Coefficient Problems in the Subclasses of Close-to-Star Functions

For two subclasses of close-to-star functions we estimate early logarithmic coefficients, coefficients of inverse functions, Hankel determinant H2,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{2,2}$$\end{document} and Zalcman functional J2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_{2,3}$$\end{document}. All results are sharp.


Introduction
A function f ∈ A is called close-to-star if there exist g ∈ S * and β ∈ R such that Re e iβ f (z) g(z) > 0, z ∈ D. (1.2) respectively. Let us add the inequality (1.4) defines the subclass of the class of functions starlike in the direction of the real axis introduced by Robertson [31]. Moreover, each function F given by (1.3) over the class ST (i) maps univalently D onto a domain F (D) convex in the direction of the imaginary axis. The concept of convexity in one direction belongs to Roberston [31] (see e.g., [12, p. 199]). Each function F given by (1.3) over the class ST (1) maps univalently D onto a domain F (D) called convex in the positive the direction of the real axis, i.e., {w + it: t ≥ 0} ⊂ f (D) for every w ∈ f (D) [2,8,9,11,20,21]. Let us remark that the condition (1.4) was generalized by replacing the expression 1 − z 2 by the expression 1 − α 2 z 2 with α ∈ [0, 1] in [13].
In this paper we find the sharp estimates of early logarithmic coefficients (Sect. 2), of the Hankel determinant H 2,2 and of Zalcman functional J 2,3 (Sect. 3) and of the early inverse coefficients (Sect. 4) of functions in the classes ST (i) and ST (1). Since both classes ST (i) and ST (1) have a representation using the Carathéodory class P, i.e., the class of functions p ∈ H of the form having a positive real part in D, the coefficients of functions in ST (i) and ST (1) have a suitable representation expressed by the coefficients of functions in P. Therefore to get the upper bounds of considered functionals our computing is based on parametric formulas for the second and third coefficients in P. However both classes are rotation non-invariant. Thus to solve discussed problems we will apply a general formula for c 3 recently found in [7]. The formula (1.7) was proved by Carathéodory [3] (see e.g., [10, p. 41]). The formula (1.8) can be found in [28, p. 166]. The formula (1.9) was shown in a recent paper [7], where the extremal functions (1.11) and (1.12) were computed also. For c 1 ≥ 0 the formula (1.9) is due to by Libera and Zlotkiewicz [22,23]. and For ζ 1 ∈ T, there is a unique function p ∈ P with c 1 as in (1.7), namely, (1.10) For ζ 1 ∈ D and ζ 2 ∈ T, there is a unique function p ∈ P with c 1 and c 2 as in (1.7)-(1.8), namely, (1.11) For ζ 1 , ζ 2 ∈ D and ζ 3 ∈ T, there is a unique function p ∈ P with c 1 , c 2 and c 3 as in (1.7)-(1.9), namely, (1.12)

Logarithmic Coefficients
γ n z n , z ∈ D\{0}, log 1 := 0. The numbers γ n are called logarithmic coefficients of f. Differentiating (2.1) and using (1.1) we get (2.2) As it well known, the logarithmic coefficients play a crucial role in Milin conjecture ( [26], see also [10, p. 155]). It is surprising that for the class S the sharp estimates of single logarithmic coefficients S are known only for γ 1 and γ 2 , namely, and are unknown for n ≥ 3. Logarithmic coefficients is one of the topic recently being of interest by various authors (e.g., [1,18,33]). Logarithmic coefficients can be considered for functions f from the class A however under the assumption that the branch of logarithm needs not be necessarily a simply connected domain. Therefore, let ST 0 (i) and ST 0 (1) be the subclasses of ST (i) and ST (1) respectively, of all functions f for which the branch D z → log f (z)/z with log 1 := 0 exists.
All inequalities are sharp.
All inequalities are sharp.

Zalcman Functional and Hankel Determinant
Now we compute the sharp upper bound of the Zalcman functional J 2,3 (f ) := a 2 a 3 −a 4 being a special case of the generalized Zalcman functional J n,m (f ) := a n a m − a n+m−1 , n, m ∈ N\{1}, which was investigated by Ma [24] for f ∈ S (see also [29] for relevant results on this functional). We will find also the sharp bound of the second Hankel determinant H 2,2 (f ) = a 2 a 4 − a 2 3 . Both functionals J 2,3 and H 2,2 have been studied recently by various authors (see e.g., [5,6,14,16,17,19,27]). The inequality is sharp with the extremal function (3.1) Proof. From (2.4) by using (1.7)-(1.9) it follows that with sharpness for the function (3.1).
To find sharp estimate for H 2,2 over ST (i) we use the following lemma.

Inverse Coefficients
Since ST (i) is a compact class and f (0) = 1 for every f ∈ ST (i), there exists r 0 ∈ (0, 1) such that every f ∈ ST (i) is invertible in the disk D r0 . Thus there exists δ > 0 such that the inverse functionf of f |Dr 0 has a series expansion in the disk D δ of the form Proof. Substituting (2.4) into (4.2) we get and Now we prove (iii). By (4.5) and (1.7)-(1.9) we have Let now x ∈ [0, 1). Then for y ∈ [0, 1] we have ∂F ∂y  Hence and from (4.12) it follows that the inequality in (iii) is true. All inequalities are sharp with the extremal function (4.9).