On Coefficients Problems for Typically Real Functions Related to Gegenbauer Polynomials

We solve problems concerning the coefficients of functions in the class T(λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}(\lambda )$$\end{document} of typically real functions associated with Gegenbauer polynomials. The main aim is to determine the estimates of two expressions: |a4-a2a3|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a_4-a_2 a_3|$$\end{document} and |a2a4-a32|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a_2 a_4 -a_3{}^2|$$\end{document}. 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}(\lambda )$$\end{document}. Furthermore, we find the upper and the lower bounds of functionals of Fekete–Szegö type. Finally, we present some conclusions for the classes T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document} and T(1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}(1/2)$$\end{document}.


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