Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions
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Hermitian Toeplitz determinants of the second and third-order for classes of close-to-star functions 1 · Bogumiła Kowalczyk1 · Oh Sang Kwon2 · Adam Lecko1 Piotr Jastrzebski ¸ 2 Young Jae Sim
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Received: 22 October 2019 / Accepted: 30 June 2020 © The Author(s) 2020
Abstract For some subclasses of close-to-star functions the sharp upper and lower bounds of the second and third-order Hermitian Toeplitz determinants are computed. Keywords Hermitian Toeplitz determinant · Univalent functions · Close-to-star functions · Carathéodory class Mathematics Subject Classification 30C45 · 30C50
1 Introduction Given r > 0, let Dr := {z ∈ C : |z| < r }, Tr := {z ∈ C : |z| = r }, D := D1 , T := T1 and D := {z ∈ C : |z| ≤ 1}. Let H be the class of all analytic functions in D and A be its subclass of functions f normalized by f (0) = 0 and f (0) = 1, i.e., of the form f (z) =
∞
an z n , a1 := 1, z ∈ D.
(1)
n=1
Let S be the subclass of A of all univalent functions.
B
Adam Lecko [email protected] Piotr Jastrz¸ebski [email protected] Bogumiła Kowalczyk [email protected] Oh Sang Kwon [email protected] Young Jae Sim [email protected]
1
Department of Complex Analysis, Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Słoneczna 54, 10-710 Olsztyn, Poland
2
Department of Mathematics, Kyungsung University, Busan 48434, Korea 0123456789().: V,-vol
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P. Jastrz¸ ebski et al.
Given q, n ∈ N, the Hermitian Toeplitz determinant Tq,n ( f ) of f ∈ A of the form (1) is defined by an an+1 . . . an+q−1 a n+1 an . . . an+q−2 Tq,n ( f ) := . .. .. .. , .. . . . a n+q−1 a n+q−2 . . . an where a k := ak . Recently, Ali, Thomas and Vasudevarao [2] introduced the concept of the symmetric Toeplitz determinant Tq (n) for f ∈ A in the following way: an an+1 . . . an+q−1 an+1 an . . . an+q−2 Tq (n)[ f ] := . .. .. .. . .. . . . an+q−1 an+q−2 . . . an They found estimates for T2 (n), T3 (1), T3 (2), and T2 (3) over selected subclasses of A. In recent years many papers have been devoted to the estimation of determinants whose entries are coefficients of functions in the class A or its subclasses. Hankel matrices i.e., square matrices which have constant entries along the reverse diagonal and the generalized Zalcman functional Jm,n ( f ) := am+n−1 − am an , m, n ∈ N, are of particular interest (see e.g., [3,4,9–11,18,20,22–25,28,32,34,35,39]). In [14,21], research was instigated into the study of Hermitian Toeplitz determinants which elements are coefficients of functions in subclasses of A, observing that Hermitian Toeplitz matrices play an important role in functional analysis, applied mathematics as well as in technical sciences. In this paper we continue this research by finding the sharp upper and lower bounds of the second and third-order Toeplitz determinants over subclasses of close-to-star functions. Let F be a subclass of A such that F (2) := { f ∈ F : a2 = 0} is a nonempty subfamily and
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