• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.2053-2059
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• 2013

In this paper, a boundary version of the Schwarz lemma is investigated. We obtain more general results at the boundary. Also, new inequalities of the Schwarz lemma at boundary is obtained and the sharpness of these inequalities is proved.

• Korean Journal of Mathematics
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• v.29 no.2
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• pp.293-304
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• 2021

In this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions. For new inequalities, the results of Rogosinski's lemma, Subordination principle and Jack's lemma were used.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.75-81
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• 2014

In this paper, a boundary version of the Schwarz and Carath$\acute{e}$odory inequality are investigated. New inequalities of the Carath$\acute{e}$odory's inequality and Schwarz lemma at boundary are obtained by taking into account zeros of f(z) function which are different from zero. The sharpness of these inequalities is also proved.

• Korean Journal of Mathematics
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• v.27 no.4
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• pp.1027-1041
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• 2019

In this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions. For new inequalities, the results of Jack's lemma and Hankel determinant were used. We will get a sharp upper bound for Hankel determinant H₂(1). Also, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

• The Pure and Applied Mathematics
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• v.27 no.4
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• pp.157-169
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• 2020

In this paper, we improve a new boundary Schwarz lemma, for analytic functions in the unit disk. For new inequalities, the results of Rogosinski's lemma, Subordinate principle and Jack's lemma were used. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained.

• Honam Mathematical Journal
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• v.45 no.4
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• pp.648-654
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• 2023

Schwarz lemma at the non-smooth boundary point for holomorphic self-map on the intersection of two balls in ℂ² is obtained. At the complex tangent point in the corner of the boundary of the domain, the tangential eigenvalue of the complex Jacobian of the holomorphic map is estimated if the map is transversal.

• The Pure and Applied Mathematics
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• v.23 no.1
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• pp.61-72
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• 2016

In this paper, a boundary version of the Schwarz lemma for the holom- rophic function satisfying f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1 is invetigated. Also, we estimate a modulus of the angular derivative of f(z) function at the boundary point c with ℜf(c) = a. The sharpness of these inequalities is also proved.

• The Pure and Applied Mathematics
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• v.21 no.3
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• pp.219-227
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• 2014

In this note we study the Schwarz lemma and inequalities for some holomorphic functions on the unit disc. Also, we obtain the inequality of the derivative of holomorphic maps at a boundary point of the unit disc and find a holomorphic map to satisfy the equality.

• The Pure and Applied Mathematics
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• v.18 no.3
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• pp.275-284
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• 2011

In this note we study the Schwarz lemma and its various versions. We find a condition for a holomorphic map to have fixed points only on the boundary of the unit disc and compare its derivatives at fixed points to get some relations among them.

• Kyungpook Mathematical Journal
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• v.60 no.3
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• pp.507-518
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• 2020

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 ${\frac{f{^{\prime}^{\prime}}(z)}{f(z)}}$ at the boundary point c with f'(c) = 0. The sharpness of these inequalities is also proved.