Successive Logarithmic Coefficients of Univalent Functions

Theorem 3.1 which is the main result) and in some of its subclasses (see Corollary 3.2, 123

Lemma 2.3 For all r ∈[−2; 2] and ζ ∈T, r,ζ ∈S and r,ζ(D) = C \  ζt 2 −r : t ≥1  ∪  −ζt 2 + r : t ≥1  , if |r| < 2, (2.1) as well as

Lemma 2.4 For every function f : D →C the following equivalence holds: f ∈S and | c3( f ) −c2( f )2| = 1 if and only if f (z) = z 1 −| c2( f )|ζz + ζ 2z2 , z ∈D, (2.4) for a certain ζ ∈T.

Theorem 3.1 For every f ∈S, − √ 2 2 ≤|γ2( f )| −|γ1( f )| ≤1 2. (3.5) Moreover, both inequalities are sharp and for every function f : D →C the following two equivalences hold: (i) f ∈S and the equality in the second inequality in (3.5) holds if and only if f (z) = z 1 + ζ 2z2 , z ∈D,

Corollary 3.2 For every class S′ ⊂S, if S∗⊂S′, then Theorem 3.1 holds with S replaced by the class S′. We recall that a function f ∈H is close-to-convex if f (0) = 0 = f ′(0) −1 and there exist a function g ∈S∗and δ ∈(−π/2; π/2) such that Re eiδzf ′(z) g(z) > 0, z ∈D \ {0}. (3.27) The class C of all close-to-convex functions was introduced by Kaplan [15]. By the condition (3.26), each f ∈S∗satisfies the condition (3.27) with g := f and δ := 0, and so S∗⊂C. As was shown by Ozaki [22] and Kaplan [1

Theorem 3.1.

Corollary 3.4 For all ζ ∈T and f ∈CV(ζ) the inequalities (3.5) hold. Moreover, both the inequalities are sharp and for every function f : D →C: (i) f ∈CV(ζ) and the equality in the second inequality in (3.5) holds if and only if f = 0,ζ or f = 0,−ζ; (ii) f ∈CV(ζ) and the equality in the first inequality in (3.5) holds if and only if f = √ 2,ζ or f = √ 2,−ζ. Funding Not applicable. Data Availability Statement Not applicable. Code Availability Not applicable. Declarations Conflict of interest Th