Integral means inequalities, convolution, and univalent functions

Theorem 1. Suppose that f, F ∈Hol(D) with F being bound preserving. Then (1.2) Mp(r, f ⋆F) ≤Mp(r, f), 0 < r < 1, whenever 1 ≤p ≤∞.

Theorem 2. There exist two functions F1, F2 ∈Hol(D) with log |Fj|
∗≤ log |I|
∗, for j = 1, 2, and such that (2.1) the inequality log |F1 ⋆F2|
∗≤ log |I ⋆I|
∗does not hold. Here, I is the identity element of the convolution defined in (1.1), that is, I(z) = 1

Theorem 3. There exist f, F analytic and univalent in D such that F is convexity preserving and with the property that the inequality (log |f ⋆F|)∗≤(log |f|)∗does not hold. The following lemma will be used in the proof of Theorem 3.

Lemma 1. Let f, F ∈Hol((D) and suppose that F(0) = 1, F is convexity preserving, and that f and f ⋆F are zero-free in D and satisfy the inequality (log |f ⋆F|)∗≤(log |f|)∗. Then we also have that (2.5)  log

Theorem 4. Suppose that f ∈Z and let F be an analytic function in D which is convexity preserving. We have, for every p > 0, (2.7) Mp(r, f ⋆F) ≤Mp(r, f), 0 < r < 1.