Abstract. In this paper, a boundary version of the Schwarz lemma is investigated. We obtain more general results at the boundary. Also, new inequalities of the Schwarz lemma at boundary is obtained and the sharpness of these inequalities is proved.

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In this study, a boundary analysis is carried out for the derivative of driving point impedance (DPI) functions, which is mainly used for the synthesis of networks containing resistor-inductor, resistor-capacitor and resistor-inductor-capacitor circuits. It is known that DPI function, Z(s), is an analytic function defined on the right half of the s-plane. In this study, the authors present four theorems using the modulus of the derivative of DPI function, |Z′(0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis and finally, the sharpness of the inequalities obtained in the presented theorems are proved. It is also shown that simple inductor-capacitor tank circuits and higher-order filters are synthesised using the unique DPI functions obtained in each theorem.