Stud. Univ. Babe¸s-Bolyai Math. 69(2024), No. 3, 517–534

Lemma 2.1. [30] Let w(z) = c1z + c2z2 + · · · be a Schwarz function. Then |c3 + µc1c2 + νc3 1| ≤1, where 1/2 ≤|µ| ≤2, 4(|µ| + 1)3/27 −(|µ| + 1) ≤ν ≤1. Let B be the class of functions f ∈A satisfying |f(z)| < 1 for all z ∈D.

Lemma 2.2. [8] Let f(z) = a0 + ∞ P n=1 anzn be in B. Then |a2n+1| ≤1 −|a0|2 −|a1|2 −· · · −|an|2, n = 0, 1, · · · (2.1) and |a2n| ≤1 −|a0|2 −|a1|2 −· · · −|an−1|2 − |an|2 1 + |a0|, n = 1, 2, · · · . (2.2) Equality in (2.1) holds for f(z) = a0 + a1z + · · · + anzn + εz2n+1

Lemma 2.3. [33] Let f(z) = z + ∞ P n=2 anzn be univalent and starlike with respect to symmetric points in D. Then |an| ≤1, n ≥2 equality being attained by the function z/(1 + εz), |ε| < 1.

Lemma 2.4. [31] If p(z) = 1 + p1z + p2z2 + p3z3 + · · · ∈P then for all n, m ∈N |µpnpm −pm+n| ≤  2, 0 ≤µ ≤1 2|2µ −1|, elsewhere. If 0 < µ < 1, the inequality is sharp for the function p(z) = (1 + zn+m)/(1 −zn+m). In other cases, the inequality is sharp for the function p(z) = (1 + z)/(1 −z). 3. Zalcman conjecture In this section, we first prove Zalcman conjecture (n = 2) for starlike functions with respect to the symmetric space.

Theorem 3.1. If the function f ∈S∗ S is of the form f(z) = z + a2z2 + a3z3 + · · · . Then |a2 2 −a3| ≤1. The inequality is sharp.

Theorem 3.2. Let the function f(z) = z + a2z2 + a3z3 + · · · ∈S∗ S. Then |a2a4 −a5| ≤1. The inequality is sharp.

Theorem 4.1. Let the function f ∈S∗ S be of the form f(z) = z + a2z2 + a3z3 + · · · . Then |H3(1)(f)| ≤329 400 ≃0.8225.

Theorem 4.3. If f ∈S∗ S is of the form f(z) = z + a2z2 + a3z3 + · · · . Then |H3(2)(f)| < 83 24 ≃3.45.

Theorem 4.4. If f(z) = z + ∞ P n=2 anzn ∈S∗ S, then |H3(3)(f)| ≤89 24 ≃3.7.

Theorem 4.5. Let f ∈S∗ S be of the form f(z) = z + a2z2 + a3z3 + · · · . Then |H4(1)(f)| ≤1.84.

Theorem 5.1. If f ∈S∗ S be of the form f(z) = z + a2z2 + a3z3 + · · · . Then |T2(2)(f)| ≤2. The inequality is sharp.

Theorem 5.2. Let f ∈S∗ S be of the form f(z) = z + a2z2 + a3z3 + · · · . Then |T2(3)(f)| ≤2. The inequality is sharp.

Theorem 5.3. If f ∈S∗ S be of the form f(z) = z + a2z2 + a3z3 + · · · . Then |T3(1)(f)| ≤4. The inequality is sharp.

Theorem 5.4. Let f ∈S∗ S be of the form f(z) = z + a2z2 + a3z3 + · · · . Then |T3(2)(f)| = |(a2 −a4)(a2 2 −2a3 2 + a2a4)| ≤6.

Theorem 5.5. Let f ∈S∗ S be of the form f(z) = z + a2z2 + a3z3 + · · · . Then |T4(2)(f)| =|(a2 2 −a2 3)2 + 2(a2 3 −a2a4)(a2a4 −a3a5) −(a2a3 −a3a4)2 + (a2 4 −a3a5)2 −(a3a4 −a2a5)2| ≤15.12.