Coefficient Behavior of a Class of Meromorphic Functions

THEOREM 1. Suppose k > 2 and f £ Ak is given by (2). Then co = lim (1 - r)kl2-lM(r,n exists, is finite, and equals 0 unless s2 is of the form 2/(1 — ze~id)2. If œ > 0, Received September 9, 1974 and in revised form, February 6, 1975. This research was supported in part by NSF Grant GP-43774. 1157 https://doi.org/10.4153/CJM-1975-121-6 Published online by Cambridge University Press

THEOREM 2. Suppose k > 2 and f G Afc is given by (1). Then r \a„] co hm n^nW2)-3 r ( ( * / 2 ) - l ) - Define Ft € A* by 1 1 + 2 - 22 I (4) F^ = -P ( f T ^ and set

THEOREM 3. Suppose k > 2, / G A* is given fr;y (1), and Fk is as above. Then lim |a„|/|i4w| exists, is at most 1, a?zd equals 1 if and 0w/;y if f{z) = eieFk(eidz) for some 6. We note that for fixed / G A*, there exists w(/), depending only on /, such that \an\ ^ \An\ for « _• w(/). This result is clearly false for k = 2, since then ^2(2) = s - 1 + A0 + 2; (and ^4n = 0 for n =• 2). It is also interesting to note that although Fk is not the solution (for all n) of the problem of determining max {\

LEMMA 1. // {zn}iœ approaches 1 strictly tangentially, then lim(l- k|)*/i-l|/'^)l =0.

LEMMA 2. / / coi > 0, /&ew o>i = co2.

Theorem 1 now follows immediately. If coi = 0, then clearly coi = co2 = co = 0. If coi > 0, then Lemma 2 states that co exists and is finite. In the course of the proof of Lemma 2, we showed that if co > 0, then co = lim (1 -r)k,2-1\f'(r)\. If in place of $2(3) = z/(l — z)2 we have s2(z) = 2 / ( 1 — ze~id)2, then « = lim(l-r)*/*-V(«")|. r->l 3. Proof of Theorem 2. Suppose first that co = 0. Since n2\an\ = 0(l)M(rn,f) where rn = 1 — \/n [8], we have an = o(l)nk/2~*y as required. If co > 0, we app$

LEMMA 3. Suppose k > 2 o«rf a> = lim (1 - r ^ V W I > 0. r->l £ + 2 -ï_j/(?l 4 2 (1-*)*'' I T \ 5 1 ( 2 ) / (A+6)/4 https://doi.org/10.4153/CJM-1975-121-6 Published online by Cambridge University Press

LEMMA 4. Suppose k > 2, / Ç Ak, œ > 0. For n ^ 2, ^ r„. = 1 — l/n, con = (k/2 - l)Sl(rn)-w\ gn'{z) = co„(l - z)~k'\ Put In = {6:0 è \0\ g c(l - r n)}. TTzen w ^ z = rne7'*, g'OsVg/^) —> 1 uniformly for 6 Ç 7W, as n —> oo .

LEMMA 5. Suppose f £ Afc(& > 2) is g w n 6^ (2). Assume that co = lim (1 - r)kl2-l\f{r)\ > 0, and suppose that Si in (2) is given by i r \ si(z) = zexp ) — I log (1 — ze u)da(t) f where a is increasing on [ — T, w] with J da(t) = 2, J e-uda(t) = 2(* - 2)/(ft + 2).

Lemma 5 and the fact that the step functions are dense in the class of in- creasing functions will allow us to conclude that Fk is the unique solution (up to rotation) of the constrained optimization problem: max {œ : f £ Ak\. We first give a rather awkward preliminary technical lemma.

LEMMA 6. With k > 2, N ^ 2, X = (xu . . . , xN), and A = (au • • , (IN), set h(x,A) = n (i-^r i / 2. Let e > 0 be given. Suppose that X* = (xi*, . . . , xN*) and A* = (ai*, . . . , aN*) are such that h(X*, A*) = max h(X, A), where the maximum is taken subject to the conditions — 1 ^ Xj ^ 1 and a$ =• 0 (1 ^ j ^ N), |*i -$

Lemma 6, there exists ei > 0 depending only o n / i such that o)N ^ (S/(& + 2) - 2ei)(*+2)/4. Therefore «(fi) = lim œN < (8/(* + 2)) (* + 2 ) / 4 = co*. A7"-*» This completes the proof of Theorem 3. REFERENCES 1. E. Hille, Analytic function theory, Vol. II (Ginn, Boston, 1962). 2. J. W. Noonan, Meromorphic functions of bounded boundary rotation, Michigan Math. J. 18 (1971), 343-352. https://doi.org/10.4153/CJM-1975-121-6 Published online by Cambridge University Press