On Coeﬃcients Problems for Typically Real Functions Related to Gegenbauer Polynomials

. We solve problems concerning the coeﬃcients of functions in the class T ( λ ) of typically real functions associated with Gegenbauer polynomials. The main aim is to determine the estimates of two expressions: | a 4 − a 2 a 3 | and | a 2 a 4 − a 32 | . The second one is known as the second Hankel determinant. In order to obtain these bounds, we consider the regions of variability of selected pairs of coeﬃcients for functions in T ( λ ). Furthermore, we ﬁnd the upper and the lower bounds of functionals of Fekete–Szeg¨o type. Finally, we present some conclusions for the classes T and T (1 / 2).


Introduction
Let Δ denote the unit disk {z ∈ C : |z| < 1} and A be the class of all functions f analytic in Δ, normalized by the condition f (0) = f (0) − 1 = 0. This means that f ∈ A has the expansion f (z) = z + ∞ n=2 a n z n . (1) In 1994 Szynal [15] introduced the class T (λ), λ ≥ 0 as the subclass of A consisting of functions of the form where 99 Page 2 of 12 P. Zaprawa et al. MJOM and μ is a probability measure on the interval [−1, 1]. The collection of such measures on [a, b] is denoted by P [a,b] . The function k(z, t) has the Taylor series expansion k(z, t) = z + C (λ) where C (λ) n (t) denotes the Gegenbauer polynomial of degree n (for details about the Gegenbauer polynomials, see [1,14]).
First polynomials of this type are following: If f ∈ T (λ) is given by (2), then the coefficients of this function can be written as follows: Note that T (1) = T is the well-known class of typically real functions (for details, see e.g., [2,8,9]). For λ = 1/2 we obtain the class of typically real functions related to Legendre polynomials P n (t) = C (1/2) n (t). As it was shown in [15] (see also, [12]), this class is a proper superclass for S * R (1/2), the class of starlike functions of order 1/2 and for K R (i), the class of convex functions in the direction of the imaginary axis; both of them consist of functions with real coefficients. More precisely, we have two chains of inclusions: This observation is significant because it happens that the results in T (1/2) can be transferred to its subclasses (see [12,16]). In this paper we consider various problems concerning the coefficients of functions in T (λ). We discuss the Fekete-Szegö functional a 3 − μa 2 2 and its modification, i.e., a 5 − μa 3 2 . Moreover, we find the sharp bounds of two expressions: |a 4 −a 2 a 3 | and |a 2 a 4 −a 3 2 |. The second is called the second Hankel determinant. It was Pommerenke the first who studied this determinant in the geometric theory of analytic functions ( [6,7]).
Recently, the Hankel determinant has been studied by many mathematicians. They discussed the second Hankel determinants for various classes of univalent functions. Some results in this direction can be found in [3][4][5]10,11].

Regions of Variability of a Pair of Coefficients for T (λ)
Let A n,m denote the set of variability of the point (a n , a m ), where a n and a m are the coefficients of a given function f ∈ T (λ) with the series expansion (1). The set A n,m coincides with the closed convex hull of the curve . By the Caratheodory theorem we conclude that it is sufficient to discuss only the functions Now, we shall establish a few simple lemmas concerning the sets A n,m for initial integers n and m.
Hence, the boundary of the convex hull of γ 2,3 ([−1, 1]) consists of this curve and the line segment that connects two endpoints of this arc.
The similar result can be obtained for A 3,5 . It is enough to determine the convex hull of the arc of the parabola γ 3,5 ([−1, 1]). In this way we get the following lemma.

Remark 1.
In both lemmas the boundaries of the discussed sets consist of an arc of a parabola and a line segment. The points lying on the parabolas correspond to the coefficients of k(z, t), t ∈ [−1, 1]. On the other hand, the points of the line segments are the coefficients of a function αk(z, −1)
Applying this lemma we can derive the bounds of the product of the second and the fourth coefficients of f ∈ T (λ). Theorem 2.1. For f ∈ T (λ), λ > 0, the following sharp bounds hold: and Proof. The upper bound is obvious. In order to calculate the lower bound of a 2 a 4 , we consider only these points (a 2 , a 4 ) which belong to the boundary of A 2,4 . Since A 2,4 is symmetric with respect to the origin, it is enough to discuss the points lying on the curve y = g(x), where g is the function described in Lemma 2.3. Let us denote by h 1 (x) and h 2 (x) two functions corresponding to the values of a 2 a 4 on each part of the curve y = g(x). Namely, Only one critical point It is easy to check that where x 2 = −(λ + 2)/3. Combining (9) and (10) we obtain the desired result.
For λ = 1 this result reduces to the one proved in [17] in Theorem 13.

Functionals of the Fekete-Szegö Type
Now, we are ready to find the sharp bounds of two functionals defined for f ∈ T (λ) of the form (1): a 3 − μa 2 2 and a 5 − μa 3 2 . The first one is very well known and often discussed the so-called Fekete-Szegö functional. Theorem 3.1. If f ∈ T (λ), λ > 0, then the following sharp bounds hold: and Proof. Let f ∈ T (λ). From Lemma 2.1, On the other hand, by Lemma 2.1, with equality for k(z, 0) = z − λz 3 + · · · in first case and for k(z, ±1) in the second one.

Bounds of |a 4 − a 2 a 3 |
Let us denote by Ω n (T (λ)), n ≥ 1 the region of variability of three succeeding coefficients of functions in T (λ), i.e., the set {(a n (f ), a n+1 (f ), a n+2 (f )) : f ∈ T (λ)}. Therefore, Ω n (T (λ)) is the closed convex hull of the curve Let X be a compact Hausdorff space and J μ = X J(t)dμ(t). Szapiel in [13] (Thm.1.40) proved the following theorem. In the above, the symbol u, v means the scalar product of vectors u and v, whereas the symbols P X and |supp(μ)| describe the set of probability measures on X and the cardinality of the support of μ, respectively.
According to Theorem 4.1, the boundary of the convex hull of γ 2 ([−1, 1]) is determined by atomic measures μ for which support consists of at most