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

Abstract

For an analytic function f defined on the unit disk |z| < 1, let ∆(r, f ) denote the area of the image of the subdisk |z| < r under f , where 0 < r ≤ 1. In 1990, Yamashita conjectured that ∆(r, z/f ) ≤ πr 2 for convex functions f and it was finally settled in 2013 by Obradović and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation zf ′ (z)/f (z) ≺ (1 + (1 − 2β)αz)/(1 − αz) for 0 ≤ β < 1 and 0 < α ≤ 1. We prove Yamashita's conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.

Maximal Area Integral Problem for Certain Class of Univalent Analytic Functions

“…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…”

“…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 f ∈ S * ((1 − 2β)α, −α) ≡ T (α, β) ( 0 < α ≤ 1, 0 ≤ β < 1 ), is established (see Corollary 2.7). 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 the recent paper [19], the maximum area problem for the functions of type z/f (z) when f ∈ S * ((1 − 2β)α, −α) ≡ T (α, β) ( 0 < α ≤ 1, 0 ≤ β < 1 ), is established (see Corollary 2.7). 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.…”

“…Set for an abbreviation p(z) := zf ′ (z)/f (z). [18] 1978 Silverman [16] S [19] := T (α, β), < α α ∈ (0, 1], β ∈ [0, 1) Table 1 From (1.1), we note that…”