The second Hankel determinant of the logarithmic coefficients of strongly starlike and strongly convex functions

Sharp bounds are given for the second Hankel determinant of the logarithmic coefficients of strongly starlike and strongly convex functions.


Introduction
Denote by H the class of analytic functions in D := {z ∈ C : |z| < 1} with Taylor expansion f (z) = ∞ n=1 a n z n , z ∈ D, (1) and let A be the subclass of f normalized by f (0) = 1. Let S denote the subclass of univalent functions in A. For f ∈ S, logarithmic coefficients γ n := γ n ( f ) of f are defined by γ n ( f )z n , z ∈ D, log 1 := 0. (2) and play a crucial role in the theory of univalent functions, and in articular to prove the Milin conjecture ( [19], see also [7, p. 155]). We note that for the class S sharp estimates are known only for γ 1 and γ 2 , namely, Estimating the modulus of logarithmic coefficients for f ∈ S and various subclasses has been considered recently by several authors (e.g., [1,2,5,8,12,24]). For q, n ∈ N, the Hankel determinant H q,n ( f ) of f ∈ A of the form (1) is defined as H q,n ( f ) := a n a n+1 · · · a n+q−1 a n+1 a n+2 · · · a n+q . . . . . . . . . . . . a n+q−1 a n+q · · · a n+2(q− 1) , and in particular many authors have examined the second and the third Hankel determinants H 2,2 ( f ) and H 3,1 ( f ) over selected subclasses of A, (see e.g., [4,11] with further references). We note that H 2,1 ( f ) = a 3 − a 2 2 is the well known coefficient functional which for S was studied first in 1916 by Bieberbach (see e.g., [9,Vol. I,p. 35]).
Based on the these ideas, in this paper and in [10] we propose research study of the Hankel determinants H q,n (F f /2) which entries are logarithmic coefficients of f . We are therefore concerned with . Differentiating (2) and using (1) we obtain and so Note that when f ∈ S, then for f θ (z) := e −iθ f (e iθ z), θ ∈ R, so |H 2,1 (F f θ /2)| is rotationally invariant. In this paper we find sharp upper bounds for H 2,1 (F f /2) in the case when f is strongly starlike or strongly convex function of order α, defined respectively as follows. Given α ∈ (0, 1], a function f ∈ A is called strongly starlike of order α if Also, a function f ∈ A is called strongly convex of order α if We denote these classes by S * α and S c α respectively, noting that S * 1 =: S * and S c 1 =: S c are the classes of starlike and convex functions, respectively. The class of strongly starlike functions was introduced by Stankiewicz [21,22], and independently by Brannan and Kirwan [3] (see also [9, Vol. I, pp. 137-142]). Stankiewicz [22] found an external geometrical characterization of strongly starlike functions and Brannan and Kirwan gave a geometrical condition called δ-visibility, which is sufficient for functions to be strongly starlike. Subsequently Ma and Minda [16] proposed an internal characterization of functions in S * α based on the concept of k-starlike domains. Further results regarding the geometry of strongly starlike functions were given in [14, Chapter IV], [15] and [23].
In view of (6) and (7) both classes S * α and S c α can be represented using the Carathéodory class P, i.e., the class of analytic functions p in D of the form having a positive real part in D. Thus the coefficients of functions in S * α and S c α have a convenient representation in terms of the coefficients of functions in P. Therefore obtaining the upper bound of H 2,1 (F f /2), we base our analysis on well-known expressions for c 2 (e.g., [20, p. 166]), and c 3 (Libera and Zlotkiewicz [17,18]), and c 4 obtained recently in [13], all of which are contained in the following lemma [13]. Let D := {z ∈ C : |z| ≤ 1} and T := {z ∈ C : |z| = 1}.
For ζ 1 ∈ T, there is a unique function p ∈ P with c 1 as in (9), namely, For ζ 1 ∈ D and ζ 2 ∈ T, there is a unique function p ∈ P with c 1 and c 2 as in (9)-(10), namely, 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 (9)-(11), namely, We will also use the following lemma.

Strongly starlike functions
We prove the following sharp inequality for |H 2,1 (F f /2)| for the class S * α .
The inequality is sharp.
For α = 1 we obtain the following result for the class S * of starlike functions [10].
Corollary 1 If f ∈ S * , then The inequality is sharp.

Strongly convex functions
We prove the following sharp inequality for |H 2,1 (F f /2)| in the class S c α .
For α = 1 we obtain the sharp inequality for the class S c of convex functions [10].
Corollary 2 If f ∈ S c , then The inequality is sharp.