Estimation of a Stieltjes function expanded to Taylor series at complex conjugate points

Taylor series expansions of a Stieltjes function f in various complex conjugate points are used to construct the so called unified continued fractions (UCF) terminated on P-th step by a remainder f P U named tail of f. We prove that, if f is a Stieltjes function then its tail f P U is also a Stieltjes function. The estimations of f are obtained in what follows. Numerical calculations of the new complex bounds on f generated by complex conjugate input data are carried out.