Bounds on Hermitian-Toeplitz determinant for starlike, convex and bounded turning functions associated with the exponential function

Abstract

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 $\mathbb{D}=\{z\colon |z|<1\}$ function $f$ is said to be subordinate to an analyic in $\mathbb{D}$ function $g$ (denoted by $f \prec g$), if there exists an analytic in  $\mathbb{D}$  function $w$ with $|w(z)| \leq |z|$ and $w(0) = 0$ such that $f(z) = g(w(z))$. Let $\mathcal{A}$ be the class of analytic functions $f$ in   $\mathbb{D}$ of the form $f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$\ $z \in \mathbb{D},$ and $\mathcal{S}^{\ast}_{e}$ be the class of functions $f \in \mathcal{A}$ such that ${z f'(z)}/{f(z)} \prec e^z.$    In particular, the following statement has been proven (Theorem 1): If $f \in \mathcal{S}^{\ast}_{e}$, then   $-\frac{1}{15}\leq T_{3,1}(f) \leq 1$,  where $T_{3,1}(f)$ is the third-order Hermitian-Toeplitz determinant  of the form $T_{3,1}(f) := 2 \, \Re\!\left(a_{2}^{2}\cdot\overline{a_{3}}\right) - 2|a_{2}|^{2} - |a_{3}|^{2} + 1.$  The upper and lower bounds are sharp. The article also obtained similar (sharp) estimates in the class $\displaystyle \mathcal{C}_{e} := \Big\{ f \in \mathcal{A} \colon   1 + \frac{z f''(z)}{f'(z)} \prec e^z \Big\}\colon  \quad \frac{9}{16}\leq T_{3,1}(f) \leq 1\quad$ (Theorem 2), and in the class $\displaystyle  \mathcal{R}_e:=\left\{ f'(z)\prec e^z \right\}\colon  \quad \frac{5}{9}\leq T_{3,1}(f) \leq 1\quad$ (Theorem 3).