Consider the set \(\mathcal {V}_p(\lambda )\) , which is defined as the collection of those functions f on the open unit disc \(\mathbb {D}\) in the plane of complex numbers which have a simple pole at \(z=p\) , provided p is in the interval (0, 1). They are analytic in \(\mathbb {D}\) except at \(z=p\) . Additionally, they satisfy the normalizations \(f(0)=0, f'(0)-1=0\) , and for all z in \(\mathbb {D} \) and \(\lambda \) in the interval (0, 1], the inequality \(\left| (z/f(z))^2 f'(z)-1\right| < \lambda \) is satisfied. Every function \(f\in \mathcal {V}_p(\lambda )\) can be expressed as a Taylor series expansion: \( f(z)=z+\sum _{n=2}^{\infty }a_n z^n, \quad |z|<p. \) This paper initially determines the regions of variability of the difference of subsequent coefficients \((a_{n+1}-a_n)\) for \(n\ge 1\) within a specific range of values of \(p\in (0,1)\) for functions in the class \(\mathcal {V}_p(\lambda )\) . As a result of this analysis, we also obtain sharp upper bounds for the determinants of Toeplitz and Hermitian Toeplitz matrices, where the entries of these matrices correspond to the Taylor coefficients of functions in \(\mathcal {V}_p(\lambda )\) .
