Results & Lemmas (10)
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THEOREM 1.
THEOREM 1. A function /(a) = z - £ |a is in S* a, 3) if n=2 n and only if 00 X (n-l)+B(n+i-aa) |a I « 20(l-a). n=2 n This result is sharp.
THEOREM 1. A function /(a) = z - £ |a \zn is in S*{a, 3) if n=2 n and only if 00 X {(n-l)+B(n+i-aa)}|a I « 20(l-a) . n=2 n This result is sharp.
THEOREM 2.
THEOREM 2. A function f(z) = z - £ ie in C*(a, 3) if M=2 " a?id only if https://doi.org/10.1017/S0004972700025326 Published online by…
THEOREM 2. A function f(z) = z - £ \a \zn ie in C*(a, 3) if M=2 " a?id only if https://doi.org/10.1017/S0004972700025326 Published online by Cambridge University Press
THEOREM 3.
THEOREM 3. A function f(z) = z - £ is in S*(a, 6) if n=2 n and only if (3.1) fiz) = z exp|2(l-a) '0 where y(z) is analytic and satisfies…
THEOREM 3. A function f(z) = z - £ \a \zn is in S*(a, 6) if n=2 n and only if (3.1) fiz) = z exp|2(l-a) '0 where y(z) is analytic and satisfies |q>(3)| £ 8 , for \z\ < 1 .
THEOREM 4.
THEOREM 4. If f € S* a, 6), then /k i) r 2B(i-a) r2, i n.. ^ 28(l-a) 2 ( i a i = 1. ) t 4 > i; r l+8(3-2a) - | J l 2 j | -
THEOREM 4. If f € S*{a, 6) , then /k i) r 2B(i-a) r2 , i n .. ^ 28(l-a) 2 ( i a i = 1 . ) t 4 > i ; r l+8(3-2a) - | J l 2 j | -
THEOREM 5.
THEOREM 5. If f € C*(a, g), then and itfe equalities for f(z) = s - B(l-ot) [l+B(3-2a) I"1*2, (2 = ±r). THEOREM 6. Let f € S * ( a, g).…
THEOREM 5. If f € C*(a, g) , then and itfe equalities for f(z) = s - B(l-ot) [l+B(3-2a) I"1*2 , (2 = ±r) . THEOREM 6. Let f € S * ( a , g) . 2%ew *fo? d i s k | s j < 1 is
THEOREM 7.
THEOREM 7. Let f € C*(a, g). Then the disk < 1 is mapped onto a domain that contains the disk < (l+g(2-a))/l+g(3-2a). The result is sharp…
THEOREM 7. Let f € C*(a, g) . Then the disk \z\ < 1 is mapped onto a domain that contains the disk \w\ < (l+g(2-a))/l+g(3-2a) . The result is sharp vrlth extremal function z - g(l-a) [l+B(3-2a) ]~ z . Proofs of Theorems 6 and 7. Proofs follow by letting r -»• 1 in (U.I) and (4.3). 5. Closure theorems In this section, we shall prove that the classes S*(a, B) and C*(a, g) are closed under 'arithmetic mean' and "convex linear combinations'.
THEOREM 8.
THEOREM 8. If f(z) = z - £ and g(z) = z - £ n=2 n n=2 n GO are in S*(a, B) 3 then h z) = z - -~ Y. +b is also in S*(a, B) • n=2 n n
THEOREM 8. If f(z) = z - £ \a \zn and g(z) = z - £ \b \zn n=2 n n=2 n GO are in S*(a, B) 3 then h{z) = z - -~ Y. \a +b \z is also in S*(a, B) • n=2 n n
THEOREM 9.
THEOREM 9. Let - 2. 3.... ). TTien / € S*(a, 6) i / and cmli/ i / i t can be expressed in the form I 7 1 = 1 ' where X n - ° and I 7 J = 1
THEOREM 9. Let - 2 . 3. . . . ) . TTien / € S*(a, 6) i / and cmli/ i / i t can be expressed in the form I 7 1 = 1 ' where X n - ° and I 7 J = 1
THEOREM 10.
THEOREM 10. if f(z) = z - £ and g(z) = z - £ n=2 n=2 " OO are in C*(a, &), then h(z) = z - -~ £ +b is also in C*(a, 6) ^71=2 n
THEOREM 10. if f(z) = z - £ \a\zn and g(z) = z - £ \b \zn n=2 n=2 " OO are in C*(a, &) , then h(z) = z - -~ £ \a +b \zn is also in C*(a, 6) ^71=2 n
THEOREM 11. · coeff
THEOREM 11. Let ^ f e ^ ' (» = 2, 3,...). / € C*(a> 3) i / and on^y if it can be expressed in the form I /J ^M 2 0 and X n=l " " " n=l…
THEOREM 11. Let ^ f e ^ ' (» = 2, 3, ...). / € C*(a> 3) i / and on^y if it can be expressed in the form I \/J ^M 2 0 and X n=l " " " n=l References [J] A. Schild, "On a class of functions schlicht in the unit circle", Proa. Amer. Math. Soa. 5 (195U), 115-120. [2] Herb Silverman, "Univalent functions with negative coefficients", Proa. Amer. Math. Soa. 51 (1975), 109-116.
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