…
COROLLARY 1. 1/ Σ ^ ^ Ί β J > 1> ίAβw ίfeβrβ exist real numbers azi '' Ί aN ( — π < w3- <> π) and an ε > 0 such that for each λ, — ε ^ λ ^ ε, ,Λ(3) - z + e u(j: ane^z n + Σ aΛz*) $ LS .$
Theorem 2.
Theorem 2.
Theorem 2.
COROLLARY 2.
COROLLARY 2. //Σί=2 w* |α*J > 1, ί/^w ίfcere e^isί rβαZ numbers a2J, aN ( — 7Γ < α y ^ TΓ) and ε > 0 s^cfc £fca£ /o? Λ eac/^ λ, — ε <; λ ^…
COROLLARY 2. //Σί=2 w* |α*J > 1, ί/^w ίfcere e^isί rβαZ numbers a2J , aN ( — 7Γ < α y ^ TΓ) and ε > 0 s^cfc £fca£ /o? Λ eac/^ λ, — ε <; λ ^ ε, fλ(z) - 2 +
LEMMA 2.
LEMMA 2. For n^S there exist real numbers alf a2, -•-,(*„ such that ΣΓk^e iakz h < n ( ^ 1).
LEMMA 2. For n^S there exist real numbers alf a2, -•-,(*„ such that \ΣΓk^e iakz h\ < n (\z\ ^ 1).
THEOREM 3.
THEOREM 3. (i) If ΣUn k n k > 1, ί^β^ fλ(z) = z + e iλ(an2z n* + ansz nή g LS /or some λ. (ii) If N> 3, ίfeew we cαw ^ ^ d αΛfc | k = 2,…
THEOREM 3. (i) If ΣUn k\a n k\ > 1, ί^β^ fλ(z) = z + e iλ(an2z n* + ansz nή g LS /or some λ. (ii) If N> 3, ίfeew we cαw ^ ^ d {αΛfc | k = 2, 3, , JV} swcfo ίfeαί Σ L 2 ^i|α nj > 1 and fλ(z) = z + e iλ Σί= 2 a>«kz* k 6 S /or αϊί λ.
LEMMA 3.
LEMMA 3. Set α(r) = 1/(1 - r 2) - εV 2/(l - ε 2r 2) and d(r) = r/(l - r 2) + er/(l — εV). For g 6 Ge, the values for zg )jg z))f ^ r, lie…
LEMMA 3. Set α(r) = 1/(1 - r 2) - εV 2/(l - ε 2r 2) and d(r) = r/(l - r 2) + er/(l — εV). For g 6 Ge, the values for {zg\z)jg{z))f \z\ ^ r, lie in a disk centered at a(r) and having radius d(r).
LEMMA 4.
LEMMA 4. A function g eGε defined by (7) is starlike if and only if s <L 1/3.
LEMMA 4. A function g eGε defined by (7) is starlike if and only if s <L 1/3.
THEOREM 4.
THEOREM 4. Suppose h(z) = z + e iλ X^=2 « n e Jϊ£, ε ^ 1/3, feF9 with a(r) and d(r) defined in Lemma 3. Then h*f is sta like in a disk <…
THEOREM 4. Suppose h(z) = z + e iλ X^=2 « n e Jϊ£, ε ^ 1/3, feF9 with a(r) and d(r) defined in Lemma 3. Then h*f is sta like in a disk \z\ < r0, where r0 is ίfeβ jftrβί positive root of 9 ) a(r) -
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