Mediterr. J. Math. (2017) 14:19

Theorem 1.1. f ∈S∗⇒|H3(1)| ≤16, (2) f ∈K ⇒|H3(1)| ≤0.714 . . . , (3) f ∈R ⇒|H3(1)| ≤0.742 . . . . (4) All results are sharp. Moreover, Babalola claimed that the extremal functions for S∗are the rotations of f(z) = z (1−z)2 . The estimates given in Theorem 1.1 are true, but rather weak, and so, not sharp! We improve these estimates in the subsequent section. There we also discuss particular subclasses of S∗, K and R consisting of functions with so-called n-fold symmetry. The results for these cla

Lemma 1.1 [13]. If p ∈P, then the sharp estimate |pn| ≤2 holds for n = 1, 2, . . ..

Lemma 1.2 [8]. If p ∈P, then the sharp estimate |pn −pkpn−k| ≤2 holds for n, k = 1, 2, . . . , n > k.

Lemma 1.3 [2]. If p ∈P, then the sharp estimate |pn −μpkpn−k| ≤2 holds for n, k = 1, 2, . . . , n > k and μ ∈[0, 1].

Lemma 1.4. If p ∈P, then the sharp estimate |pn −pk2pn−2k| ≤6 holds for n, k = 1, 2, . . . , n > 2k. The last lemma immediately follows from Lemmas 1.1 and 1.2. It can easily be seen when we write pn −pk2pn−2k = (pn −pkpn−k) + pk(pn−k − pkpn−2k). Moreover, Libera and Zlotkiewicz proved that

Lemma 1.5 [6]. If p ∈P, then 2p2 = p12 + x(4 −p12) for some x such that |x| ≤1. 2. Bounds of |H3(1)| for S∗, K and R At the beginning, observe that H3(1) can be written in the form H3(1) = (a3a5 −a4 2) + a2(a3a4 −a2a5) + a3(a2a4 −a3 2), (5) or equivalently, H3(1) = H2(3) + a2J2 + a3H2(2), (6) where H2(k), k = 2, 3 are the second Hankel determinants defined by (1) and J2 = a3a4 −a2a5. The expression J2 is a particular case of Jn = an+1an+2 −anan+3. (7)

Theorem 2.1. f ∈S∗⇒|H3(1)| ≤1, (17) f ∈K ⇒|H3(1)| ≤49 540 = 0.090 . . . , (18) f ∈R ⇒|H3(1)| ≤41 60 = 0.683 . . . . (19)

Theorem 2.2. If f ∈S∗then |J2| ≤2.

Theorem 3.1. f ∈S∗(3) ⇒|H3(1)| ≤4 9, (23) f ∈K(3) ⇒|H3(1)| ≤1 36, (24) f ∈R(3) ⇒|H3(1)| ≤1 4. (25) All these bounds are sharp.

Theorem 3.2. If f ∈S∗(2), then Φf(μ) ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Lemma 1.5 and (12) result in a3 = 3 4p1 2 + 1 4(4 −p1 2)x. (29) Combining (12) and (28–29), we obtain Φf(μ) = 1 16 p1  2(1 −μ)p1 2 + (4 −p1

Lemma 1.5 it follows that p2 = 2. Hence, equality in (27) holds for f(z) = z 1−z2 and its rotations if μ ≤2/3. On the other hand, if μ ∈[2/3, 1], then the extremal functions are f(z) = z (1 −z2)t(1 + z2)1−t , t = (1 + 1/  3(2μ −1))/2 (34) and its rotations. The Taylor series expansion of this function, in terms of μ, is as follows: f(z) = z + 1

Corollary 1. If f ∈S∗(2) then α3

Theorem 3.3. f ∈S∗(2) ⇒|H3(1)| ≤ 1 3 √ 3 = 0.192 . . . , (36) f ∈K(2) ⇒|H3(1)| ≤ 4 135 = 0.029 . . . , (37) f ∈R(2) ⇒|H3(1)| ≤2 √ 6 45