On the integral means of univalent, meromorphic functions

PROPOSITION 1 ([1, Theorems A and A' and Proposition 5]). (i ) Let D be a domain containing r0 > 0 and having a clas- sical Green's function. Let u be the Green's function of D with pole at r0. (It is assumed here that u is defined on the extended plane by defining it to be zero on the complement of 2λ) Then u\re ίθ) = u*(re iβ) + 2π log + — r0 is subharmonic in the upper half-plane. (ii) Let D and u be as in (i) and suppose further that D is circularly symmetric. Let D

PROPOSITION 2 ([1, Proposition 2]). For g^L ι[—πfπ\9 g*(θ) = [' G(x)dx , 0 ^ θ ^ π , where G(x) is the symmetric nonincreasing rearrangement of g. (For the definition of G(x) see [1] and [2].)

PROPOSITION 3 ([1, Proposition 3]). For g,heL ι[—π,π) the following are equivalent. (a) For every convex nondecreasing function Φ on ( — 00,00), Γ Φ(g(β))dθ ^ Γ Φ{h{θ))dθ . (b) For every ίe( — «, oo), \*_ [g{θ) - t] +dθ ^ Γ [h(θ) - t] +dθ . ( c) g*(θ) ^ h*(θ), O^θ^π.

THEOREM 1. Let Φ be a convex nondecreasing function on (—00, oo). Then for all fe Σ(p) and 0 < r < 1, \'χΦ(±

Theorem 1. T H E O R E M 2 . L e t feΣ(p), then for all λ , — oo < λ < oo, and 0 < r < 1, (2.14) \'r \f{re ίθ)\ λdθ ^ 3* Applications of Theorem 2*

THEOREM 3. Let fe Σ(p) and 0 < r < 1, then for \z\ — r. (3.1) Fp(-r)£\

THEOREM 4. Let feΣ(p) and f(z) = 1 + M < then (3.2) p V The inequalities are sharp.

Theorem 2 directly. In [4] sharp estimates on the quantity \f'(z)/f(z)\ were obtained for feΛ*(p). Making use of Theorem 4, we can now extend the results to the class Σ(p).

THEOREM 5. Let feΣ(p) and we A, w Φ p, then (3.3) ( l - M) - | w I 2) f\w) f(w) (1 + \a\Y \a\ where a — (p — w)/(l — pw). Moreover, given we A, w Φ p, there exists a function feΣ(p) for which equality is obtained on the right side o/(3.3) and similarly for the left side of (3.3).

THEOREM 6. Let Φ be a convex nondecreasing function on (-oo, oo). Then for all feΣ(p, q) and 0 < r < 1, {re iθ)\)dθ S \ π πΦ(±log \G{p>q)(re iθ)\)dθ .

THEOREM 7. Let fe Σ(p, q), 0 < r < 1, — oo < λ, < oo, then \f{re iΘ)\Hθ ^ 5* Applications of Theorem 7* Arguing as in Theorem 3 we obtain the following.

THEOREM 8. Let feΣ(p, q), then for \z\ = r.

THEOREM 9. Let feΣ(p, q) and f(z) = 1 + Σ ϊ ^ α ^ i | s | < P» (5.1) l p ^"^K 1 "" Pg) ^ | t t ι | ^ (P + g)^ + Pg) . Pg ~~ ί>g jBoίA inequalities are sharp.

THEOREM 10. Let feΣ(p, q) with f(z0) = 0, \zo\ = q, then for