Starlike functions with a fixed coefficient

LEMMA 1. For a given point z in A, let F be regulas in a neighbourhood of each point p(z), p £ Pk(l, — 1). Then the functional ReF(p(z)), z £ A, attains its maximum and minimum over the class P/b(l, —1) only for functions of the form {l + e-iezk)/(l-e-iezk). PROOF: See Pfaltzgraff and Pinchuk [6, Theorem 7.3]. | THEOREM I . If p{z) e Pk,b{A> B)> tllen °n \z\ = r < 1> l+b(l-A)rk-Ar*k , , _ „ Re{P(z)}- ' ^ - ° ^ - ° ^ * " • * - ! . ». 6 . -

LEMMA 2. If w(z) e Bk, then for 2 e A, (2.8) \zw\z) - kw(z)\ ^ V

THEOREM 2. Let a ^ 0, /3 ^ 0, k = 1 , 2 , 3 , . . . , \z\ = r < 1, L = /?fc(l -A)(l + Ar2k), K = a(A-B)(l-r2k) + (3k(l - B)(l + Br2k). If p(z) € Pk,biA> B)> then °n \z\=r,

THEOREM 3. Let a ^ 0 ,/? ^ 0, k - 1,3, 5, . . . , \z\ = r < 1, £> = (r* + 6)/(l + 6rfc) , C = rfcl>. Under tie following conditions: (i) A + B>Q,AB<A + B, (ii) i _ r2t + (A + 5)r2fc - 2rfc(l + ABr2k)D + r2k(A + B- AB(l - r2*)) D 2 > 0 , 0 < r < l , w e iave for p(z) E Pk,b(A, B) that (A-B)(3krk rk + (l-r2k)D-rkD2 The result is sharp. PROOF: Write Then, for p(z) 6 Pk,b(A, B), we have which yields

COROLLARY 1. Let A, B, b be such that (3.1) + Ar2fc)(l - BC)2 < [(A - B)(l - r2k) + jfe(l - B)(l + Br2k)](l - AC)2 for 0 < r < 1. Tiien tiie radius of convexity of S£ b(A, B) is given by the smallest root in (0, 1] of the equation (3.2) [{A + B) (1 - r2k) — 2Jfe(l — ABr2k)} (l + 6(1 - A)rk - Ar2k) (l + 6(1 - B)rk - Br2k) + k(l - A)(l + Ar2k) (1 + 6(1 - B)rk - Br2k)2 + [(A - B)(l - r2k) + fc(l - £)(1 + Br2k)](l + 6(1 - A)rk - Ar2k)2 = 0.

COROLLARY 2. The radius of convexity of SjJ>6(l, I/a - 1) is given by the smallest root in (0, 1] of equation (3.2) with A = l, B = I/a - 1. For the class 5jJ6(a, 0), condition (2.15) becomes a ^ 2C/(l + C2) . Thus, for this class, we have

COROLLARY 3. The radius of convexity of SjJ 6(a, 0) is given by the smallest root in (0, 1] of equation (3.2) with A = a, B = 0 and a ^ 2C/(l + C2) . Using Theorem 3, we have

COROLLARY 4. Let A, B , b satisfy conditions (i) and (ii) of Theorem 3. Then the radius of convexity of S£ b(A, B) is given by the smallest root in (0, 1] of the equation (3-3) (1 - r2k) (l + 6(1 - A)rk - Ar2k) [1 + (26 - A - B)rk + (62(1 - A)(l -B)~A- B)r2k -b{A + B- 2AB)r3k + ABr4k] -{A- B)krk(l + 6(1 - B)rk - Br2k)(l + 2rk - (1 - 6)r2fc - 2r3k - brik) = 0. Again, it can be checked that the LHS is equal to 1 at r — 0 and 0 at r = 1. Thus, the equation has at least one root within (0, 1]. For t

COROLLARY 5. The radius of convexity of 5£ b(a, 0) is given by the smallest root TQ in (0, 1] of equation (3.3) with A = a, B = 0 for such a that 1 - 2rk + (2a - l)rj* > 0. For the class S^b(a, —a), we note that condition (i) of Theorem 3 is always sat- isfied, while conditions (ii) becomes 1 _ 2r* - (1 - a2)r2k + 2a2rik - a V * > 0, 0 < r < 1. Thus for this class, we get

COROLLARY 6. The radius of convexity of 5£ b(a, —a) is given by the smallest root rx in (0, 1] of equation (3.3) with A = a, B = —a for such a that 1 _ 2 r* - (1 - ec2)r\k + 2oc2r\k - «2r\k > 0. REFERENCES [1] V.V. Anh, 'fe-fold symmetric starlike univalent functions', Bull. Austral. Math. Soc. 32 (1985), 419-436. [2] W. Janowski, 'Extremal problems for a family of functions with positive real part and for some related families', Ann. Polon. Math. 23 (1970), 159-177. [3] T.H. MacGregor, 'The rad