Applications of the Schwarz Lemma related to Boundary Points
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Different versions of the boundary Schwarz lemma for the $\mathcal{N}\left( \rho \right) $ class are discussed in this study. Also, for the function $% g(z)=z+b_{2}z^{2}+b_{3}z^{3}+...$ defined in the unit disc $D$ such that $% g\in \mathcal{N}\left( \rho \right) $, we estimate a modulus of the angular derivative of $g(z)$ function at the boundary point $1\in \partial D$ with $% g^{\prime }(1)=1+\sigma \left( 1-\rho \right) $, where $\rho =\frac{1}{n}% \sum\limits_{i=1}^{n}g\left( c_{i}\right) =\frac{g^{\prime }(c_{1})+g^{\prime }(c_{2})+...+g^{\prime }(c_{n})}{n}\in g^{\prime }(D)$ and $\rho \neq 1$, $\sigma >1$ and $c_{1},c_{2},...,c_{n}\in \partial D$. That is, we shall give an estimate below $\left\vert g^{\prime \prime }(1)\right\vert $ according to the first nonzero Taylor coefficient of about two zeros, z=0 and z_1≠0. Estimating is made by using the arithmetic average of n different derivatives g^{\prime }(c_{1})+g^{\prime }(c_{2})+...+g^{\prime }(c_{n})}.
Abstract
Different versions of the boundary Schwarz lemma for the $\mathcal{N}\left( \rho \right) $ class are discussed in this study. Also, for the function $% g(z)=z+b_{2}z^{2}+b_{3}z^{3}+...$ defined in the unit disc $D$ such that $% g\in \mathcal{N}\left( \rho \right) $, we estimate a modulus of the angular derivative of $g(z)$ function at the boundary point $1\in \partial D$ with $% g^{\prime }(1)=1+\sigma \left( 1-\rho \right) $, where $\rho =\frac{1}{n}% \sum\limits_{i=1}^{n}g\left( c_{i}\right) =\frac{g^{\prime }(c_{1})+g^{\prime }(c_{2})+...+g^{\prime }(c_{n})}{n}\in g^{\prime }(D)$ and $\rho \neq 1$, $\sigma >1$ and $c_{1},c_{2},...,c_{n}\in \partial D$. That is, we shall give an estimate below $\left\vert g^{\prime \prime }(1)\right\vert $ according to the first nonzero Taylor coefficient of about two zeros, z=0 and z_1≠0. Estimating is made by using the arithmetic average of n different derivatives g^{\prime }(c_{1})+g^{\prime }(c_{2})+...+g^{\prime }(c_{n})}.
참고문헌 (16)
- Tuğba Akyel, "Some remarks on Schwarz lemma at the boundary", Filomat, 31(13), pp.4139~4151 (2017), National Library of Serbia[DOI]
- T. Aliyev Azeroǧlu, "A refined Schwarz inequality on the boundary", Complex Variables and Elliptic Equations, 58(4), pp.571~577 (2012), Informa UK Limited[DOI]
- H.P. Boas, "Julius & Julia: Mastering the Art of the Schwarz lemma", Amer. Math. Monthly, 117, pp.770~785 (2010)[DOI]
- V. N. Dubinin, "The Schwarz Inequality on the Boundary for Functions Regular in the Disk", Journal of Mathematical Sciences, 122(6), pp.3623~3629 (2004), Springer Science and Business Media LLC[DOI]
- G.M. Golusin, "Geometric Theory of Functions of Complex Variable" (1966)
- I. S. Jack, "Functions Starlike and Convex of Order α", Journal of the London Mathematical Society, s2-3(3), pp.469~474 (1971), Wiley[DOI]
- M. Mateljevic, "Note on some classes of holomorphic functions related to Jack’s and Schwarz’s lemma", Applicable Analysis and Discrete Mathematics, 16(1), pp.111~131 (2022), National Library of Serbia[DOI]
- Peter R. Mercer, "Sharpened Versions of the Schwarz Lemma", Journal of Mathematical Analysis and Applications, 205(2), pp.508~511 (1997), Elsevier BV[DOI]
- Peter R. Mercer, "Boundary Schwarz inequalities arising from Rogosinski's lemma", Journal of Classical Analysis, 12(2), pp.93~97 (2018), Element d.o.o.[DOI]
- Peter R. Mercer, "An improved Schwarz Lemma at the boundary", Open Mathematics, 16(1), pp.1140~1144 (2018), Walter de Gruyter GmbH[DOI]
- S.S. Miller, "Second-order differential inequalities in the complex plane", J. Math. Anal. Appl, 65, pp.289~305 (1978)[DOI]
- R. Osserman, "A sharp Schwarz inequality on the boundary", Proc. Amer. Math. Soc, 128, pp.3513~3517 (2000)[DOI]
- Bülent Nafi ÖRNEK, "On a Class of Analytic Function related to Schwarz Lemma", 순수 및 응용수학, 29(1), pp.113~124 (2022), 한국수학교육학회[DOI]
- Bulent Nafi Ornek, "Boundary Analysis for the Derivative of Driving Point Impedance Functions", IEEE Transactions on Circuits and Systems II: Express Briefs, 65(9), pp.1149~1153 (2018), Institute of Electrical and Electronics Engineers (IEEE)[DOI]
- Ch. Pommerenke, "Boundary Behaviour of Conformal Maps" (1992), Springer-Verlag
- Helmut Unkelbach, "ÄUber die Randverzerrung bei konformer Abbildung", Mathematische Zeitschrift, 43(1), pp.739~742 (1938), Springer Science and Business Media LLC[DOI]