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In this paper, a boundary version of Carath\'{e}odory's inequality on the right half plane for $p$-valent is investigated. Let $Z(s)=1+c_{p}\left( s-1\right) ^{p}+c_{p+1}\left( s-1\right) ^{p+1}+\cdots$ be an analytic function in the right half plane with $\Re Z(s)\leq A$ $\left( A>1\right) $ for $\Re s\geq 0$ . We derive inequalities for the modulus of $Z(s)$ function, $|Z^{\prime }(0)|$, by assuming the $Z(s)$ function is also analytic at the boundary point $s=0$ on the imaginary axis and finally, the sharpness of these inequalities is proved.
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참고문헌 (16)
참고문헌 신청
T. Aliyev Azeroglu / 2013 / A refined 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 / 2004 / The Schwarz inequality on the boundary for functions regular in the disc / J. Math. Sci. (N.Y.) 122(6):3623~3629
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 / 2004 / The Schwarz inequality on the boundary for functions regular in the disc / J. Math. Sci. (N.Y.) 122(6):3623~3629
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