In this paper we discuss holomorphic mappings f of the unit disc U and corresponding index defined as If (z) = zf?(z) f(z). We are interested in finding bounds on the growth of functions f and related issues, if there are known some properties of If on U. Our main tool in accomplishing this connection is Jack?s lemma. As a special case, we got estimates on the growth of some classes of ?-starlike functions as well as some interesting generalisations.

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, , 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, , by assuming the function is also analytic at the boundary point 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.

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.