On Coefﬁcient Functionals for Functions with Coefﬁcients Bounded by 1

: In this paper, we discuss two well-known coefﬁcient functionals a 2 a 4 − a 32 and a 4 − a 2 a 3 . The ﬁrst one is called the Hankel determinant of order 2. The second one is a special case of Zalcman functional. We consider them for functions in the class Q R ( 12 ) of analytic functions with real coefﬁcients which satisfy the condition Re f ( z ) z > 12 for z in the unit disk ∆ . It is known that all coefﬁcients of f ∈ Q R ( 12 ) are bounded by 1. We ﬁnd the upper bound of a 2 a 4 − a 32 and the bound of | a 4 − a 2 a 3 | . We also consider a few subclasses of Q R ( 12 ) and we estimate the above mentioned functionals. In our research two different methods are applied. The ﬁrst method connects the coefﬁcients of a function in a given class with coefﬁcients of a corresponding Schwarz function or a function with positive real part. The second method is based on the theorem of formulated by Szapiel. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold. The obtained estimates signiﬁcantly extend the results previously established for the discussed classes. They allow to compare the behavior of the coefﬁcient functionals considered in the case of real coefﬁcients and arbitrary coefﬁcients.


Introduction
Let ∆ be the unit disk {z ∈ C : |z| < 1} and A denote the class of all functions f analytic in ∆ with the typical normalization f (0) = f (0) − 1 = 0. This means that the function f ∈ A has the following representation f (z) = z + ∞ ∑ n=2 a n z n . (1) Additionally, we denote by A R the class of those functions f ∈ A whose all coefficients are real. In this paper we discuss two functionals and a 4 − a 2 a 3 (3) considered for functions of the form (1) in a given class A ⊂ A.
Recently, these functionals have been widely discussed. The research mainly focused on estimating so called Hankel determinants. Pommerenke (see [1,2]) defined the k-th Hankel determinant for a function f of the form (1) and n, k ∈ N as H k (n) = a n a n+1 . . . a n+k−1 a n+1 a n+2 . . . In a view of this definition, a 2 a 4 − a 2 3 is the second Hankel determinant (more precisely, H 2 (2)). The sharp bounds of |a 2 a 4 − a 3 2 | for almost all important subclasses of the class S of analytic univalent functions were found (see, for example, [3][4][5][6][7][8]). It is worth noting that we still do not know the exact bound of this expression for S, nor for C consisting of all close-to-convex functions (see [9]). On the other hand, finding the bounds, upper and lower, for classes of analytic functions with real coefficients is a much more complicated task. For this reason, only a few papers were devoted to solving this problem. Such result for univalent starlike functions was obtained by Kwon and Sim ([10]). Furthermore, similar problems for functions which are typically real were discussed in [11].
The functional a 4 − a 2 a 3 is a special case of the so-called generalized Zalcman functional which was studied, among others, in [12] and [13]. The generalized version of this functional, that is, a 4 − µa 2 a 3 , was discussed in [14].
We start with considering (2) and (3) in the class Q R ( 1 2 ) of analytic functions given by (1) which satisfy the condition It is known that all coefficients of f ∈ Q R ( 1 2 ) are real and bounded by 1. The class Q R ( 1 2 ) contains three well-known, important subclasses of univalent functions: K R of convex functions, S * R ( 1 2 ) of starlike functions of order 1/2, K R (i) of functions that are convex in the direction of the imaginary axis. Two other classes T ( 1 2 ) and W consisting of functions defined by specific Riemann-Stieltjes integrals are also included in Q R ( 1 2 ). The precise definitions of these classes will be given in Section 3. In this section we show the partial ordering of the mentioned above subclasses of Q R ( 1 2 ) with respect to the relation of inclusion. Clearly, the coefficients of functions in each subset of Q R ( 1 2 ) are bounded by 1. What is interesting, this number cannot be improved. Finding the estimates of a 2 a 4 − a 3 2 and a 4 − a 2 a 3 gives additional information about the richness of these classes (compare [15]).
All functions in Q R ( 1 2 ) and in other classes discussed in this paper have real coefficients. For this reason, it is interesting to find not only the bounds of moduli of (2) and (3), but also their upper and lower bounds. On the other hand, it is clear that if the following property holds for all functions f in a given class A, then The same property is not true for the functional a 2 a 4 − a 3 2 .

