Bounds on Hermitian-Toeplitz determinant for starlike, convex and bounded turning functions associated with the exponential function - Sarkar, Nabadwip (Matematychni Studii, 2025)
In the article we derive sharp bounds for the third-order Hermitian-Toeplitz determinant of starlike and convex functions associated with the exponential function, as well as for their inverse classes. An analytic in the unit disk D={z|z|<1} function f is said to be subordinate to an analyic in D function g (denoted by f g), if there exists an analytic in D function w with |w(z)| ≤ |z| and w(0) = 0 such that f(z) = g(w(z)). Let A be the class of analytic functions f in D of the form f(z) = z + _n=2^∞ a_n z^n, z D, and S^*_e be the class of functions f A such that z f’(z)/f(z) e^z. In particular, the following statement has been proven (Theorem 1): If f S^*_e, then -115≤ T_3,1(f) ≤ 1, where T_3,1(f) is the third-order Hermitian-Toeplitz determinant of the form T_3,1(f) := 2 \!(a_2^2·a_3 ) - 2|a_2|^2 - |a_3|^2 + 1. The upper and lower bounds are sharp. The article also obtained similar (sharp) estimates in the class C_e := { f A 1 + z f’’(z)f’(z) e^z } 916≤ T_3,1(f) ≤ 1 (Theorem 2), and in the class R_e:={ f’(z)e^z } 59≤ T_3,1(f) ≤ 1 (Theorem 3).