Sharp inequalities for the coefficients of concave schlicht functions

Abstract

Abstract. Let D denote the open unit disc and let f : D → C be holomorphic and injective in D. We further assume that f (D) is unbounded and C \ f (D) is a convex domain. In this article, we consider the Taylor coefficients a n (f ) of the normalized expansionand we impose on such functions f the second normalization f (1) = ∞. We call these functions concave schlicht functions, as the image of D is a concave domain. We prove that the sharp inequalitiesare valid. This settles a conjecture formulated in [2].

Coefficient Discs and Generalized Central Functions for the Class of Concave Schlicht Functions

“…We call such functions concave univalent functions with opening angle πα at infinity and we refer to the articles [1,2,4,5] for a detailed discussion on functions in this class. We now recall the following characterization for functions in Co(α) (compare [ The present article is dedicated to establish the sets of variability of the functionals a n (f ), f ∈ Co(α) for n ≥ 2.…”

“…Now, the fact that Co(α) ⊆ Co(2) for α ∈ (1, 2] implies that the set of variability of a n (f ), f ∈ Co(α), is contained in the set of variability of a n (f ), f ∈ Co(2). Indeed, from [1], we recall that the following set is the exact set of variability for a n (f ), f ∈ Co (2):…”

The punishing factors for convex pairs are $2^{n-1}$

“…proved in [7], the triangle inequality, and |u + 1| ≤ 1 together with Fejér's inequality. Now, we only have to replace the last formula of the proof of C n (Ω, Π) ≤ 2 n−1 above by the formula Since |f (n) 1 (x)| = x −(n+2) (n + 1)!, the constant 2 n−2 is attained at all points x ∈ (0, ∞).…”

The Pre-Schwarzian Norm Estimate for Analytic Concave Functions

“…For more investigation of concave functions, we may refer to [3][4][5][6][7]. We say that is subordinate to in D, written as ≺ , if and only if ( ) = ( ( )) for some Schwarz functions ( ), (0) = 0, and By using the subordination, we define a subclass of concave functions as follows.…”