Presented at the 2nd International Symposium on Innovative Approaches in Scientific Studies (ISAS2018-Winter), Samsun, Turkey, Nov 30, 2018

SETSCI Conference Proceedings, 2018, 3, Page (s): 196-201 , https://doi.org/

Published Date: 31 December 2018    | 1389     13

In this paper, a boundary version of the Schwarz lemma has been considered for driving point impedance functions
at s = 0 point of the imaginary axis. Accordingly, under Z(0) 0 = condition, the modulus of the derivative of the Z s ( )
function has been considered from below. Here, Z a ( ) and
( ) ( )
!
pZ a p
coefficients of the Taylor expansion of the
Z s b c s a ( ) ... = + - + p( )p
function have been used in the obtained inequalities. The sharpness of these inequalities has also
been proved. It is also shown that simple LC tank circuits and higher order filters are synthesized using the unique DPI
functions obtained in each theorem.  

Keywords - Schwarz lemma, Analytic function, Circuit

[1] F. M. Reza, “A bound for the derivative of positive real functions,” SIAM Review, vol. 4, no. 1, pp. 40–42, 1962.
[2] G. Fernandez-Anaya, B. Aguirre, R. Suarez, and J. J. Flores-Godoy, “On Operators on Positive Real Functions and Related Issues,” IEEE
Transactions on Circuits and Systems II: Express Briefs, vol 55, no. 11, pp. 1203–1207, 2008.
[3] G. Fernandez-Anaya, J. C. Martinez-Garcia, V. Kucera, and D. Aguilar-George, “MIMO systems properties preservation under SPR Substitutions,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol 51, no. 5, pp. 222–227, 2004.
[4] B. Van Der Pol, “A new theorem on electrical networks,” Physica, vol. 4, no. 7, pp. 585–589, 1937.
[5] R. J. Krueger and D. P. Brown, “Positive real derivatives of driving point functions,” Journal of the Franklin Institute, vol. 287, no. 1, pp. 51–60, 1969.
[6] D. Hazony, Elements of network synthesis, New York, NY, USA, Reinhold, 1963.
[7] T. Huang, “Some mapping properties of RC and RL driving-point impedance functions,” IEEE Transactions on Circuit Theory, vol. 12, no. 2, pp. 257–259, 1965.
[8] P.I. Richards, "A special class of functions with positive real part in a half-plane,"Duke Math. J., Vol. 14, No. 3, pp. 777-789, 1947.
[9] V. N. Dubinin, “The Schwarz inequality on the boundary for functions regular in the disc,” J. Math. Sci., vol. 122, no. 6, pp. 3623–3629, 2004.
[10] G. M. Golusin, Geometric Theory of Functions of Complex Variable [in Russian], 2nd edn., Moscow, 1966.
[11] R. Osserman, “A sharp Schwarz inequality on the boundary,” Proc. Amer. Math. Soc., vol. 128, no. 12, pp. 3513–3517, 2000.
[12] T. A. Azeroğlu and B. N. Örnek, “A refined Schwarz inequality on the boundary,” Complex Variables and Elliptic Equations, vol. 58, no. 4, pp. 571–577, 2013.
[13] B. N. Örnek, “Sharpened forms of the Schwarz lemma on the boundary,” Bull. Korean Math. Soc., vol. 50, no. 6, pp. 2053–2059, 2013.
[14] H. P. Boas, "Julius and Julia: Mastering the Art of the Schwarz lemma,"American Mathematical Monthly, vol. 117, no. 9, pp. 770- 758, 2010.