On the Fekete–Szegö Type Functionals for Close-to-Convex Functions

: In this paper, we consider two functionals of the Fekete–Szegö type Θ f ( µ ) = a 4 − µ a 2 a 3 and Φ f ( µ ) = a 2 a 4 − µ a 32 for a real number µ and for an analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + . . . , | z | < 1. This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re { f ( z ) / z } > α or Re { f (cid:48) ( z ) } > α , α ∈ [ 0, 1 ) . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss Θ f ( µ ) and Φ f ( µ ) for close-to-convex functions such that f (cid:48) ( z ) = h ( z ) / ( 1 − z ) 2 , where h is an analytic function with a positive real part. Many coefﬁcient problems, among others estimating of Θ f ( µ ) , Φ f ( µ ) or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals deﬁned for different subclasses of analytic univalent functions.


Introduction
Let A be the family of all functions analytic in ∆ = {z ∈ C : |z| < 1|} having the power series expansion: f (z) = z + a 2 z 2 + a 3 z 3 + . . . , (1) and let S * denote the class of univalent starlike functions in A (for the definitions and properties of S * and other classes, see [1]). For a given real argument β ∈ (−π/2, π/2) and a given function g ∈ S * , a function f ∈ A is called close-to-convex with argument β with respect to g if: Re e iβ z f (z) g(z) > 0, z ∈ ∆ .
All functions in C are univalent.
In this paper, we consider the class C 0 (k), where k is the Koebe function: The class C 0 (k) is sometimes denoted by CR + . Such functions have a well known geometrical meaning. Namely, for each function f in this class, the set f (∆) is a domain such that {w + t : t ≥ 0} ⊂ f (∆) for every w ∈ f (∆). Such functions f are convex in the positive direction of the real axis.
For a function f analytic in ∆ of the form (1), we define two functionals for a fixed real µ: and: Φ f (µ) = a 2 a 4 − µa 3 2 .
The functionals Θ f (µ) and Φ f (µ) are the generalizations of two well known expressions: a 4 − a 2 a 3 and a 2 a 4 − a 3 2 . Both functionals are symmetric, or invariant, under rotations. The first one is a particular case of the generalized Zalcman functional. It was investigated, among others, by Ma [4] and Efraimidis and Vukotić [5]. The second functional is known as the second Hankel determinant, and it was studied in many papers. The investigation of Hankel determinants for analytic functions was started by Pommerenke (see [6,7]) and continued by many mathematicians in various classes of univalent functions (see, for example [8][9][10][11][12][13][14][15][16]). The functional Φ f (µ) was first studied by Hayami and Owa [17]. They discussed an even more general functional a n a n+2 − µa n+1 2 for the classes Q(α) and R(α), α ∈ [0, 1), of functions f ∈ A such that Re { f (z)/z} > α and Re { f (z)} > α, respectively. The functionals Φ f (µ) and Θ f (µ) for the classes S * and K of starlike and convex functions, respectively, were discussed in [18]. It is worth pointing out a particularly interesting property of Φ f (µ). The sharp estimates of this functional are often symmetric with respect to a certain point. It was shown in [18] that such points for S * and K are 8/9 and one, respectively. We have: and: |Φ f (µ)| ≤ max{|µ − 1|, 1/8} for K.
A similar situation occurs for Q(1/2) and for the class C 0 (h), where h(z) = z/(1 − z 2 ); this point is 1/2 (see [17,19]). This situation appears even in the class T of typically real functions, which do not necessarily have to be univalent (see [19]). In this work, we derive bounds of Θ f (µ) and Φ f (µ) for functions in C 0 (k).

Preliminary Results
Let P denote the class of all analytic functions h with a positive real part in ∆ satisfying the normalization condition h(0) = 1. Let h ∈ P have the Taylor series expansion: We shall need here three results. The first one is known as Caratheodory's lemma (see, for example, ref. [1]). The second one is due to Libera and Złotkiewicz ([20,21]), and the third one is the result of Hayami and Owa.
The following lemma was proven by Lecko (see Corollary 2.3 in [23]).

Lemma 6.
If h ∈ P is given by (6), then: The inequality is sharp.
Proof. By Lemma 2, Applying the invariance of |p 1 p 3 − p 2 2 | under rotation, we can assume that p 1 is a non-negative real number. Writing r = |x| ∈ [0, 1] and p = p 1 ∈ [0, 2], we get by the triangle inequality and the assumption |y| ≤ 1: which gives the desired bound. The equality (9) holds for: which means that there is equality in (9) for rotations of (10).
The next lemma is a special case of more general results due to Choi et al. [24] (see also [9]). Let ∆ = {z ∈ C : |z| ≤ 1}. Define: Lemma 7. If ac < 0, then: where: If ac ≥ 0, then: Applying the correspondence between the functions in C 0 (k) and P: and Expansions (1) and (6) we get: Moreover, by Lemma 1, Re{a 2 } ≥ 0 with equality if and only if p 1 = −2. The equality is possible only for the function h(z) = 1−z 1+z ∈ P, and then, f (z) = 1 2 log 1+z 1−z ∈ C 0 (k). Hence, we can express Θ f (µ) and Φ f (µ) for f ∈ C 0 (k) as coefficients of a corresponding function h ∈ P in the following way: and:

Example
Let us consider the function: which has the following Taylor series expansion: so F ∈ C 0 (k). Moreover, For F, we have: For µ < 0, we have: Let us denote: for µ ∈ (6/17, 6) and: Similarly, the critical point for µ ∈ (0, 1) and: Finally, for a function F given by (15), we obtain: and:

Bounds of |Θ(µ)| for the Class C 0 (k)
In the main theorem of this section, we establish the sharp bounds of |Θ(µ)| for the class C 0 (k). The proof is divided into six lemmas. The first one is a particular case of the result obtained in [22] (Theorem 3.1 or Theorem 3.3 in [22]), and the second one is obvious.
The result is sharp.
The result is sharp.
Taking into account (13) and Lemma 2, we can write Θ f (µ) as follows: From the above formula, we can obtain bounds of |Θ f (µ)|, while µ ∈ (0, 1) and f ∈ C 0 (k), but only with an additional assumption that a 2 is a positive real number. The assumption of Lemma 2 enforces that given by (2), and we have: . is in C 0 (k), and so: To shorten notation, we write p instead of p 1 . One can observe that Θ f (µ) can be written as: where: From (18), the triangle inequality, |y| ≤ 1, and Lemma 2, we get: where a, b, and c are given by (19).
The result is sharp.
The result is sharp.

Bounds of |Φ(µ)| for the Class C 0 (k)
At the beginning of this section, we will quote the well known theorem of Marjono and Thomas [14].
In the next step, we shall prove that the Koebe function (2) is the extremal function for µ ≤ 63/92. Proof. At the beginning, let us discuss the case µ = 63/92. From (14), it follows that: Now, applying Lemmas 1 and 4 for µ = 1/2, Lemma 5 (remembering that 2(2 + Rep 1 ) ≤ 2(2 + |p 1 |)), Lemma 6, and the triangle inequality and writing p instead of |p 1 |, we obtain: where: Hence, It is worth adding that the function H given by (31) is decreasing for p > 2, so the choice µ = 63/92 is important. Now, we will find the exact bound of Φ f (µ) for µ close to one. Namely, we will discuss the case µ ∈ [µ 0 , 1], where: In this result, we need in addition that the coefficient a 2 should be real and a 2 ∈ [0, 2]. From (12), we get p = p 1 ∈ [−2, 2]. In the proof, we are going to apply Lemma 7.
Taking into account (14) and Lemma 2, we can write Φ f (µ) as follows: where: If p = −2 and p = 2, then f (z) = 1 2 log 1+z 1−z and f (z) = z (1−z) 2 , respectively, so: We will show that these values are less than or equal to the real bound of |Φ f (µ)| for all f ∈ C 0 (k). Now and on, we assume that p ∈ (−2, 2). Taking into account (14) and Lemma 2, by the triangle inequality and the assumption |y| ≤ 1, we get: where: Now, we are ready to establish the main theorem of this section.
It is easy to check that both values of Φ f (µ) for f (z) = 1 2 log 1+z 1−z and f (z) = k(z), which are given in (33), are less than or equal to µ/(9 − 8µ). This completes the proof. Applying the triangle inequality, we obtain our claim.
The results presented in Theorems 2-6 can be collected as follows.

Concluding Remarks
In this paper, we estimated two functionals Θ f (µ) = a 4 − µa 2 a 3 and Φ f (µ) = a 2 a 4 − µa 3 2 for the family C 0 (k), where µ is a real number. This family is a subset of the class C of all close-to-convex functions.
The results presented above broaden our knowledge about the behavior of the coefficient functionals defined for functions not only in C, but also generally in the class S of univalent functions. Unfortunately, there are no good estimates of the discussed functionals in the whole classes C and S. It seems that further research on the classes of the type C 0 ( f ), where f is different from k, may result in obtaining some conclusions about S.
In our opinion, the most important problem to be solved now is the estimating of the second Hankel determinant, or in other words Φ f (1) for f ∈ S. Even in the class C 0 , the exact bound is unknown. It is only known that for C 0 , there is |a 2 a 4 − a 3 2 | < 1.242 . . . (see [25]). On the other hand, the conjecture posed by Thomas [26] about 30 years ago that |a n a n+2 − a n+1 2 | ≤ 1 for S and n ≥ 2 was disproven. This means that there are functions in S for which |a n a n+2 − a n+1 2 | > 1. Finding (even non-sharp) estimates of Φ f (1) for f ∈ S remains an interesting open problem.