• 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.

• 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.30 no.2
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• pp.351-360
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• 2022

In this paper, we will introduce the class of analytic functions called 𝓡 (𝛼, λ) and explore the different 5properties of the functions belonging to this class.

• The Pure and Applied Mathematics
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• v.28 no.3
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• pp.235-246
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• 2021

In this study, a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions, is considered.The results of Rogosinskis lemma and Jacks lemma have been utilized to derive novel inequalities. Also, these inequalities have been strengthened by considering the critical points which are different from zero.

• The Pure and Applied Mathematics
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• v.22 no.2
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• pp.169-178
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• 2015

In this paper, a boundary version of Carathéodory’s inequality is investigated. Also, new inequalities of the Carathéodory’s inequality at boundary are obtained and the sharpness of these inequalities is proved.

• 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.

• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1205-1215
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• 2018

In this paper, a boundary version of the $Carath{\acute{e}}odory^{\prime}s$ inequality is investigated. We shall give an estimate below ${\mid}f^{\prime}(b){\mid}$ according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and $z_1{\neq}0$. The sharpness of these estimates is also proved.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.25-33
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• 2023

This paper's objectives are to present the $\mathcal{H}$ class of analytical functions and explore the many characteristics of the functions that belong to this class. Some inequalities regarding the angular derivative have been discovered for the functions in this class. In addition, the symmetry points on the unit disc are used for the obtained inequalities.

• The Pure and Applied Mathematics
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• v.22 no.3
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• pp.263-273
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• 2015

In this paper, a boundary version of Schwarz lemma is investigated. For the function holomorphic f(z) = a + c_pz^p + c_p+₁z^p+1 + ... defined in the unit disc satisfying |f(z) − 1| < 1, where 0 < a < 2, we estimate a module of angular derivative at the boundary point b, f(b) = 2, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.

• The Pure and Applied Mathematics
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• v.30 no.3
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• pp.337-345
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• 2023

Different versions of the boundary Schwarz lemma for the 𝒩 (𝜌) class are discussed in this study. Also, for the function g(z) = z+b₂z²+b₃z³+... defined in the unit disc D such that g ∈ 𝒩 (𝜌), we estimate a modulus of the angular derivative of g(z) function at the boundary point 1 ∈ 𝜕D with g'(1) = 1 + 𝜎 (1 - 𝜌), where ${\rho}={\frac{1}{n}}{\sum\limits_{i=1}^{n}}g(c_i)={\frac{g^{\prime}(c_1)+g^{\prime}(c_2)+{\ldots}+g^{\prime}(c_n)}{n}}{\in}g^{\prime}(D)$ and 𝜌≠1, 𝜎 > 1 and c₁, c₂, ..., c_n ∈ 𝜕D. That is, we shall give an estimate below |g"(1)| according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z ≠ 0. Estimating is made by using the arithmetic average of n different derivatives g'(c₁), g'(c₂), ..., g'(c_n).