The fourth-order Hermitian Toeplitz determinant for convex functions
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The fourth-order Hermitian Toeplitz determinant for convex functions Adam Lecko1
1 ´ · Young Jae Sim2 · Barbara Smiarowska
Received: 23 October 2019 / Revised: 23 October 2019 / Accepted: 3 August 2020 © The Author(s) 2020
Abstract The sharp bounds for the fourth-order Hermitian Toeplitz determinant over the class of convex functions are computed. Keywords Hermitian Toeplitz determinant · Univalent functions · Convex functions · Carathéodory class Mathematics Subject Classification 30C45 · 30C50
1 Introduction Let H be the class of analytic functions in D := {z ∈ C : |z| < 1}, A be its subclass normalized by f (0) := 0, f (0) := 1, that is, functions of the form f (z) =
∞
an z n , a1 := 1, z ∈ D,
(1)
n=1
and S be the subclass of A of univalent functions. Let S c denote the subclass of S of convex functions, that is, univalent functions f ∈ A such that f (D) is a convex domain in C. By the well-known result of Study [11] (see also [5, p. 42]), a function
B
Adam Lecko [email protected] Young Jae Sim [email protected] ´ Barbara Smiarowska [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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f is in S c if and only if
z f (z) Re 1 + f (z)
> 0, z ∈ D.
(2)
Given q, n ∈ N, the Hermitian Toeplitz matrix Tq,n ( f ) of f ∈ A of the form (1) is defined by ⎡
an
⎢ a n+1 ⎢ Tq,n ( f ) := ⎢ . ⎣ .. a n+q−1
an+1 an .. . a n+q−2
... ... .. . ...
⎤ an+q−1 an+q−2 ⎥ ⎥ .. ⎥ , . ⎦ an
where a k := ak . Let |Tq,n ( f )| denote the determinant of Tq,n ( f ). In particular, the third Toeplitz determinant |T3,1 ( f )| is given by 1 |T3,1 ( f )| = a 2 a 3
a2 1 a2
a3
a2 = 1 + 2 Re a22 a 3 − 2|a2 |2 − |a3 |2 1
(3)
and the fourth Toeplitz determinant |T4,1 ( f )| is given by a4 a3 a2 a1
=1 − 2 Re a23 a 4 + 4 Re a22 a 3 − 2 Re a2 a 23 a4
a1 a |T4,1 ( f )| = 2 a 3 a 4
a2 a1 a2 a3
a3 a2 a1 a2
(4)
+ 4 Re (a2 a3 a 4 ) + |a2 |4 − 3|a2 |2 + |a3 |4 − 2|a3 |2 + |a2 |2 |a4 |2 − 2|a2 |2 |a3 |2 − |a4 |2 . In recent years a lot of papers has been devoted to the estimation of determinants built with using coefficients of functions in the class A or its subclasses. Hankel matrices i.e., square matrices which have constant entries along the reverse diagonal (see e.g., [3] with further references), and the symmetric Toeplitz determinant (see [1]) are of particular interest. For this reason looking on the interest of specialists in [4] and [7] the study of the Hermitian Toeplitz determinants on the class A or its subclasses has begun. Hermitian Toeplitz matrices play an important role in functional analysis, applied mathematics as well as in physics and technical sciences. In [4] the conjecture that the sharp inequalities 0 ≤ |Tq,1 ( f )| ≤ 1 for all q ≥ 2, holds over the class S c was proposed an
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