The second Hankel determinant of the logarithmic coefficients of strongly starlike and strongly convex functions

Lemma 1 If p ∈P and is given by (6) with c1 ≥0, then c1 = 2ζ1, (9) c2 = 2ζ 2 1 + 2(1 −ζ 2 1 )ζ2 (10) and c3 = 2ζ 3 1 + 4(1 −ζ 2 1 )ζ1ζ2 −2(1 −ζ 2 1 )ζ1ζ 2 2 + 2(1 −ζ 2 1 )(1 −|ζ2|2)ζ3. (11)

Lemma 2 [6] Given real numbers A, B, C, let Y(A, B, C) := max

Theorem 1 If f ∈S∗ α, α ∈(0, 1], then |H2,1(F f /2)| = |γ1γ3 −γ 2 2 | ≤1 4α2. (14) The inequality is sharp.

Corollary 1 If f ∈S∗, then |γ1γ3 −γ 2 2 | ≤1 4. The inequality is sharp. 3 Strongly convex functions We prove the following sharp inequality for |H2,1(F f /2)| in the class Sc α.

Theorem 2 If f ∈Sc α, α ∈(0, 1], then |γ1γ3 −γ 2 2 | ≤ ⎧ ⎪⎪⎨ ⎪⎪⎩ α2 36, 0 < α ≤1 3, α2(17 + 18α + 13α2) 144(4 + 6α + α2) , 1

Corollary 2 If f ∈Sc, then |γ1γ3 −γ 2 2 | ≤1 33. The inequality is sharp. Acknowledgements The authors would like to express their thanks to the referees for their constructive advices and comments that helped to improve this paper. Availability of data and material The manuscript has no associated data. Declarations Conflict of interest The authors declare that they have no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, w