Rogosinski Lemması ile ilgili Süren Nokta Empedans Fonksiyonları içinCarathéodory Eşitsizliği

Bu makalede, Carathéodory eşitsizliğinin bir sınır versiyonu, pozitif reel fonksiyonlar açısından incelenmiştir.Buna göre, Z(s) süren nokta empedans fonksiyonu; s düzleminin sağ yarı düzleminde tanımlanmış, ?(?) =$frac A2+c_1(s-1)+;c_2left(s-1right)^2$ + ⋯ olarak verilen analitik bir fonksiyondur. Z(s) fonksiyonunun sanal eksen üzerinde s= 0 sınır noktasında da analitik olduğu varsayılarak, Rogosinski lemması yardımıyla, Z(s) 'nin türevinin modülüiçin yeni eşitsizlikler elde edilmiştir. Ayrıca, sunulan eşitsizliklerin kesinliği kanıtlanmış ve elde edilen ekstremalfonksiyonların spektral özellikleri araştırılmıştır. Bu doğrultuda, çalışmada önerilen analizler kullanılarak çeşitlifiltre yapılarının elde edilmesinin mümkün olduğu gözlenmiştir.

Carathéodory's Inequality for Driving Point Impedance Functions Concerned with Rogosinski's lemma

In this paper, a boundary version of the Carathéodory’s inequality has been investigated for positive real functions.Accordingly, the driving point impedance function ?(?) where ?(?) =$frac A2+c_1(s-1)+;c_2left(s-1right)^2$+. .. is ananalytic function defined in the right half of the s-plane. With the help of Rogosinski’ lemma, novel inequalitieshave been derived for the modulus of derivative of ?(?) by assuming that the ?(?) function is also analytic at theboundary point ? = 0 on the imaginary axis. In addition, the sharpness of presented inequalities has been provedand the spectral characteristics of resulting extremal functions have been investigated. Accordingly, it has beenobserved that it is possible to obtain various filter structures using proposed analysis in the study.

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[1] Örnek, B. N., Düzenli, T., (2019). Pozitif Reel Fonksiyonlar için Devre Uygulamaları. DÜMF Mühendislik Dergisi, 10, 2, 457-465.
[2] Sharma, A., Soni, T., (2017). A review on passive network synthesis using Cauer form. World Journal of Wireless Devices and Engineering, 1, 1, 39-46.
[3] Ishida, M., Fukui, Y., & Ebisutani, K., (1984). Novel active-R synthesis of a driving-point impedance. International Journal of Electronics, 56, 1, 151-158.
[4] Ochoa, A., (2016). Driving point impedance and signal flow graph basics: a systematic approach to circuit analysis. In Feedback in analog circuits (pp. 13-34). Springer, Cham.
[5] Wunsch, A. D., Hu, S. P., (1996). A closed-form expression for the driving-point impedance of the small inverted L antenna. IEEE Transactions on Antennas and Propagation, 44, 2, 236-242.
[6] Richards, P. I., (1947). A special class of functions with positive real part in a half-plane. Duke Mathematical Journal, 14, 3, 777-786.
[7] Hazony, D., (1963). Elements of network synthesis. Reinhold, NY, USA.
[8] Reza, F. M., (1961). Schwarz’s lemma and linear passive systems. Proc. IRE, 49, 2, 17–23.
[9] Örnek, B. N., Düzenli, T., (2019). On boundary analysis for derivative of driving point impedance functions and its circuit applications. IET Circuits, Devices & Systems, 13, 2, 145-152.
[10] Örnek, B. N., Düzenli, T., (2018). Boundary Analysis for the Derivative of Driving Point Impedance Functions. IEEE Transactions on Circuits and Systems II: Express Briefs, 65, 9, 1149-1153.
[11] Dineen , S., (1989). The Schwarz Lemma. Clarendon Press, USA.
[12] Mercer, P. R., (1997). Sharpened versions of the Schwarz lemma. Journal of Mathematical Analysis and Applications, 205, 2, 508-511.
[13] Mercer, P. R. (2018). Boundary Schwarz inequalities arising from Rogosinski’s lemma. Journal of Classical Analysis, 12, 93-97.
[14] Maz’ya, V., Kresin, G., (2007). SharpReal-Part Theorems: A Unified Approach. Springer.
[15] Örnek, B. N., (2015). Carathéodory’s inequality on the boundary. The Pure and Applied Mathematics, 2, 2, 169-178.
[16] Örnek, B. N., (2016). The caratheodory inequality on the boundary for holomorphic functions in the unit disc. Journal of Mathematical Physics, Analysis, Geometry, 12, 4, 287-301.
[17] Mercer, P. R., (2018). An improved Schwarz Lemma at the boundary. Open Mathematics, 16, 1, 1140- 1144.
[18] Osserman, R., (2000). A sharp Schwarz inequality on the boundary. Proceedings of the American Mathematical Society, 128, 12, 3513-3517.
[19] Azeroǧlu, T. A., Örnek, B. N., (2013). A refined Schwarz inequality on the boundary. Complex Variables and Elliptic Equations, 58, 4, 571-577.
[20] Boas, H. P., (2010). Julius and Julia: Mastering the art of the Schwarz lemma. The American Mathematical Monthly, 117, 9, 770-785.
[21] Dubinin, V. N., (2004). The Schwarz inequality on the boundary for functions regular in the disk. Journal of Mathematical Sciences, 122, 6, 3623-3629.
[22] Mateljevic, M. (2018) Rigidity of holomorphic mappings & schwarz and jack lemma, Researchgate.