Multivalent and Meromorphic Functions of Bounded Boundary Rotation

LEMMA 1.1. Letf(z) 6 Vk{p). Then dd S pkir. r-»l~ J 0 I V dd exists.

THEOREM 1.2. Let f(z) G Ffc(£). Then f(z) is at most max [p, {pk/2 — 1}] valent, where {pk/2 — 1} denotes the smallest integer greater than pk/2 — 1. Received August 12, 1973. This research is part of a Ph.D. thesis written under the direction of Professor William E. Kirwan at the University of Maryland. 186 https://doi.org/10.4153/CJM-1975-024-9 Published online by Cambridge University Press

COROLLARY 1.3. Le/ /(s) G V(p) with k < 2 + 2/£. 77w?» / w at most p-valent in U. Our next goal is to obtain representation formulas for Vk(p). We will need to use the functions (1.3) *(*,*,) = (g - s,) (1 - g>) which have been employed by Hummel [6] and others.

LEMMA 1.4. Letf(z) = aqzQ + . . . (q è 1) belong to Vk(p) and have non-zero critical points zi, . . . zp-qi counting multiplicities. Let «(*)= Pff ^(M/r'/'(s)&. •/ o j=i JTzeft g(g) /zas £ — 1 critical points all at z — 0 aw<i g(g) G T/ A;(^).

THEOREM 1.5. Let f(z) = aqza + . . . G F*(£) a»d suppose f(z) has non-zero critical points z\> . . . gp_?, counting multiplicities. Then: re f f"(reie)\ \reid) ] fire") dd + e. dO ^ pkw + e. https://doi.org/10.4153/CJM-1975-024-9 Published online by Cambridge University Press

THEOREM 1.6. Let f(z) = aqzq + . . . G Vk(p) have non-zero critical points Zi, . . . zp-Q, counting multiplicities. Let R\ = max |z;-|, R^ = min \ZJ\. Then withz = reid, /I _ r\h>&-2) I j P-Q l/'OOl è ! , Im^-Wf-/-1 II (r - W)d - N0,*i < r < 1 (L ~r r) Up;I ^=1 l/'(*)| ^ (T+7p'^n^ir'"1 H ( N ~ ")(1 " Nr)'° < r < R*

THEOREM 1.7. Let f(z) = aqzq + . . . G ^(/O Aaue P — q non-zero critical points zi, . . . 2P_2, counting multiplicities. The f{z) is q-valently convex for \z\ < rq, where rq is the least positive root of s[(>+-i)fe)+(>~i)(**)] £i ( W - r ) d - Wr) U- https://doi.org/10.4153/CJM-1975-024-9 Published online by Cambridge University Press

COROLLARY 1.8. Let f(z) = avzv + . . . G Vk(p). Then f(z) is p-valently convex for \z\ < |(ft — (ft2 — 4)1/2) and this result is sharp. 2. Coefficient Bounds for VK(p). Goodman [4] has conjectured that if f(z) = 2n=i a,nzn is at most ^-valent in U, then ior n ^ p + 1, (2 i) \a\< Y 2j(»+f>)! ,. , (2.1) \an\ ^ ^ (w2 _f)(p + j ) [ { p _ m n

THEOREM 2.1. Letf(z) = a^v + .. . € Vk(p). Then (p + l)\ap+1\ ^p2k\aP\ (p + 2)\aP+2\^ (i-f + p)p\ap\ (P + 3)|ap+3| g & (*>V + &p + 2)10,1. .4// of these results are sharp, with equality for F'(z) = p ap[gf(z)]p 1 where *«-fer-']-

LEMMA 2.2. Let g(z) = z + b2z2 + . . . G Vk(p). Then for any integer P è 1, \Spb3 - 2p(p - l)b2 2\ ^ p2k2/2 - p, with equality for *<*>=i :(^r-i].

