Some properties of the class $\mathcal{U}$

In this paper we study the class $\mathcal{U}$ of functions that are analytic in the open unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$ and satisfy \[\left|\left [\frac{z}{f(z)} \right]^{2}f'(z)-1 \right|<1\quad\quad (z\in {\mathbb D}).\] For functions in the class $\mathcal{U}$ we give sharp estimate of the second ant the third Hankel determinant, its relationship with the class of $\alpha$-convex functions, as well as certain starlike properties.


Introduction
Let A denote the family of all analytic functions in the unit disk D := {z ∈ C : |z| < 1} and satisfying the normalization f (0) = 0 = f ′ (0) − 1. Let S ⋆ and K denote the subclasses of A which are starlike and convex in D, respectively, i.e., Geometrical characterisation of convexity is the usual one, while for the starlikeness we have that f ∈ S ⋆ , if, and only if, f (D) is a starlike region, i.e., z ∈ f (D) ⇒ tz ∈ f (D) for all t ∈ [0, 1].
The linear combination of the expressions involved in the analytical representations of starlikeness and convexity brings us to the classes of α-convex functions introduced in 1969 by Mocanu ([3]) and consisting of functions f ∈ A such that = 0 for z ∈ D and α ∈ R. Those classes he denoted by M α .
Further, let U denote the set of all f ∈ A satisfying the condition where the operator U f is defined by All this classes consist of univalent functions and more details on them can be found in [1,10].
The class of starlike functions is very large and in the theory of univalent functions it is significant if a class doesn't entirely lie inside S ⋆ . One such case is the class of functions with bounded turning consisting of functions f from A that satisfy Re f ′ (z) > 0 for all z ∈ D. Another example is the class U defined above and first treated in [5] (see also [6,7,10]). Namely, the function − ln(1 − z) is convex, thus starlike, but not in U because U f (0.99) = 3.621 . . . > 1, while the function f defined This rear property is the main reason why the class U attracts huge attention in the past decades.
In this paper we give sharp estimates of the second and the third Hankel determinant over the class U and study its relation with the class of α-convex and starlike functions.

Main results
In the first theorem we give the sharp estimates of the Hankel determinants of second and third order for the class U. We first give the definition of the Hankel determinant, whose elements are the coefficients of a function f ∈ A.
Definition 2. Let f ∈ A. Then the qth Hankel determinant of f is defined for q ≥ 1, and n ≥ 1 by Thus, the second and the third Hankel determinants are, respectively, Theorem 1. Let f ∈ U and f (z) = z + a 2 z 2 + a 3 z 3 + . . .. Then we have the sharp estimates: Proof. In [5] the following characterization for functions f in the class in U was given: where function ω is analytic in D with ω(0) = ω ′ (0) = 0 and |ω(z)| < 1 for all z ∈ D.
t 2 dt, then we easily obtain that |ω 1 (z)| ≤ |z| < 1 and gives (see relation (13) in the paper of Prokhorov and Szynal [8]): Also, from (4) we have From the last relation we have We may suppose that c 1 ≥ 0, since from (6) we have c 1 = a 3 − a 2 2 and a 3 and a 2 2 have the same turn under rotation. In that sense, from (5) we obtain If we use (3), (6) and (7), then The functions k(z) = z (1−z) 2 and f 1 (z) = z 1−z 2 show that the estimate is the best possible.
Similarly, after some calculations we also have The function f 2 (z) = z 1−z 3 /2 shows that the result is the best possible.
In the rest of the paper be consider some starlikeness problems for the class U and its connection with the class of α-convex functions.
First, let recall the classical results about the relation between the starlike functions and α-convex functions.  As an anlogue of the above theorem we have f ′ (z) = p(z) + 1 and, after some calculations that The relation (1) is equivalent to We want to prove that |p(z)| < 1, z ∈ D. If not, then according to the Clunie-Jack Lemma ( [2]) there exists a z 0 , |z 0 | < 1, such that p(z 0 ) = e iθ and z 0 p ′ (z 0 ) = kp(z 0 ) = ke iθ , k ≥ 2. For such z 0 , from (8) we have that ≤ 0 since f ∈ S ⋆ (by Theorem 2) and α ≤ −1. That is the contradiction to (1).
(b) To prove this part, by using Theorem 2(b), it is enough to find a function g ∈ K such that g not belong to the class U. Really, the function g(z) = − ln(1 − z) is convex but not in U.

Open problem.
It remains an open problem to study the relationship between classes M α and U when −1 < α < 0 and α > 1.
In the next theorem we consider starlikeness of the function where f ∈ U and a 2 = f ′′ (0) 2 = 0, i.e., its second coefficient doesn't vanish.
As for sharpness, we can also consider the function f b defined by (10) with 0 < b ≤ 1. For |z| < b 2 we have which implies that g b belongs to the class U in the disc |z| < b/2.
We believe that part (b) of the previous theorem is valid for all 0 < |a 2 | ≤ 2. In that sense we have the next Conjecture 1. Let f ∈ U. Then the function g defined by the expression (9) belongs to the class U in the disc |z| < |a 2 |/2. The result is the best possible.