Applications of Jack's lemma for certain subclasses of holomorphic functions on the unit disc
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In this paper, we give some results on $\frac{zf^{\prime }(z)}{f(z)}$ for the certain classes of holomorphic functions in the unit disc $E=\left\{ z:\left\vert z\right\vert <1\right\} $ and on $\partial E=\left\{ z:\left\vert z\right\vert =1\right\} $. For the function $ f(z)=z^{2}+c_{3}z^{3}+c_{4}z^{4}+\cdots$ defined in the unit disc $E$ such that $f(z)\in \mathcal{A}_{\alpha }$, we estimate a modulus of the angular derivative of $\frac{zf^{\prime }(z)}{f(z)}$ function at the boundary point $b$ with $\frac{bf^{\prime }(b)}{f(b)}=1+\alpha $. Moreover, Schwarz lemma for class $\mathcal{A}_{\alpha }$ is given. The sharpness of these inequalities is also proved.
Abstract
In this paper, we give some results on $\frac{zf^{\prime }(z)}{f(z)}$ for the certain classes of holomorphic functions in the unit disc $E=\left\{ z:\left\vert z\right\vert <1\right\} $ and on $\partial E=\left\{ z:\left\vert z\right\vert =1\right\} $. For the function $ f(z)=z^{2}+c_{3}z^{3}+c_{4}z^{4}+\cdots$ defined in the unit disc $E$ such that $f(z)\in \mathcal{A}_{\alpha }$, we estimate a modulus of the angular derivative of $\frac{zf^{\prime }(z)}{f(z)}$ function at the boundary point $b$ with $\frac{bf^{\prime }(b)}{f(b)}=1+\alpha $. Moreover, Schwarz lemma for class $\mathcal{A}_{\alpha }$ is given. The sharpness of these inequalities is also proved.
참고문헌 (20)
- T. A. Azeroꠘglu, "A rened 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. M. Burns, "Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary", J. Amer. Math. Soc, 7(3), pp.661~676 (1994)
- 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. (N. Y.) 122 (2004), no. 6, 3623-3629; translated from Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 286 (2002)", Anal. Teor. Chisel i Teor. Funkts, 18(74-84), pp.228~229 (2002)
- M. Elin, "Harmonic and complex analysis and its applications", pp.2014, Trends Math, Birkhäuser
- G. M. Goluzin, "Geometrical Theory of Functions of Complex Variable (Russian)" (1966), Nauka
- I. S. Jack, "Functions starlike and convex of order", J. London Math. Soc. (2), 3, pp.469~474 (1971)
- 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
- M. Mateljevic, "Note on Rigidity of Holomorphic Mappings & Schwarz and Jack Lemma (in preparation)", ResearchGate, ~ (2015)
- M. Mateljevic, "Schwarz lemma, the Caratheodory and Kobayashi Metrics and Applications in Complex Analysis", XIX GEOMETRICAL SEMINAR, pp.1~12 (2016)
- M. Mateljevic, "Hyperbolic geometry and Schwarz lemma", ResearchGate, ~ (2016)
- M. Nunokawa, "On multivalent functions in the unit disc", Tsukuba J. Math, 41(2), pp.251~263 (2017)
- 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
- B\"{u}lent Nafi \"{O}rnek, "Inequalities for the angular derivatives of certain classes of holomorphic functions in the unit disc", 대한수학회보, 53(2), pp.325~334 (2016), 대한수학회
- B. N. Örnek, "Estimates for holomorphic functions concerned with Jack's lemma", Publications de l'Institut Mathematique, In press
- Ch. Pommerenke, "Boundary Behaviour of Conformal Maps" (1992), Springer-Verlag