Results & Lemmas (8)
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THEOREM 1. · coeff
THEOREM 1. The coefficients Pn a), n = 1, 2,..., (a ¥= 0) of the function (14) have the explicit representation 05) W = 2 V,*(T.---.- J*T…
THEOREM 1. The coefficients Pn{a), n = 1, 2, . . . , (a ¥= 0) of the function (14) have the explicit representation 05) W = 2 V,*(T.---.- J*T ±TT)' k = \ a W n — k + \ ' where (>6) c,„(ei _ i = f ± ^ ) _ 2 " n ' ! ( « ) ' V 1 H — & + 1 ' l V = l *y- ^ '
THEOREM 2. · coeff
THEOREM 2. For a > 0, the coefficients Pn(a), n = 1, 2,..., of the function (14) satisfy the sharp inequalities (19) „(a) ^ 2 a I n n where…
THEOREM 2. For a > 0, the coefficients Pn(a), n = 1, 2, . . . , of the function (14) satisfy the sharp inequalities (19) \P„(a)\ ^ 2 a I n n\ where equality holds only for the function oo (20) [P*(z)]1/a = (1 - ezy2/« = 1 + 2 - e V , (|c| = 1).
Theorem 1 · coeff
Theorem 1 yields the coefficients of the function (24) in terms of the coefficients of the associated function p (z ) in (26), and Theorem…
Theorem 1 yields the coefficients of the function (24) in terms of the coefficients of the associated function p (z ) in (26), and Theorem 2 gives us the known estimates [11]-[13] m |,.(_y a£^M. ,.-,.2...., for the class S% of functions op(z) starlike of order /?; here equality holds only for the Koebe functions „ /„x f , i V (2 - 2/?), + 1 °^(Z) = (l- ez) 2<'-« = Z + „f, ~ • ^ ' Z ( W = 1X For /? = 0 the results reduce to the results for the class S* = SQ of starlike univalent functions.
THEOREM 3. · coeff
THEOREM 3. The Taylor coefficients an + x(a), n = 1,2,... of the function (11) for an arbitrary a > 0 have the explicit representation (31)…
THEOREM 3. The Taylor coefficients an + x(a), n = 1,2,... of the function (11) for an arbitrary a > 0 have the explicit representation (31) an + x(a) = Z (oL)kCnk\—— 1, & = 1 \<x 4- 1 (A? — k + l)a -f 1 / where ,™ r (1M. Pn-k + \(«) \
THEOREM 4. · coeff
THEOREM 4. The coefficients A^+1(e; a) of the general "a-convex Koebe functions" (42) for an arbitrary a > 0 have the explicit…
THEOREM 4. The coefficients A^+1(e; a) of the general "a-convex Koebe functions" (42) for an arbitrary a > 0 have the explicit representation (48) K„ + ](c a) = « „ + ,(«) (n = 1, 2 , . . . ), n (49) *„ + ,(«) = 2 (a)kCnk{cx{a\ . . . , cwHt + 1(a) ), * = i where (50) C l f,(c 1(a),...,c n_, + 1(a)) = 2',îl
Theorem 4 · coeff
Theorem 4 yields the simplest combinatorial form yet of that conjecture. CONJECTURE. The coefficients an + x(a) of the a-convex functions…
Theorem 4 yields the simplest combinatorial form yet of that conjecture. CONJECTURE. The coefficients an + x(a) of the a-convex functions (30) satisfy the sharp inequalities https://doi.org/10.4153/CJM-1987-037-2 Published online by Cambridge University Press
THEOREM 5. · coeff
THEOREM 5. The coefficients an + x(a) of the a-convex functions (30) satisfy the sharp inequalities (53) for n = 1,..., [a] + 1 if a > 0 is…
THEOREM 5. The coefficients an + x(a) of the a-convex functions (30) satisfy the sharp inequalities (53) for n = 1, . . . , [a] + 1 if a > 0 is not a positive integer ([a] denotes the greatest integer less than a), and for all n = 1, 2, . . . if a is a positive integer (a = 1, 2, . . . ) where Kn + X(a) are given by (49) and (51), respectively, with equality only for the "a-convex Koebe functions" (42).
THEOREM 6.
THEOREM 6. If the function oo (57) f ) = 2 anzn (ax = 1) « = i w analytic in A, J/ze/7 the following statements are valid. (58) 5Ù£) s l +…
THEOREM 6. If the function oo (57) f\z) = 2 anzn (ax = 1) « = i w analytic in A, J/ze/7 the following statements are valid. (58) 5Ù£) s l + 2 A*", f/œ/2/?, = A7g,? (rt = 1, 2, . . .), n (59) g„ = 2 ( - l ) * - ! ( * -