Methods and Applications of Analysis © 1996 International Press

Lemma 1. Let /ii,/i2 £ AQ and let ui G G(hi), 0J2 G G(/i2). Then UJ1UJ2 G G(hih2).

Lemma 2. Let $(z, w) = Yl™k=o an,kZn'wk be a formal power series in two variables with an,k > 0 (n, k = 0,1,...), and let g,h G AQ . Then for any u G G(h) and iV=l,2,..., |{/}iv| < {F}N (1) where f(z) = $(z,(g*u;)(z)) (2) and F(z) = $(z,(g*h)(z)). (3) The equality in (1) holds for {F}^ ^ 0 iff arg{(^ * uj)k}N-n = constant and \{(g*w)k}N-n\ = {(g*h)k}N-n provided that an,k{(g*h)k}N-.n ^ 0, 0 < fc + n< N. Received July 11, 1995, revised September 4, 1995. 1991 Mathematics Subject Clas$

Lemma 3. Let g, h G AQ and UJ G G{h). Let $(z,w) = zew, and let f and F be defined by (2) and (3), respectively. Then M(r,f)/F(r) and |/(re^)|/F(r) for any 0 G [0, 2IT) are non-increasing functions of r in (0,1). // 1/(^6^)1 = F{r) for some r G (0,1) and 6 G [0,27r), then f(z) = eieF(e-iez) for each zeE.$

Lemma 4. (see e.g., [15, Ch. 1]) // rj(x) and (p(x) are continuous and iy(x) is of bounded variation in [a, 6], and if fb n(x) = — I (p(t)dv(t), x G [a, 6], Jx then pb pb I rj(x)dfi(x) = / rj(x)(f(x)du(x). J a J a

Lemma 5. If ^{x) is of bounded variation in [0,1], then

Theorem 1. A necessary and sufficient condition that the convolution of a fixed func- tion g G AQ and an arbitrary function f G AQ have the expression {f*g)(z)= / f(zx)dn(x), Jo zeE, (9) where rl du(t) rtx) = -f ^-, ^^(0,1], [XLL(X)}X=0 = IS(0)-IS(0+), (10) and u(t) is of bounded variation in [0,1], is that the sequence i+l-Z (ii) be bounded. Each function fi defined by (10) generates such a function g G AQ by (9) with f(z) = zn,n=l,2,....

Theorem 1 that for every n = 1,2,..., an= xnd^{x) = / xn 1di/(x). Jo Jo As mentioned above, there is at most one such a function z/ which is of bounded variation and normalized in [0,1]. We conclude that for every admissible function g e AQ, there is at most one rep- resentation (9) which is valid for all functions / G AQ (or even for a fixed function / with {f}n 7^ 0, n > 1) if fi is defined by (10) and ri,)_rt«+o) + M»-o)| „,„_„_ Indeed, if there exists another function /i(a x)

Lemma 6. Let ii{x), |/x| < oo, be nondecreasing in (0,1], and let JQ xd/i(x) < oo. Then the limit value [X/J,(X)]X-O+ exists and equals 0.

Lemma 6 follows. Now let fjL(x) < 0 for 0 < x < b < 1. The formula for integration by parts gives rb rb x ix xdfi(x) = bfj,(b) - efjb(e) - / fi(x)da for any e G (0, b). It follows that rb / /jJ(x)dx > bfi(b) - / xdii(x) > -oo, Je Jo+ 0+ and therefore, the finite limit value /0 fi(x)dx exists. Hence, the finite limit value 7 = [£/jb(e)]e=o+ exists also. Clearly, 7 < 0. If 7 < 0, then for some c G (0, b) and a G (7,0), XIJ,(X). < a for each x G (0,c). Again using the formula for integratio

Theorem 2. ^4 necessary and sufficient condition that the convolution of a fixed func- tion g G AQ and an arbitrary function f G AQ has the expression (f*9)(z)= I f(zx)dfi(x)i zeE, (13) where fi G M., is that the sequence {g}n, n>l is completely monotonic. Each function /i G M generates such a function g G AQ by (13) with f(z) = zn, n = 1,2,....

