In this paper, a boundary analysis is carried out for the derivative of driving point impedance functions, which is mainly used for synthesis of networks containing RL, RC and RLC circuits. It is known that driving point impedance function, Z(s), is an analytic function defined on the right half of the s-plane. In this study, we derive inequalities for the modulus of derivative of driving point impedance function, |Z (0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis and the sharpness of these inequalities is proved. Furthermore, an equation for the driving point impedance function, Z(s), is obtained as a natural result of the proved theorem in this study.

We investigate a boundary version of the Schwarz lemma for classes H(α). Also, we estimate a modulus of the angular derivative of f (z) function at the boundary point b with f (b) = b/ β √ 2, 0 < β 1. The sharpness of these inequalities is also proved.

In this paper, a boundary version of the Schwarz lemma for meromorphic functions is investigated. The modulus of the angular derivative of the meromorphic function I n f (z) = 1 z +2 n c 0 +3 n c 1 z+4 n c 2 z 2 +... that belongs to the class of M on the boundary point of the unit disc has been estimated from below.

In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f (z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), f (0) , by assuming that the f (z) function is also holomorphic at the boundary point z = 0 on the real axis with f (0) = f (i). Let E be the unit disc and S = {z ∈ C : z > 0} the upper half plane in C. For i ∈ E, w−i 1+iw defines a conformal self-map of E carrying i to 0. Similary, for any i ∈ S, w → z − i z + i is conformal map of S onto E, i to 0. It follows in particular the S and E are conformal equivalent. Consider the function f (w) = f (z) − f (i)

A boundary version of the Schwarz lemma for meromorphic functions is investigated. For the function Inf(z) = 1/z +?? k=2 knck?2zk?2, belonging to the class of W, we estimate from below the modulus of the angular derivative of the function on the boundary point of the unit disc.