NORM ESTIMATES OF THE PRE-SCHWARZIAN DERIVATIVES FOR CERTAIN CLASSES OF UNIVALENT FUNCTIONS

Theorem 1.1. Let ϕ, ψ ∈M and suppose that ϕ is univalent and the image ϕ(D) is starlike with respect to 1. Then the inequality ∥Tf∥⩽sup |z|<1 (1 −|z|2)  ϕ(z) −1 z  + sup |z|<1 (1 −|z|2)  ψ′(z) ψ(z) 

Lemma 2.1 (Dieudonn´e). Let F be the family of analytic functions ω on the unit disc with |ω| < 1, ω(0) = 0 and ω(z0) = w0, where z0 and w0 are points in D with |w0| ⩽|z0| ̸= 0. Then the set {ω′(z0) : ω ∈F} is the closed disc centred at w0/z0 with radius (|z0|2 −|w0|2)/|z0|(1 −|z0|2). Furthermore, if ω′(z0) lies on the boundary of the disc, then ω has the form ω(z) = z λ((z −z0)/(1 −¯z0z)) + (w0/z0) 1 + λ( ¯w0/¯z0)((z −z0)/(1 −¯z0z)) (2.1) for a constant λ with |λ| = 1. In particular, we obtain

Lemma 2.2. Let F be a continuous function on the interval [0, 1). Then (1 −r2)|F(r)| ⩽(1 −r2) ˆF(r) ⩽max 0⩽s⩽r(1 −s2)|F(s)|. (2.5)

Proposition 2.4. For a continuous function F on the interval [0, 1), the maximal function ˆF satisfies lim r→1−(1 −r2) ˆF(r) = sup 0⩽r<1 (1 −r2) ˆF(r) = sup 0⩽r<1 (1 −r2)|F(r)|.

Corollary 2.5. For F, G ∈C([0, 1)), sup 0⩽r<1 (1 −r2)( ˆF(r) + ˆG(r)) = sup 0⩽r<1 (1 −r2) ˆF(r) + sup 0⩽r<1 (1 −r2) ˆG(r). 3. Proof of the main theorem For ϕ ∈M, we define the functions hϕ and kϕ in A by the relations zh′ ϕ(z) hϕ(z) = ϕ(z) and 1 + zk′′

Lemma 3.1 (see Theorem 1 in [9]). Suppose that a function ϕ ∈M is univalent and ϕ(D) is starlike with respect to 1. Then f ′ ≺k′ ϕ holds for every f ∈K(ϕ).

Theorem 3.2. Let ϕ be as in Lemma 3.1. If f ∈K(ϕ), then ∥Tf∥⩽∥Tkϕ∥holds, where kϕ is the function given in (3.2). We now prove Theorem 1.1. It is convenient below to introduce the class B of analytic functions ω on the unit disc with |ω(z)| ⩽|z|. Let f ∈C(ϕ, ψ). Then, by definition, there is a function h ∈K(ϕ) such that f ′/h′ ≺ψ. By Lemma 3.1, we see that h′ ≺k′ ϕ. Let ω1 and ω2 be analytic functions in B satisfying h′ = k′ ϕ ◦ω1 and f ′/h′ = ψ ◦ω2. Conversely, for any pair of functions ω1, ω2 ∈

Lemma 4.1. For real numbers A, B with −1 ⩽B < A ⩽1, the inequality |1 + Az| |1 + Bz| ⩾(1 + εA|z|)(1 + εB|z|) holds for every z ∈D. Here, ε = 1 when A + B ⩽0 and ε = −1 when A + B ⩾0.

Lemma 4.2. If −1 ⩽B < A ⩽1, then E(A, B) = 2 1 −AB +

Theorem 4.3. Let −1 ⩽B < A ⩽1. If f ∈K(ϕA,B), then ∥Tf∥⩽ 2(A −B) 1 + √ 1 −B2 , (4.3) and equality holds when f = kϕA,B.

Theorem 1.1 together with Lemma 4.2 now yields the following result.

Theorem 4.4. Let −1 ⩽Bj < Aj ⩽1 for j = 1, 2. If f ∈C(ϕA1,B1, ϕA2,B2), then ∥Tf∥⩽2(A1 −B1) 1 +

Corollary 4.5. For 0 ⩽k ⩽1, functions f ∈S∗(ϕ−k,k) satisfy the inequality ∥Tf∥⩽ 4k 1 + √ 1 −k2 + 2k.

Theorem 5.1. Let 1 ⩽p < ∞. Suppose that ϕ ∈M is univalent, ϕ(D) is starlike with respect to 1 and k′ ϕ ∈H1, where kϕ is given by (3.2). Then C(ϕ, ψ) ⊂Hp for every ψ ∈M ∩Hp.

Theorem 5.2 (Aleman and Siskakis [1]). Let h be an analytic function in the unit disc and let 1 ⩽p < ∞. The operator f →1 z  z 0 f(t)h′(t) dt maps Hp continuously into itself if and only if h ∈BMOA.

Corollary 5.3. Let −1 ⩽B < A ⩽1. If −1 < B or A ⩽0, then, for any number 1 ⩽p < ∞, the relation C(ϕA,B, ψ) ⊂Hp holds for all ψ ∈M ∩Hp. If B = −1 and A > 0, then, for each 1 ⩽p < ∞, there exists a function ψ ∈M ∩Hp such that the relation C(ϕA,B, ψ) ⊂Hp does not hold.

Theorem 3.16]). The above ranges for p are sharp. Acknowledgements. Y.C.K. was supported by the Korea Basic Science Research Foundation, under grant no. DP0022. T.S. was partly supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 9740056 and 14740100. References 1. A. Aleman and A. G. Siskakis, An integral operator on Hp, Complex Variables Theory Applic. 28 (1995), 149–158. 2. P. L. Duren, Theory of Hp spaces (Academic, 1970). 3. P. L. Duren, Univalent func