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AN IMPROVED LOWER BOUND FOR SCHWARZ LEMMA AT THE BOUNDARY
Bülent Nafi ÖRNEK,Tuğba AKYEL 한국수학교육학회 2016 純粹 및 應用數學 Vol.23 No.1
In this paper, a boundary version of the Schwarz lemma for the holomrophic 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)=α. The sharpness of these inequalities is also proved.
On a Class of Analytic Function related to Schwarz Lemma
Bülent Nafi ÖRNEK 한국수학교육학회 2022 純粹 및 應用數學 Vol.29 No.1
In this paper, we plan to introduce the class of the analytic functions called $\mathcal{P}\left( b\right) $ and to investigate the various properties of the functions belonging this class. The modulus of the second coefficient $c_{2}$ in the expansion of $f(z)=z+c_{2}z^{2}+...$ belonging to the given class will be estimated from above. Also, we estimate a modulus of the second angular derivative of $f(z)$ function at the boundary point $% \alpha $ with $f^{\prime }(\alpha )=1-b$, $b\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $, by taking into account their first nonzero two Maclaurin coefficients.
APPLICATIONS OF THE JACK'S LEMMA FOR ANALYTIC FUNCTIONS CONCERNED WITH ROGOSINSKI'S LEMMA
( Bülent Nafi Örnek ) 한국수학교육학회 2021 純粹 및 應用數學 Vol.28 No.3
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.
CARATHÉODORY'S INEQUALITY ON THE BOUNDARY
BÜlent Nafi Örnek 한국수학교육학회 2015 純粹 및 應用數學 Vol.22 No.2
In this paper, a boundary version of Carathéodory's inequality is in- vestigated. Also, new inequalities of the Carathéodory's inequality at boundary are obtained and the sharpness of these inequalities is proved.
Estimates for a Certain Subclass of Holomorphic Functions
BÜLENT NAFI ÖRNEK,TUĞBA AKYEL 한국수학교육학회 2019 純粹 및 應用數學 Vol.26 No.2
In this paper, a version of the boundary Schwarz Lemma for the holomorphic function belonging to N(α) is investigated. For the function f(z) = z + c2z2 + c3z3 + ... which is defined in the unit disc where f(z) ∈ N(α), we estimate the modulus of the angular derivative of the function f(z) at the boundary point b with f(b) = 1 b b ∫ 0 f(t)dt. The sharpness of these inequalities is also proved.
B\"{u}lent Nafi \"{O}rnek 대한수학회 2016 대한수학회보 Vol.53 No.2
In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$ holomorphic in the unit disc and $\left\vert \frac{f(z)}{\lambda f(z)+(1-\lambda )z}-\alpha \right\vert <\alpha $ for $\left\vert z\right\vert <1$, where $\frac{1}{2}<\alpha \leq \frac{1}{1+\lambda }$, $ 0\leq $ $\lambda <1$. If we know the second and the third coefficient in the expansion of the function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account $ c_{p+1}$, $c_{p+2}$ and zeros of $f(z)-z$. We obtain a sharp lower bound of $ \left\vert f^{\prime }(b)\right\vert $ at the point $b$, where $\left\vert b\right\vert =1$.
APPLICATIONS OF THE SCHWARZ LEMMA RELATED TO BOUNDARY POINTS
( Bülent Nafi Örnek ) 한국수학교육학회 2023 純粹 및 應用數學 Vol.30 No.3
Different versions of the boundary Schwarz lemma for the N (ρ) class are discussed in this study. Also, for the function g(z) = z+b<sub>2</sub>z<sup>2</sup>+b<sub>3</sub>z<sup>3</sup>+... 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 ρ = 1/n∑<sup>n</sup><sub>i=1</sub> g(ci) = g′(c1)+g′(c2)+:::+g′(c<sub>n</sub>)/n ∈ g′(D) and ρ ≠ 1, σ > 1 and c1; c2, ..., c<sub>n</sub> ∈ ∂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<sub>1</sub> ≠ 0. Estimating is made by using the arithmetic average of n different derivatives g′(c<sub>1</sub>), g′(c<sub>2</sub>), ..., g′(c<sub>n</sub>).
APPLICATIONS OF SUBORDINATION PRINCIPLE FOR ANALYTIC FUNCTIONS CONCERNED WITH ROGOSINSKI'S LEMMA
SELIN AYDINOĞLU,BÜLENT NAFI ÖRNEK 한국수학교육학회 2020 純粹 및 應用數學 Vol.27 No.4
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 bound- ary point, the estimations below of the modulus of angular derivative have been obtained.
A SHARP SCHWARZ LEMMA AT THE BOUNDARY
Tuğba AKYEL,Bülent Nafi ÖRNEK 한국수학교육학회 2015 純粹 및 應用數學 Vol.22 No.3
In this paper, a boundary version of Schwarz lemma is investigated. For the function holomorphic f(z)=a+c_{p}z^{p}+c_{p+1}z^{p+1}+... defined in the unit disc satisfying |f(z)-1|<1, where 0 < a < 2, we estimate a module of angular