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Ornek, Bulent Nafi Korean Mathematical Society 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 $\|\frac{f(z)}{{\lambda}f(z)+(1-{\lambda})z}-{\alpha}\|$ < ${\alpha}$ for ${\mid}z{\mid}$ < 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 ${\mid}f^{\prime}(b){\mid}$ at the point b, where ${\mid}b{\mid}=1$.
Sharpened forms of analytic functions concerned with Hankel determinant
Bulent Nafi Ornek 강원경기수학회 2019 한국수학논문집 Vol.27 No.4
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 $H_{2}(1)$. 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.
Bounds of Hankel determinants for analytic function
Bulent Nafi Ornek 강원경기수학회 2020 한국수학논문집 Vol.28 No.4
In this paper, we give estimates of the Hankel determinant $H_{2}(1)$ in a novel class $\mathcal{N}\left( \varepsilon \right) $ of analytical functions in the unit disc. In addition, the relation between the Fekete-Szegö function $H_{2}(1)$ and the module of the angular derivative of the analytical function $p(z)$ at a boundary point $b$ of the unit disk will be given. In this association, the coefficients in the Hankel determinant $b_{2}$, $b_{3}$ and $b_{4}$ will be taken into consideration. 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.
On bounds for the derivative of analytic functions at the boundary
Bulent Nafi Ornek,Tugba Akyel 강원경기수학회 2021 한국수학논문집 Vol.29 No.4
In this paper, we obtain a new boundary version of the Schwarz lemma for analytic function. We give sharp upper bounds for $\left\vert f^{\prime}(0)\right\vert $ and sharp lower bounds for $\left\vert f^{\prime}(c)\right\vert $ with $c\in \partial D=\left\{ z:\left\vert z\right\vert=1\right\} $. Thus we present some new inequalities for analytic functions. Also, we estimate the modulus of the angular derivative of the function $f(z) $ from below according to the second Taylor coefficients of $f$ about $z=0$ and $z=z_{0}\neq 0.$ Thanks to these inequalities, we see the relation between $\vert f^{\prime }(0)\vert$ and $\Re f(0).$ Similarly, we see the relation between $\Re f(0)$ and $\vert f^{\prime }(c)\vert$ for some $c\in\partial D.$ The sharpness of these inequalities is also proved.
Applications of the Schwarz Lemma and Jack’s Lemma for the Holomorphic Functions
Bulent Nafi Ornek,Batuhan Catal 경북대학교 자연과학대학 수학과 2020 Kyungpook mathematical journal Vol.60 No.3
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 f''(z)/f(z) at the boundary point c with f'(c) = 0. The sharpness of these inequalities is also proved.
INEQUALITIES FOR THE NON-TANGENTIAL DERIVATIVE AT THE BOUNDARY FOR HOLOMORPHIC FUNCTION
Ornek, Bulent Nafi Korean Mathematical Society 2014 대한수학회논문집 Vol.29 No.3
In this paper, we present some inequalities for the non-tangential derivative of f(z). For the function $f(z)=z+b_{p+1}z^{p+1}+b_{p+2}z^{p+2}+{\cdots}$ defined in the unit disc, with ${\Re}\(\frac{f^{\prime}(z)}{{\lambda}f{\prime}(z)+1-{\lambda}}\)$ > ${\beta}$, $0{\leq}{\beta}$ < 1, $0{\leq}{\lambda}$ < 1, we estimate a module of a second non-tangential derivative of f(z) function at the boundary point ${\xi}$, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.
A SHARP SCHWARZ AND CARATHÉODORY INEQUALITY ON THE BOUNDARY
Ornek, Bulent Nafi Korean Mathematical Society 2014 대한수학회논문집 Vol.29 No.1
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.
BEHAVIOR OF HOLOMORPHIC FUNCTIONS ON THE BOUNDARY OF THE UNIT DISC
Bulent Nafi Ornek 한국수학교육학회 2017 純粹 및 應用數學 Vol.24 No.3
In this paper, we establish lower estimates for the modulus of the non- tangential derivative of the holomorphic functionf(z) at the boundary of the unit disc. Also, we shall give an estimate below jf00(b)j according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z0 6= 0.
Sharpened forms of the Schwarz lemma on the boundary
Bulent Nafi Ornek 대한수학회 2013 대한수학회보 Vol.50 No.6
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.
A SHARP CARATHÉODORY'S INEQUALITY ON THE BOUNDARY
Ornek, Bulent Nafi Korean Mathematical Society 2016 대한수학회논문집 Vol.31 No.3
In this paper, a generalized boundary version of $Carath{\acute{e}}odory^{\prime}s$ inequality for holomorphic function satisfying $f(z)= f(0)+a_pz^p+{\cdots}$, and ${\Re}f(z){\leq}A$ for ${\mid}z{\mid}$<1 is investigated. Also, we obtain sharp lower bounds on the angular derivative $f^{\prime}(c)$ at the point c with ${\Re}f(c)=A$. The sharpness of these estimates is also proved.