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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.
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참고문헌 신청
T. A. Azeroꠘglu / 2013 / A rened Schwarz inequality on the boundary / Complex Var. Elliptic Equ 58(4):571~577
H. P. Boas / 2010 / Julius and Julia : mastering the art of the Schwarz lemma / Amer. Math. Monthly 117(9):770~785
D. M. Burns / 1994 / Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary / J. Amer. Math. Soc 7(3):661~676
D. Chelst / 2001 / A generalized Schwarz lemma at the boundary / Proc. Amer. Math. Soc 129(11):3275~3278
V. N. Dubinin / 2002 / 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):228~229
H. P. Boas / 2010 / Julius and Julia : mastering the art of the Schwarz lemma / Amer. Math. Monthly 117(9):770~785
D. M. Burns / 1994 / Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary / J. Amer. Math. Soc 7(3):661~676
D. Chelst / 2001 / A generalized Schwarz lemma at the boundary / Proc. Amer. Math. Soc 129(11):3275~3278
V. N. Dubinin / 2002 / 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):228~229
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