Coefficient estimates for certain multivalent functions

LEMMA 1. If fk{z) = Σ?-,-* ^z n = Σί«p-» C™M* where u{z) = z(l - z)~ 2 and zk+1 is real, then fk+1(z) = φ(z, zk+1)fk(z) = ΣSW-*-i αϊ +1)«* is αZso α polynomial ΣS=?.-ft-i^ +1)^ m w. 1/ sgnαj* } = ( — l) p" j for p — k ^* j <: p, then for fixed n and fixed zk+1 > 0, the signs of af +1) alternate for p — k — l<^j<^>p and for n > p, (2.1)

LEMMA 2. Suppose zm = zm. Then there are real numbers δ0, blf and &2 so that β u(z) = «(1 —

THEOREM 1. Suppose f(z) = Σo° anz n £ M(p) and that each an is real. Then for 1 ^ j ^ p < n, (ooλ \n I < v 2j(Ή, + p)\ , , h (p + j)l (P - Λ! (n - p - 1)! (^ 2 - ?) l* il '

LEMMA 3. Let gk(z) = Σ?-* Vn k)z n, with \bp%\, , | δ ^ | fixed and let zk+1 < 0. Then \ b {k+1) \ is maximal when the signs of Vp alternate for p — k <* j <^ p.

Lemma 1 that for n > p, K + Ί ^ Σ D(p,n,j)\a?^\. j=p-k-l The result now follows if / has only real zeros. If / has a nonreal zero zm, since / has real coefficients, / must also have zm as a zero. If we use Lemma 2 in place of Lemma 1, the argument in this case is similar to that in the case of real zeros. We omit the straightforward modification. This completes the proof of the theorem. 4* Applications* In the proof of Theorem 1 we showed that the only possible extremal functions were of the for

THEOREM 2. Let f(z) = Σanz n eK(p) and suppose each an is real. Then for n ^ p, I «• I ^ Σ D(P, n, j) \as\ , where the D(p, n, j) are given by (3.2). A similar result holds for the class Vk(p) of multivalent func- tions of bounded boundary rotation. Recall that fe Vk(p) if there is a p, 0 < p < 1 so that if 0 < p < r < 1, (4.i) and (4.2) lim sup i JΓ Re {l + ^ / " ( r ^ ) | \ ^ pkπ , Jo

THEOREM 3. Let f(z) = Σanz n e Vk(p) and suppose each an is real. Then for n > p, fn, j)\ad\ , where c(p, n, j) is defined by c(p, n, j) = 0 (j > p) , c(ny p) = JL x {coefficient of ^"^ in ( — k — 1) = c(p, n + l,p — k) + c(p, ^ — 1, p — fc) + c(p, n, p — k + 1) p + 1, p - k) T h i s e x t e n d s a r e s u l t i n [ 8 ] f o r t h e c a s e n = p + lifaί= = ap^2 — 0. We note that this technique works for any class of functions having a representation similar to (1.1) and where the coefficients o