Estimates for Second non-tangential derivatives at the boundary
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function $f(z)$ holomorphic in the unit disc and $ f(0)=0 $, $f^{\prime }(0)=1$ such that $\Re f^{\prime }(z)>\frac{1-\alpha }{2 }$, $-1<\alpha <1$, we estimate a modulus of the second non-tangential derivative of $f(z)$ function at the boundary point $z_{0}$ with $\Re f^{\prime }(z_{0})=\frac{1-\alpha }{2}$, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below $ \left\vert f^{\prime \prime }(z_{0})\right\vert $ according to the first nonzero Taylor coefficient of about two zeros, namely $z=0$ and $z_{1}\neq 0$ . The sharpness of these inequalities is also proved.
Abstract
In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function $f(z)$ holomorphic in the unit disc and $ f(0)=0 $, $f^{\prime }(0)=1$ such that $\Re f^{\prime }(z)>\frac{1-\alpha }{2 }$, $-1<\alpha <1$, we estimate a modulus of the second non-tangential derivative of $f(z)$ function at the boundary point $z_{0}$ with $\Re f^{\prime }(z_{0})=\frac{1-\alpha }{2}$, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below $ \left\vert f^{\prime \prime }(z_{0})\right\vert $ according to the first nonzero Taylor coefficient of about two zeros, namely $z=0$ and $z_{1}\neq 0$ . The sharpness of these inequalities is also proved.
참고문헌 (20)
- T. Aliyev Azero˘glu, "A refined Schwarz inequality on the boundary", Complex Var. Elliptic Equ, 58(4), pp.571~577 (2013)
- H. P. Boas, "Julius and Julia: Mastering the Art of the Schwarz lemma", Amer. Math. Monthly, 117(9), pp.770~785 (2010)
- D. Chelst, "A generalized Schwarz lemma at the boundary", Proc. Amer. Math. Soc, 129(11), pp.3275~3278 (2001)
- V. N. Dubinin, "The Schwarz inequality on the boundary for functions regular in the disc", J. Math. Sci, 122(6), pp.3623~3629 (2004)
- V. N. Dubinin, "Bounded holomorphic functions covering no concentric circles", J. Math. Sci, 207(6), pp.825~831 (2015)
- G. M. Golusin, "Geometric Theory of Functions of Complex Variable" (1966)
- 정문자, "THE SCHWARZ LEMMA AND BOUNDARY FIXED POINTS", 순수 및 응용수학, 18(3), pp.275~284 (2011), 한국수학교육학회
- 정문자, "THE SCHWARZ LEMMA AND ITS APPLICATION AT A BOUNDARY POINT", 순수 및 응용수학, 21(3), pp.219~227 (2014), 한국수학교육학회
- S. G. Krantz, "Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary", J. Amer. Math. Soc, 7(3), pp.661~676 (1994)
- T. Liu, "The Schwarz lemma at the boundary of the Egg domain Bp1,p2 in Cn", Canad. Math. Bull, 58(2), pp.381~392 (2015)
- J. Lu, "Schwarz lemma at the boundary of the unit polydisk in Cn", Sci. China Math, 58(8), pp.1639~1652 (2015)
- M. Mateljevi´c, "Ahlfors-Schwarz lemma and curvature", Kragujevac J. Math, 25, pp.155~164 (2003)
- M. Mateljevi´c, "Distortion of harmonic functions and harmonic quasiconformal quasi-isometry", Rev. Roumaine Math. Pures Appl, 51(5-6), pp.711~722 (2006)
- M. Mateljevi´c, "The Lower Bound for the Modulus of the Derivatives and Jacobian of Harmonic Injective Mappings", Filomat, 29(2), pp.221~244 (2015)
- M. Mateljevi´c, "Note on Rigidity of Holomorphic Mappings & Schwarz and Jack Lemma (in preparation)", ResearchGate, ~
- R. Osserman, "A sharp Schwarz inequality on the boundary", Proc. Amer. Math. Soc, 128(12), pp.3513~3517 (2000)
- Bulent Nafi Ornek, "Sharpened forms of the Schwarz lemma on the boundary", 대한수학회보, 50(6), pp.2053~2059 (2013), 대한수학회
- Bulent Nafi Ornek, "Inequalities for the non-tangential derivative at the boundary for holomorphic function", 대한수학회논문집, 29(3), pp.439~449 (2014), 대한수학회
- Ch. Pommerenke, "Boundary Behaviour of Conformal Maps" (1992), Springer-Verlag
- D. Shoikhet, "Harmonic and Complex Analysis and its Applications", pp.135~230 (2014), Springer International Publishing