Corollary 1. Under the conditions of Theorem 2, /i G Mo iff the sequence {g}n, n > 1, is minimal completely monotonic.

Theorem 3. Let h G ^4j, // G M, and let F be defined by (21). Then, for any f(z) = 2:exp{/0 uj(xz)dfi(x)} G ECh(^) with UJ G G(/I), (a) |{/(z)}iv|<{}iv,iV>2, (b) \f(z)\<F(\z\),zzE. For some natural N > 2, let {h}n > 0 (n — 1,..., k) where k = min{A^ - 1, L(/x)} and L(IJL) ^ 0 is defined by (22). Then the equality in (a) holds iff {w}n = {Hze 7 '6)}^, for every n = 1,..., k and for some 9 G [0,2ir). If for some z = reie G E - {0} \f(z)\ = F(r), then for each z G E, f(z) = ei0F(e-i0z).

Lemma 2 gives equality in (a) if and only if arg{}iv_1 = constant and {tp 1}^! ={4>1}N-1 for / = l,...,iV-1. Let I = N - 1. Then we get |ai| = h > 0 and arg{(pz}Ar_1 = (JV-l)argai for I = 1,..., AT-l. The case k = 1 is trivial. If k > 1, then for every 1 = 2,..., iV-2, W 1}M-I ^S a sum 0^ Products of coefficients ai, a2,. •. and of a positive constant such

Theorem 4. EC (logx2) = 5*.

Theorem 5. For every q G (0,1), PSq = EC(nq) where }iq G Mo and is defined by 'N{x) = \ogq2\ xe[q\qk-1) (30) /orfc = 0,l,....

Theorem 6. Let /JL G Mo, f G EC(ii), and let F^ be defined by (27). Then (a) \{f}n\<{Ffi}n,n>2, (b) M(r, f)/F^(r) and \f (re19) l/F^r) for any 9 G [0,27r) are nonincreasing func- tions of r in (0,1). If\z\=r<l, (c) -FM(-r)<|/(z)|<^(r); (d) arg < max |C|=r -.^ ^^ / ^ nJ^ ^V, „9„9^(^)- Vo 1 xr sin 0 ^GIO^TT) J0 1 — 2xr cos 0 + :E2r2 c T/ie equality sign in (a) occurs iff f(z) = e-i$F4eiez) (32)$

Theorem 3 and Lemma 3. If J0 xdfi(x) = 0, then f(z) = F^{z) = z. Otherwise, according to Theorem 3, the equality in (a) for some n > 2 implies that |{u;}i| = 1, where u G W is defined by (23). It follows that in this case LJ(Z) = uj*(el9z) for some 9 G [0,27r). The representation (25) implies that for each u G W < Re{uj(z)}, z G E, 1 + 1 . with equality iff w(z) = w*(eiez) where 9 = TT - argz (see e.g., [4, Ch. 7]). Using this, (23), and (27), we obtain the left-hand inequality in (c). O

Corollary 2. The Koebe domain for the class EC(fi) is the open disk centered at zero with the radius r(M)=exp{-^ j^M*)}- In particular, we have the well-known value r(logx2) = 1/4 for the class 5* and a new result r(fiq) = exp < 2(logg) Sfclo T+^ \ for the class PSq where fj,q is defined by (30).

Theorem 7. Let /i G Mo, then EC(fi) C GS iff f1 ^M < oo. (33) Jo 1 — x

Theorem 8. Let fj, G Mo and for some real numbers Xk, k = 1,..., n liny^Xklk xkdfi(x)) sinfc0 = O. (39) For each f G EC(fi), let n M(/) = X>*|2 (40) k=i where the logarithmic coefficients /?& are defined by (38). Finally, let F^ be defined by (27). Then F^ maximizes the functional M(/) on the class EC(fi). If the sum in (39) is not identically zero, then only the function F^ and its rotations maximize this functional on EC(fi). mm 0€p