Sharp bounds of logarithmic coefficient problems for functions with respect to symmetric points

Theorem 8.4]). The best upper bounds for univalent functions’ logarithmic coefficients for n ≥3 remain a conundrum, though, apparently. If f is given by (1) then, by matching the coefficients of zn in (2) for n = 1, 2, 3 it concludes 2γ1 = a2, 2γ2 = a3 −1 2a2 2, 2γ3 = a4 −a2a3 + 1 3a3 2. (3) Further, due to the significance of the recent studies about the logarithmic coefficients, the problem obtaining the sharp bounds for the second Hankel determinant of these coefficients, that is H2,1(Ff/2) w

Lemma 1 ([6], Lemma 2.1). If ϑ(z) = P∞ n=1 ϑnzn ∈Ω, then for some p, q, with |p| ≤1 and |q| ≤1 ϑ2 = p 1 −ϑ2 1
, ϑ3 = 1 −ϑ2 1
1 −|p|2
q −ϑ1 1 −ϑ2 1

Theorem 1. If f ∈S∗ s(ψ), with ψ(z) = 1 + P∞ n=1 Gnzn, then |H2,1(Ff/2)| ≤|G1| 16 ·    4MO −N 2 4M , if M < 0 and M ≤−N 2 ≤0; max {O; M + N + O} ,

Corollary 1. If f ∈S∗ s(σ), then |H2,1(Ff/2)| ≤(1 −σ)2 4 . This result is sharp for f(z) = z (1−z2)(1−σ) = z + (1 −σ)z3 + . . . , z ∈D.

Corollary 2. If f ∈S∗ s,L, then |H2,1(Ff/2)| ≤1 64. This bound is sharp for the function f2 ∈A given by 2zf ′ 2(z) f2(z) −f2(−z) = √ 1 + z2, that is for f2(z) = z + z3/4 + . . . .

Corollary 3. If S∗ s 1+z 1−z
α
, then |H2,1(Ff/2)| ≤α2 4 . This bound is sharp for the function fα ∈A given by 2zf ′ α(z) fα(z) −fα(−z) = 1 + z2 1 −z2 α

Theorem 2. If f ∈Cs(ψ) with ψ(z) = 1 + ∞ P n=1 Gnzn, then |H2,1(Ff/2)| ≤1 4 · ( 4WY −X2 4W , if W < 0 and W ≤−X 2 ≤0; max {Y ; W + X + Y } , otherwise,

Corollary 4 ([1], Theorem 2.4). If f ∈Ks, then |H2,1(Ff/2)| ≤ 1 36. This result is sharp for f(z) = 1 2 log 1+z 1−z. REFERENCES 1. V. Allu, V. Arora, A. Shaji, On the second Hankel determinant of logarithmic coefficients for certain univalent functions, Mediterr. J. Math., 20 (2023), 81. https://doi.org/10.1007/s00009-023-02272-x 2. S.Z.H. Bukhari, T. Bulboac˘a,