Estimates for the class
The coefficients of f ∈ Q R ( 1 2 ) can be expressed in terms of the coefficients of a relative function p in the class P R or in terms of the coefficients of a relative function ω in the class (B 0 ) R . Recall that P and (B 0 ) denote the class of functions with positive real part and the class of functions such that ω(0) = 0 and |ω(z)| < 1.
If f ∈ Q R ( 1 2 ) is of the form (1), p ∈ P R and ω ∈ (B 0 ) R are of the form and then a n = 1 2 p n−1 and For this reason, we have and To obtain our results we need the estimates for initial coefficients of Schwarz functions.

Corollary 1.
If ω ∈ (B 0 ) R is of the form (8), then We also need the generalized Livingstone result obtained by Hayami and Owa.
Observe that for f (z) = z 1−z 2 we have a 2 a 4 − a 3 2 = −1, which means that min a 2 a 4 − a 3 2 : f ∈ Q R 1 2 = −1 . Now, we shall derive an upper bound of this functional for Q R ( 1 2 ).

Other Classes with Coefficients Bounded by 1
We know a few other subclasses of A R consisting of functions with real coefficients bounded by 1: the class K R of convex functions, the class S * R ( 1 2 ) of starlike functions of order 1/2, and the class K R (i) of functions that are convex in the direction of the imaginary axis. The same property also holds for T ( 1 2 ) and W defined as follows and where P [−1,1] is the set of probability measures on the interval [−1, 1]. For convenience of the reader, let us recall that an analytic and normalized function belongs to K R and S * R ( 1 2 ) when the following conditions are satisfied, respectively and Re The class K R (i) is related to the class T of typically real functions. Namely, for all z ∈ ∆ there is The classes defined above can be ordered in the following chains of inclusions: The first inclusion in the first chain is the famous theorem of Marx-Strohhäcker ( [18]). In [19] Hallenbeck proved that S * R (1/2) ⊂ T (1/2). In fact, he proved that T (1/2) is a closed convex hull fo S * R (1/2). Robertson proved in [20] that if f ∈ K R (i) then Re( f (z)/z) > 1/2, or, in other words, The proof of the relation K R (i) ⊂ T (1/2) can be found in [21].
To prove the third chain of inclusions, observe that The successive inclusions have already been shown.
Moreover, W ⊂ K R and K R ⊂ W .
The first statement follows from the fact that f 1 (z) = z 1−z 2 , as a convex combination of the functions z 1−z and z 1−z , belongs to W, but it does not belong to K R . To show the second statement, it is enough to consider f 2 (z) = z − 1 9 z 3 . Since this function is in K R . From the formula for the coefficients of functions in W it follows that a 3 ≥ 0 for each f ∈ W, but a 3 = −1/9 for f 2 . Consequently, f 2 does not belong to W. It is easy to check that . The image set f 3 (∆) coincides with the domain lying between two branches of the hyperbola (Re w) 2 − (Im w) 2 < 1/2. This means that f 3 is not in K R (i).
From the argument given above and Theorem 1 we obtain Corollary 2. Let A denote one of the classes: T ( 1 2 ), K R (i) and W and let f ∈ A. Then Equality holds for f (z) = z 1−z 2 .
In our research two different methods are applied. The first method connects the coefficients of a function in a given class with coefficients with a corresponding Schwarz function or a function with positive real part. The second method is based on the Szapiel theorem. According to this theorem, we can point out the extremal functions in this problem, that is, functions for which equalities in the estimates hold.

