On maximal area integral problem for analytic functions in the starlike family

Maximal Area Integral Problem for Certain Class of Univalent Analytic Functions

“…Note that Theorems 2.1 and 2.3 generalize the results proved in [8,19,21]. To see the bounds for the Dirichlet finite function, we write…”

“…A general problem on the Yamashita conjecture for the class S * (A, B) was suggested in [11] (see also [8,9]), and partially it is solved in [19]. In this paper, we solve the problem in complete generality for the full class S * (A, B), and the main results are stated in Section 2.…”

“…All polynomials and, more generally, all functions f ∈ A for which f ′ is bounded on D are Dirichlet finite. Our work in this paper is motivated by the work of Yamashita [21] and recent works [8,9,11,19]. In 1990, Yamashita [21] conjectured that (1.4) max…”

“…Suppose that f is a function analytic in D. We denote by ∆(r, f ), the area of the multisheeted image of the disk D r := {z ∈ C : |z| < r} (0 < r ≤ 1) under f . Thus, in terms of the coefficients of f , f ′ (z) = ∞ n=1 na n z n−1 , one gets with the help of the classical Parseval-Gutzmer formula (see [19]) the relation…”

“…In 2013, the Yamashita conjecture was settled in [8] (see Corollary 2.4) in a more general setting including functions for functions from S α (β) (see [11]). In the recent paper [19], the maximum area problem for the functions of type z/f (z) when…”