**1. Introduction**

Let A be the family of functions of the form

$$\log(z) = z + \sum\_{n=2}^{\infty} a\_n z^n \tag{1}$$

which are analytic in the open unit disk <sup>D</sup> <sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> <sup>C</sup> : <sup>|</sup>*z*<sup>|</sup> <sup>&</sup>lt; <sup>1</sup>}. Let <sup>S</sup> denote the subfamily of <sup>A</sup> consisting of all univalent functions in D.

Let *C*(*r*) denote the image curve of the |*z*| = *r* < 1 under the function *g* ∈ A which bound the area *<sup>A</sup>*(*r*). Furthermore, let *<sup>L</sup>*(*r*) be the length of *<sup>C</sup>*(*r*) and *<sup>M</sup>*(*r*) = max|*z*|=*r*<<sup>1</sup> |*g*(*z*)|.

If *g* ∈ A satisfies

$$\mathfrak{Re}\left\{\frac{zg'(z)}{g(z)}\right\} > 0, \; z \in \mathbb{D}\_\*$$

then *<sup>g</sup>* is said to be starlike with respect to the origin in <sup>D</sup> and we write *<sup>g</sup>* ∈ S∗. It is known (for details, see [1,2]) that S<sup>∗</sup> ⊂ S.

The aim of the present paper is to prove, using a modified methodology, that in the following implication

$$\log \in \mathcal{S}^\* \quad \Rightarrow \quad L(r) = \mathcal{O}\left(M(r) \log \frac{1}{1 - r}\right) \quad \text{as} \quad r \to 1,\tag{2}$$

where O denotes the Landau's symbol, the assumption that *g* is starlike univalent can be changed by a weaker one. Result (2) was proved by Keogh [3]. Moreover, some other length problems for analytic functions are investigated. Several interesting developments related to length problems for univalent functions were considered in [4–15].
