Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ESTIMATES FOR SECOND NON-TANGENTIAL DERIVATIVES AT THE BOUNDARY

  • Received : 2016.09.21
  • Accepted : 2017.02.02
  • Published : 2017.07.31
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Abstract

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f(z) holomorphic in the unit disc and f(0) = 0, f'(0) = 1 such that ${\Re}f^{\prime}(z)$ > ${\frac{1-{\alpha}}{2}}$, -1 < ${\alpha}$ < 1, we estimate a modulus of the second non-tangential derivative of f(z) function at the boundary point $z_0$ with ${\Re}f^{\prime}(z_0)={\frac{1-{\alpha}}{2}}$, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below ${\mid}f^{{\prime}{\prime}}(z_0){\mid}$ according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and $z_1{\neq}0$. The sharpness of these inequalities is also proved.

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References

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