Acta Univ. Sapientiae, Mathematica, 7, 2 (2015) 210–219

Lemma 1 (Carath´eodory’s Lemma [4], see also [5, p. 41]). Let the function p ∈P be given by the series then the sharp estimate |cn| ≤2, n = 1, 2, · · · holds. The inequality is sharp for each n.

Lemma 2 (cf. [14, p. 254], see also [15]). Let the function p ∈P be given by (2), then 2c2 = c2 1 + x(4 −c2 1) for some x, |x| ≤1, and 4c3 = c3 1 + 2c1(4 −c2 1)x −c1(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)z for some z, |z| ≤1.

Lemma 3 ([5, p. 44]). If f ∈S∗ be given by (1), then |an| ≤n (n = 2, 3, . . . ). Strict inequality holds for all n unless f is rotation of the Koebe function k(z) = z/(1 −z)2.

Lemma 4 ([23]). If f ∈C be given by (1), then |an| ≤n (n = 2, 3, . . . ). Equality holds for all n when f is rotation of the Koebe function.

Lemma 5 ([10]). If f ∈S∗be given by (1), then for any real number μ, we have |a3 −μa2 2| ≤ ⎧ ⎨ ⎩ 3 −4μ, if μ ≤1 2 1, if 1 2 ≤μ ≤1

Lemma 6 ([11]). If f ∈C be given by (1), then |a3 −a2 2| ≤1. There is a function in C such that equality holds.

Lemma 7 ([12]). If f ∈S∗be given by (1), then |a2a4 −a2 3| ≤1. Equality is attended for the the Koebe function.

Lemma 8 ([1]). If f ∈S∗be given by (1), then |a2a3 −a4| ≤2. Equality is attained by Koebe function. 2 Main results Our first main result is contained in the following theorem:

Theorem 1 Let the function f ∈C be given by (1), then |a2a3 −a4| ≤3. (6)

Theorem 2 Let the function f ∈C be given by (1), then H2(2) = |a2a4 −a2 3| ≤85 36. (14)

Theorem 3 Let the function f ∈C be given by (1), then |H3(1)| ≤ 289 12 . (18)