4.
Estimates for K R and S * R ( 1 2 ) We know that the estimate given in Corollary 2 is true but not sharp for K R and S * R ( 1 2 ), because the function f (z) = z 1−z 2 does not belong to either of them. In [5] it was shown that the sharp bound of |a 2 a 4 − a 3 2 | in K R is 1/8 and the extremal function is On the other hand, |a 2 a 4 − a 3 2 | ≤ 1/4 in S * R ( 1 2 ) (see [3]) and the extremal function is Let f ∈ K R and ω ∈ (B 0 ) R be given by (1) and (8), respectively. From the correspondence between K R and (B 0 ) R , that is, we obtain Consequently, Putting these formulae into (9) and (10) we have and Theorem 4. If f ∈ K R , then Proof. Let f ∈ K R . I. By Lemma 1, The critical points of h 1 are the solutions of the system From this system we have This means that a 2 a 4 − a 3 2 ≤ 0.066 . . ..
From the definition of the class S * R ( 1 2 ) we can represent a function f of this class as follows Let f and ω be given by (1) and (8), respectively. Comparing the coefficients of both sides in we obtain Putting these formulae into (9) and (10)  Proof. Let f ∈ S * R ( 1 2 ). I. By Lemma 1, The critical points of h 3 are the solutions of the system We consider only the points lying in the set Lemma 1). Hence, we obtain the equation The

II. From Lemma 1,
The point (0, 0) is the only critical point of h 4 and h 4 (0, 0) = 1/3. Moreover, The relation in (6) and the above results give the declared bound.

5.
Preliminary results for T ( 1 2 ), K R (i) and W Let X be a compact Hausdorff space and J µ = X J(t)dµ(t). Szapiel in [22] proved the following theorem. Theorem 6 ([22], Thm.1.40). Let J : [α, β] → R n be continuous. Suppose that there exists a positive integer k, such that for each non-zero p in R n the number of solutions of any equation J(t), p = const, α ≤ t ≤ β is not greater than k. Then, for every µ ∈ P [α,β] such that J µ belongs to the boundary of the convex hull of J([α, β]), the following statements are true: | supp(µ)| = m + 1 and one of the points α, β belongs to supp(µ) .
In the above the symbol u, v denotes the scalar product of vectors u and v, whereas the symbols P X and | supp(µ)| denote the set of probability measures on X and the cardinality of the support of µ, respectively.
Observe that the coefficients a n of a function f belonging to the classes T ( 1 2 ), K R (i) and W can be expressed by where A n (t) is a polynomial of degree n.
Taking into account the fact that we estimate the functionals a 2 a 4 − a 3 2 and |a 4 − a 2 a 3 |, depending only on 3 coefficients of f , it is enough to consider the vectors J(t) = [A 1 (t), A 2 (t), A 3 (t)], t ∈ [−1, 1] and p = [p 1 , p 2 , p 3 ]. We can observe that is a polynomial equation of degree 3. Therefore, the equation (29) has at most 3 solutions. In particular, for the classes T ( 1 2 ), K R (i) and W, it is known that A n (t) are the Legendre polynomials P n (t), the Chebyshev polynomials U n (t) and the monomial t n , respectively.
The results are sharp.
The results are sharp.

Proof.
Observe that the function given by (32) has the following Taylor series expansion Let a 4 − a 2 a 3 = g 3 (α, t) and a 2 a 4 − a 3 2 = g 4 (α, t). Using the same reasoning as in the proof of Theorem 7, we consider the function given by (32).
In this way we have proved the following theorem.
The results are sharp.

Concluding Remarks
In this paper we derived the upper estimates of the functionals a 2 a 4 − a 3 2 and |a 4 − a 2 a 3 | for the functions in the subclasses of Q R ( 1 2 ). In the paper two different methods were applied. In the first method we expressed the coefficients of a function in a given class by coefficients with a corresponding Schwarz function or a function with positive real part. The second method was based on the Szapiel theorem. This theorem allowed us to obtain the sharp bounds of the functionals (9) and (10) and to point out the extremal functions.
The obtained results satisfy the above inequalities and coincide with the inclusions presented in Section 3.