Inequalities for the angular derivatives of certain classes of holomorphic functions in the unit disc
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In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$ holomorphic in the unit disc and $\left\vert \frac{f(z)}{\lambda f(z)+(1-\lambda )z}-\alpha \right\vert <\alpha $ for $\left\vert z\right\vert <1$, where $\frac{1}{2}<\alpha \leq \frac{1}{1+\lambda }$, $ 0\leq $ $\lambda <1$. If we know the second and the third coefficient in the expansion of the function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account $ c_{p+1}$, $c_{p+2}$ and zeros of $f(z)-z$. We obtain a sharp lower bound of $ \left\vert f^{\prime }(b)\right\vert $ at the point $b$, where $\left\vert b\right\vert =1$.
Abstract
In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$ holomorphic in the unit disc and $\left\vert \frac{f(z)}{\lambda f(z)+(1-\lambda )z}-\alpha \right\vert <\alpha $ for $\left\vert z\right\vert <1$, where $\frac{1}{2}<\alpha \leq \frac{1}{1+\lambda }$, $ 0\leq $ $\lambda <1$. If we know the second and the third coefficient in the expansion of the function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account $ c_{p+1}$, $c_{p+2}$ and zeros of $f(z)-z$. We obtain a sharp lower bound of $ \left\vert f^{\prime }(b)\right\vert $ at the point $b$, where $\left\vert b\right\vert =1$.
참고문헌 (9)
- T. Aliyev Azeroglu, "A refined Schwarz inequality on the boundary", Complex Var. Elliptic Equ, 58(4), pp.571~577 (2013)
- V. N. Dubinin, "The Schwarz inequality on the boundary for functions regular in the disc", J. Math. Sci. (N. Y.), 122(6), pp.3623~3629 (2004)
- V. N. Dubinin, "Bounded holomorphic functions covering no concentric circles", J. Math. Sci. (N. Y.), 207(6), pp.825~831 (2015)
- G. M. Golusin, "Geometric Theory of Functions of Complex Variable" (1966)
- 정문자, "THE SCHWARZ LEMMA AND ITS APPLICATION AT A BOUNDARY POINT", 순수 및 응용수학, 21(3), pp.219~227 (2014), 한국수학교육학회
- Bulent Nafi Ornek, "Sharpened forms of the Schwarz lemma on the boundary", 대한수학회보, 50(6), pp.2053~2059 (2013), 대한수학회
- R. Osserman, "A sharp Schwarz inequality on the boundary", Proc. Amer. Math. Soc, 128(12), pp.3513~3517 (2000)
- Ch. Pommerenke, "Boundary Behaviour of Conformal Maps" (1992), Springer-Verlag
- H. Unkelbach, "Uber die Randverzerrung bei konformer Abbildung", Math. Z, 43(1), pp.739~742 (1938)