Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions

Theorem 1. Let p ∈(0, 1). (i) X0(Bp) = {c0 ∈C : |c0 −P −1| ≤P −1} . For a function ϕ(z) = c0 + c1z + · · · in Bp, c0 ∈∂X0(Bp) if and only if ϕ is an analytic automorphism of D. (ii) X1(Bp) = n (c0, c1) ∈C2 : |c1 −(1 −Pc0 + c2 0)| ≤P h P −2 −|c0 −P −1|2io . In other words, a pair (c0, c1) of complex numbers is contained in X1(Bp) if and only if (1.2) c0 = P −1(1 −σ0) and c1 = P −2

Theorem 2. Let 0 < p < 1 and µ ∈R and put P = p + 1/p. Then the maximum Φ(µ) of the Fekete-Szegő functional |Λµ(f)| over f ∈Cop is given as follows: Φ(µ) =            

Corollary 3. Let f(z) = z + a1z + a2z2 + · · · be a function in Cop. Then the following sharp inequality holds: |a3| ≤P 2 −1 = p2 + 1 + 1 p2. Indeed, the above inequality is still valid as long as f is univalent meromorphic on D with a pole at p (see Jenkins [9]). Avkhadiev, Pommerenke and Wirths [1] (see also [5]) proved the even stronger result that the variability region of a3 over f ∈Cop is given as |a3 −P 2 + 2| ≤1. (This can also be proved by our method given below.) Since Φ(1) = 1 by Theo

Corollary 4. Let 0 < p < 1 and suppose that f(z) = z + a1z + a2z2 + · · · is a function in Cop. Then the following sharp inequality holds: |a3 −a2 2| ≤1. Recall that 6(a3 −a2 2) = Sf(0) is the Schwarzian derivative of f evaluated at z = 0. The inequality |a3 −a2 2| ≤1 is valid for a univalent holomorphic function f(z) = z + a2z2 + a3z3 + · · · on D (see, for instance, [8, Ex. 1 in p. 70]). Indeed, it is obtained by a simple application of Gronwall’s area theorem for the function 1/f(1/w). Since

Lemma 5 (Dieudonné’s Lemma). Let z0, w0 ∈D with |w0| < |z0|. Then the region of values of w = g′(z0) for holomorphic functions g : D →D with g(0) = 0 and g(z0) = w0 is given as the closed disk (2.2) w −w0 z0 ≤|z0|2 −|w0|2 |z0|(1 −|z0|2). Equality holds if and only if g is a Blaschke product of degree 2 fixing 0.

Proposition 6. Let Y (a, b, c) be the quantity defined in (3.1) for real numbers a, b, c. When ac ≥0, Y (a, b, c) =    |a| + |b| + |c| if |b| ≥2(1 −|c|), 1 + |a| + b2 4(1 −|c|) if |b| < 2(1 −|c|). When ac < 0, (3.2) Y (a, b, c) = 

Theorem 7. Let 0 < p < 1. The variability region Wp of a3 −a2 2 for Cop satisfies Ωp ⊂Wp ⊂D. Moreover, Wp = Ωp for 0 < p ≤p0 and Wp ̸= Ωp for p0 < p < 1, where p0 = 1 + √ 37 − q 2(1 + √ 37) 6 ≈0.553175.