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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 $\left\vert b\right\vert =1$.
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참고문헌 (9)
참고문헌 신청
T. Aliyev Azeroglu / 2013 / A refined Schwarz inequality on the boundary / Complex Var. Elliptic Equ 58(4):571~577
V. N. Dubinin / 2004 / The Schwarz inequality on the boundary for functions regular in the disc / J. Math. Sci. (N. Y.) 122(6):3623~3629
V. N. Dubinin / 2015 / Bounded holomorphic functions covering no concentric circles / J. Math. Sci. (N. Y.) 207(6):825~831
G. M. Golusin / 1966 / Geometric Theory of Functions of Complex Variable
정문자 / 2014 / THE SCHWARZ LEMMA AND ITS APPLICATION AT A BOUNDARY POINT / 순수 및 응용수학 21(3):219~227
V. N. Dubinin / 2004 / The Schwarz inequality on the boundary for functions regular in the disc / J. Math. Sci. (N. Y.) 122(6):3623~3629
V. N. Dubinin / 2015 / Bounded holomorphic functions covering no concentric circles / J. Math. Sci. (N. Y.) 207(6):825~831
G. M. Golusin / 1966 / Geometric Theory of Functions of Complex Variable
정문자 / 2014 / THE SCHWARZ LEMMA AND ITS APPLICATION AT A BOUNDARY POINT / 순수 및 응용수학 21(3):219~227
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