THEOREM 2.3. Le*/(z) = a„_12p-1 + . . . € 7*0). r*ew (p + 1)K+1| ^ ^*lo,| + (p - l ) ^ ^ 2 ^ - /» + 1). https://doi.org/10.4153/CJM-1975-024-9 Published online by Cambridge University Press

LEMMA 2.4. Let g(z) = z + b2z2 + . . . Ç Vk have real coefficients. Then if p ^ 2, |1 + 3£fr3 - 2p(p + 1)62 2| ^ p2k2/2 - p - 1, and / t e r««ft w ^ar/?.

THEOREM 2.5. Le/ /(s) = ap-izp~l + . . . G Vk(p)(p > 2) Aaz/e raz/ a?e$- cients. Then (p + l)\aP+1\ ^ p2k\ap\ + (p- l)\ap^\(p2k2/2 - p - 1) awd Z/itfre is a function in Vk(p) for which equality holds.$

Theorem 2.3 (2.3) (p + l)ap+i = papd+ (p - l)ap-i[z0/z0 + c2 - Ci2] = 2pb2 ap+ (p - l)op_i[l + Spbz -2p(p + l)b2 2]. Since z0 and the an are real, the cn and hence the bn are real. By Lemma 2.4, since the bn are real, |1 + 3pbz - 2p(p + l)b2 2\ ^ p2k2/2 - p - 1. Since g(z) Ç Vk, \b2\ ^ k/2 and the result follows. To see that this result is sharp we consider

THEOREM 3.1. Letfiz) G Vk(p). Then a = lim (1 - rt{k+2) M(r,f') r-»l exists. If a > 0, there is a unique 0O so that a = lim(l-r)hpik+2)\f'(reid°)\. r-»l

THEOREM 3.2. Letf(z) = TA anzn € F*(/>). Then Sw^^ = r ( ^ + 2))' where a is the constant of Theorem 3.1.

THEOREM 4.1. Letf(z) G Vk*(p). Then: P + 1 ^ N(oo,f) S (pk + 2p + 4)/4 and 0 £ N(0,f) g £(* - 2)/4.

LEMMA 4.2. Let f{z) be meromorphic for \z\ < R,f'(z) j* 0 on \z\ = R. If />{ 1 + f ^ \ | dd < 2AM - N(co,f ) + 1], where [ ] denotes the greatest integer function, then f is at most M valent and at least max [27V(oo,/) - M, 1] valent for \zl g J?.

COROLLARY 4.3. Let f(z) Ç F*(£) have q poles in U. Then f(z) is at least max [q + 1 — £&/2, 1] valent and at most pk/2 + q — 1 valent in U. We note that if k < 2 + 2/£, then for f sufficiently near 1, 2T JO I I / (re ) ) I and hence f(z) belongs to the class K*(p) of meromorphic close-to-convex functions of order p defined by Livingston [10]. The following result is similar to Theorem 1.5 and its proof will be omitted.

THEOREM 4.4. Let f{z) £ Vk*(p) and suppose f (z) has zeros at ai . . . an and poles at jSi, . . . Pn+p+i, counting multiplicities. Then there are two univalent starlike functions S\(z) and s^iz) such that /'(2) = ?4î[n KZ,B,)\ n VKWL^J p(k-2) [¥ï ipa+2) We note that Theorem 4.4 gives distortion theorems analogous to Theorems 1.6 and 1.7, but we do not state them here.

THEOREM 4.5. Let f(z) £ Vk*(p). Then a = l i m ^ (1 - r)**<*-2> M(r,f) exists. For k > 2, if a > 0, there is a unique 0O such that a = lira (1 - r)^k~2)\f '(rei6°)\. https://doi.org/10.4153/CJM-1975-024-9 Published online by Cambridge University Press

Theorem 1.5. We now turn to the problem of estimating the coefficients of a function /(*) 6 Vk*(p).

THEOREM 4.6. Let f{z) = Y,n=-Q anzn, with k > 2 + 2/p. Then if a denotes the constant of Theorem 4.5, T M _ a nni jP(*-2)-2 — Y\\p{k - 2)]