On arclength problem for analytic functions

Let A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} be the class of functions g(z)=z+∑n=2∞anzn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(z)=z+\sum _{n=2}^{\infty }a_nz^n \end{aligned}$$\end{document}which are analytic in the unit disk D={z:|z|<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}=\{z:|z|<1\}$$\end{document}. We denote by C(r, g) the closed curve which is the image of |z|=r<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|=r<1$$\end{document} under the mapping w=g(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=g(z)$$\end{document}, furthermore we denote by L(r, g) the length of C(r, g). In this paper we are interested in finding the maximum of the length L(r, f) as f(z) runs through all members of a fixed class of functions.


Introduction
Denote by A a class of functions g(z) = z + ∞ n=2 a n z n (1.1) which are analytic in the unit disk D = {z ∈ C : |z| < 1}. Let S be a subclass of A consisting of all univalent functions in D. Furthermore, let C(r , g) be the closed curve which is the image of |z| = r < 1 under the mapping w = g(z). Denote by L(r , g) the length of C(r , g) and let A(r , g) be the area enclosed by C(r , g).
then g(z) is said to be starlike with respect to the origin in D and we write g(z) ∈ S * . It is known that S * ⊂ S. We say that f (z) ∈ A is convex of order α, where 0 ≤ α < 1, provided The class of convex functions of order α has been introduced by Robertson in [14].
In what follows this class is denoted K(α). It is well known that K(α) ⊂ S for all α, 0 ≤ α < 1. A function f (z) ∈ A is said to be close-to-convex if it satisfies the following condition Re z f (z) e iα g(z) > 0, z ∈ D for some g(z) ∈ S * and α ∈ (−π/2, π/2). The class of close-to-convex functions is denoted by C. Let us recall that C ⊂ S. An univalent function f (z) ∈ S belongs to C if and only if a complement E of the image-region F = { f (z) : |z| < 1} is a union of rays that are disjoint (except that the origin of one ray may belong to another ray). It is known that, for any r ∈ (0, 1), the inequality holds for every f (z) ∈ S * (see [6, p.65]) and every f (z) ∈ C (see [1]). Moreover, Marx proved in [6, p.65] that for all r ∈ (0, 1) and every f (z) ∈ K(0). In [2], Eenigenburg extended this result to the class K(α), where α ∈ [0, 1). Namely, for all r ∈ (0, 1) and every f (z) ∈ K(α), we have In the other sets, the extremal functions are more complicated or the length problem is unsolved as is in the class S. Let In [5] F. R. Keogh has proved the following result.
Theorem 1.1 [5] Assume that g(z) ∈ S * . Then we have where O denotes the Landau's symbol.
In [13] Ch. Pommerenke extended the first part of Theorem 1.3 as follows Theorem 1.4 [13] If g(z) ∈ C, then In [4] W. F. Hayman has shown that, if g(z) ∈ S * and A(r , g) < A for some constant A, then Hayman gave an example a bounded starlike function g(z) satisfying lim sup r →1 L(r , g) This example clearly shows that the O in (1.3) cannot in general be replaced by small o.
The similar problems have been considered in [7,8]. The following result has been proved in [8].
Note that the above hypothesis zg (z) ∈ S * is equivalent to that g is convex univalent in D. Some related length problems were considered in [9][10][11]. In [15], D. K. Thomas considered L(r , g) for the class of bounded close-to-convex functions and asked the following question. Does there exist a starlike function g(z) for which In [12] a negative answer to the open problem (1.4) was given under some additional condition. Some related problems were considered in [9,10].
Therefore, we have (1.5) In [17] M. Tsuji has shown that It is known that for all real α. It is a purpose of this paper to prove, using a modified method, that a result related to (1.2), holds also under some another assumptions on g(z).

Main results
Applying (1.6) for z f (z)/ f (z) and the hypothesis of Theorem 2.1, we have because of the equality (1.7). For the second integral (2.3), we have Therefore, from (2.3), we have where V (R) has the form (2.2).

Theorem 2.2 If f (z) is analytic in
Proof Write z = re iθ . Then, we have Applying (1.6) for z f (z) = u(z) + iv(z), we get Thus, in view of (1.7)-(1.9), we obtain For R = 1 Theorem 2.2 becomes the following corollary.

Theorem 2.4 Let g(z) be of the form (1.1) and suppose that
for some β, 1 < β. Then we have Proof From (2.4) and (2.5), it follows that This completes the proof of Theorem 2.4.
Proof From (2.6) and (2.7), we get Then, we have It suffices to apply Hayman's lemma 2.5.
The next result is an extension of Theorem 1.5.
Putting 0 < r 1 < r , and t = (1 + ρ 2 )/2, we get Thus, we have where C is a bounded positive constant. On the other hand, letting t → 1 − , we have Using (1.7), we get This completes the proof of (2.8).

Remark
In the above theorem we do not suppose that g(s) is univalent in |z| < 1 and therefore, L(r ) and A(r ) are not necessarily the length of the image curve C(r ) and the area enclosed by the image curve C(r ) which is the image curve of the circle under the mapping w = g(z).

Author Contributions
The authors declare they have equal contribution in the paper.
Funding Not applicable.
Data availability Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Declarations
Ethical Approval Not applicable.

Conflict of interest
The authors declare they have no financial interests. The authors have no conflicts of interest to declare that are relevant to the content of this article.
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