• Title/Summary/Keyword: Julia-Wolff-lemma

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SOME REMARKS ON THE SUBORDINATION PRINCIPLE FOR ANALYTIC FUNCTIONS CONCERNED WITH ROGOSINSKI'S LEMMA

  • Akyel, Tugba
    • 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.

APPLICATIONS OF THE JACK'S LEMMA FOR ANALYTIC FUNCTIONS CONCERNED WITH ROGOSINSKI'S LEMMA

  • Ornek, Bulent Nafi
    • 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.

A SHARP SCHWARZ AND CARATHÉODORY INEQUALITY ON THE BOUNDARY

  • Ornek, Bulent Nafi
    • 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.

SOME RESULTS OF THE CARATHÉODORY'S INEQUALITY AT THE BOUNDARY

  • Ornek, Bulent Nafi
    • 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.

SOME RESULTS FOR THE CLASS OF ANALYTIC FUNCTIONS CONCERNED WITH SYMMETRIC POINTS

  • Ayse Nur Arabaci;Bulent Nafi Ornek
    • 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.

A SHARP SCHWARZ LEMMA AT THE BOUNDARY

  • AKYEL, TUGBA;ORNEK, NAFI
    • 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 + cpzp + cp+1zp+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.

APPLICATIONS OF THE SCHWARZ LEMMA RELATED TO BOUNDARY POINTS

  • Bulent Nafi Ornek
    • 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, ..., cn ∈ 𝜕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'(cn).