Integral means and Dirichlet integral for analytic functions

Abstract

For normalized analytic functions f in the unit disk, the estimate of the integral meansis important in certain problems in fluid dynamics, especially when the functions f (z) are non-vanishing in the punctured unit disk 0 < |z| < 1. We consider the problem of finding the extremal function f which maximizes the integral means L 1 (r, f ) for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses F of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functionaldenotes the area of the image of |z| < r under z/ f (z). The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990. Preliminaries and main resultsDenote by H the family of all functions f which are analytic in the unit disk D := {z ∈ C : |z| < 1} and by A the subfamily of H with the normalization f (0) = 0 = f (0) − 1. Also, let S = { f ∈ A : f is univalent in D} and S * := S * (0) ⊂ S denote the class of all starlike (univalent) functions in D. Here S * (β) denotes the family of starlike functions of order β, i.e., functions f ∈ S such that [7]

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

“…In [15], the Yamashita conjecture problem for the class of φ-spirallike functions of order β (0 ≤ β < 1) and φ ∈ (−π/2, π/2) have also been settled (see [8] for the definition of φ-spirallike function). Recent work in this direction can also be found in [12]. There are several other classes of analytic univalent functions having interesting geometric properties for which solution of the Yamashita conjecture problem would be of interesting to readers working in this field.…”

Maximal Area Integral Problem for Certain Class of Univalent Analytic Functions

“…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.…”

Concave univalent functions and Dirichlet finite integral

“…For the family of convex functions, Yamashita [, p. 439] conjectured that where the maximum is attained only by the rotations of the function . In , the authors proved this conjecture in a more general form and in , , the authors obtained analog result for some other classes of functions and spirallike functions, respectively. In this section, we obtain a similar result (Theorem ) for concave functions but with the generalized Yamashita functional (i.e.…”