K-Fold symmetric starlike univalent functions

THEOREM 2.1. [74] Let F(u,v) be regular in the v - plane and in the half - plane Be u > 0; then for every r, 0 < r < 1} the value of mtn min Ke{F(p(z), zp'(z))} p(z) e P \z\ = r ocaurs only for a function

THEOREM 2.2. [7S] Let p(z) be as given by (2.2); then zp'(z) can be written in the form (2.3) zp'(z) = h(p(z)2 - 1) + %fp2 - pp eHi>, ity. . _.

LEMMA 2 . 3 . If w(z) e B^ then for z e A ,

THEOREM 2.4. If p(z) e P]<(A,B)i a > 0, B > 0, then on \z\ = r < 1, - B) + 2aAlrk + aA2r2k „ ^ „ - Ark)(l - Brk) * A - B (A - B) (1 — - T : UUL)2- Bfe (l-ABr M ] 3 R <R , -r2K) 2 where R x = (L/K) \ R 2 = (1 - Ar)/(1 - Br), L = &k(l - A)(l + A T ) , K = a(A -

THEOREM 3.1. The radius of convexity of S*(A,B) is given by the smallest root in (0,1] o/ (i) A2r2k - 1(2 + k)A - kBlrk +1=0, if R1 <RZ, (ii) Lk(A - B)+ 4A(1 - A)lr*k+2lk (A - B)+ 2(1 -

THEOREM 3.2. Let f(z) e S*(A,B); then on \z\ = r < 1, (i) r(l-Brk)(A-B)/kB < \f(z) | <r(l+Brk)(A-B)/kB , i f \ * O , k k r exp(- ^|_J < \f(z) | < r exp(^-) , if B' = 0 ; (U) (l-Avk)(l-Brk)U-(Ukm/B < \f'(z)\ < if B £ 0 , k k

COROLLARY 3.4. The image of the unit disc under a function f(z) e 5*Mj B) contains the disc of centre 0 and radius (l-B)(A~B)/kB if B ? 0, exp(-A/k) if B = 0. 4. Coefficient Bounds for S£(A,B) It is known that if p(z) = 1 + p z + p z2 + ... belongs to P, then \p | < 2 for n = 1,2,3,... E>r the next theorem of this section, we generalise this result to the class P(A,B). The method of proof is essentially due to Clunie [2].

THEOREM 4.1. If p(z) = 1 + p^z + p2zz + ... belongs to P(A,B), then \p | <A - B for n = 1,2,3,... The estimates are sharp for each n.

THEOREM 4.2. If f(z) = z + ak+1zk+1 + a2k+1z2k+1 + ... belongs to S*( 1 - 2a, - 1) , v ] , n = 1,2,3,... v=0 n! .._/, k The estimates are sharp for each n.

THEOREM 4. 3. Let f(z) = z + \ + 1 z k + 1 + a - l k + 1 z l k + 1 + ••• be in A — B Sf(A,B) and put M = [, ., g. ] ^ tTze largest integer not greater than (A - B)/k(l + B). (a) If A - B > k(l + B), then <bTT l^ir-- VB], n = i,2,...,