Weakly Close-to-Convex Meromorphic Functions

THEOREM 1. Iff is in I(p), there exists g in A*(/?), an a, |a| ^ 7r and a P{z) in &* such that for z in A — {/?}, (1.3) f\z) = -g(z)P(z). (z-p)(l -pz)

COROLLARY 1. Iff(z) is in I(p), thenf\z) 7^ 0 for z^p. Because of the corollary, there is no loss in generality in assuming that/r(0) = 1 for f(z) in /(/?), 0 < p < 1. In the sequel we therefore make the added assumption that/'(()) = 1 for/ in l(p),0<p < 1. Because of Theorem 1, we also define another class of functions I*(p). We will say that/(z) is in /*(/?), 0 < p < 1, if it is analytic in A — {p} with a simple pole at z = p and/'(()) = 1, and there exists g(z) in A*(p), a P(z) in <P* and an

LEMMA 1. If g is in A*(p) and (3P — Res(g;/?) then l-p2 (z+p\ is in Z* and (3P — (1 — p2)/G(—p). Conversely, if G is in X* and (3P = (1 — p2)/G(-p), then X-p1 \l-pzj is in A*(p) and (3P = Res(g;p).

THEOREM 2. Iff is in J*(p)[I*(p)] and ap = —Res((z — p)f'\p)[ap = Res(/";/?)] then there exists h in /*(0)[/*(0)] such that \-p2 \l-pzj Conversely, if h is in /*(0)[/*(0)] and ap = l/h\—p), thenf{z), defined by (2.1) isinJ*(p)[I*(p)].

Theorem 2, that functions in I*(p) need not be univalent. 3. Integral means. In this section we use a technique of Baernstein [1] as employed by Leung [7] to obtain bounds on the integral means of |/'|. For this purpose we first mention some results that will be used. For g(x), a real valued integrable function on [—7r, 7r], the Baernstein *- function is defined by g*(&)= sup [ g, \E\=26JE \E\=26JE for 0 < 0 ^ 7T, where \E\ is the Lebesque measure of the set E in [—7r, IT]. Statements A, B and C

LEMMA 2. (A) For g,h in Ll[—IT, TT], the following are equivalent: (i) For every convex non-decreasing function O on (-co, oo) <3>(g(x))dx ^ / &(h(x))dx. •IX J —TT (ii) For every t in (—oo, oo) / 7T pit [g(x) - t]+dx ^ / [h(x) - t]+dx. •IX J —IX

THEOREM 3. For any f in J*(p) and every convex non-decreasing function O on (—oo, oo), / " <P(± log \f'{rew)\)dQ ^ f 0(± log \Ff 2(rew)\)d6 J—IK J— 7T where F2 is in J*(p) and is defined by (1.9).

COROLLARY 2. For any f in J*(p) and any real A, (3.5) f \f'{re'etdO ik f \F'2(rei9)\xd6. J—TT J— TT

COROLLARY 3. Iff is in /*(/?), then for r ^ p (3.6) F^(-r) è \f'(rew)\ ^ Ff 2(r) where F'2(z) is defined by (1.9). https://doi.org/10.4153/CJM-1989-027-7 Published online by Cambridge University Press

LEMMA 3. If g is in A*(p) and CO g{z)=\+YJbnZ" n=\ for \z\ <p, then for n ^ 1 , , ^ (l+p)(l -P 2") f4 n \h \ < - —- K (1 - / ? ) / ? " Equality is attained in (4.1) /?_y the function -p(\+z)2 g(z) (z-p)(l

THEOREM 4. Iff is in J*(p) and oo f(z) = a0 + z + ] T anzn n=2 for \z\ <p, then for n^2 \un\ = t*M j _ n n-\ r

LEMMA 4. For/ m J*(p) (4.2) />2(l-/>) Of„ <^o+z) (1+P)3 " ' ^ - (1-/7)3- 77zese bounds are sharp, being attained by F\ and F2 given by (1.8) and (1.9).

LEMMA 5. If h is in /*(0) and 1 oo h(z) - - + aflogz + y^c„z" n=0 forO< \z\ < 1, then \d\ ^ 4. Equality is attained by h in /*(0), defined by -(1+z) 3 h\z) z2(l-z)'

LEMMA 6. 7/"/ /s /« J*(p) and oo /(z)=7-^-r+rf iog(Z - /»>+y; c„(Z - p)« for {z : |z — p\ < 1 — p} — {z :/? ^ z < 1}, tfie« and the bound is sharp.

Lemma 6 now follows by applying Lemma 5. For sharpness, consider the function X-p1 \l-pzj where h in /*(0) is the function given in Lemma 5 and ap = l/h'(—p). For this function we have rf/ap = - 4 / ( l - p 2 ) .

THEOREM 5. Iff is in J*(p) and oo /(z) = tf0 + z + ] T a^z" for \z\ <p, then suP/e/'Cp) lim «—+00 where a^ and ap 1^ are the coefficients in the expansion about z = 0 and the residue at z — p of /i(z) -p\\-l? d - P)Hz - P)(i - Pzy