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APPLICATIONS OF JACK'S LEMMA FOR CERTAIN SUBCLASSES OF HOLOMORPHIC FUNCTIONS ON THE UNIT DISC
Catal, Batuhan,ornek, Bulent Nafi Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.2
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 $E=\{z:{\mid}z{\mid}<1\}$ and on ${\partial}E=\{z:{\mid}z{\mid}=1\}$. For the function $f(z)=z^2+c_3z^3+c_4z^4+{\cdots}$ 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.
Applications of the Schwarz Lemma and Jack’s Lemma for the Holomorphic Functions
Bulent Nafi Ornek,Batuhan Catal 경북대학교 자연과학대학 수학과 2020 Kyungpook mathematical journal Vol.60 No.3
We consider a boundary version of the Schwarz Lemma on a certain class of functions which is denoted by N. For the function f(z) = z + a2z2 + a3z3 + ... which is defined in the unit disc D such that the function f(z) belongs to the class N, we estimate from below the modulus of the angular derivative of the function f''(z)/f(z) at the boundary point c with f'(c) = 0. The sharpness of these inequalities is also proved.