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On Differential Subordinations Connected with Convex Functions Related to a Sector
Adam Lecko KYUNGPOOK UNIVERSITY 1999 Kyungpook mathematical journal Vol.39 No.2
Let k = 2, 3,…, α ∈ [0, 1] and β ∈ (0, 1/(k-1)] be fixed. In this paper the author calculates the constant C_(k)(α,β) such that the diffierential subordination of the form ◁수식 삽입▷(원문을 참조하세요) imlies p(z) < ((1+z)/(1-z))^(β), z∈U.
THE SHARP BOUND OF THE THIRD HANKEL DETERMINANT FOR SOME CLASSES OF ANALYTIC FUNCTIONS
Kowalczyk, Bogumila,Lecko, Adam,Lecko, Millenia,Sim, Young Jae Korean Mathematical Society 2018 대한수학회보 Vol.55 No.6
In the present paper, we have proved the sharp inequality ${\mid}H_{3,1}(f){\mid}{\leq}4$ and ${\mid}H_{3,1}(f){\mid}{\leq}1$ for analytic functions f with $a_n:=f^{(n)}(0)/n!$, $n{\in}{\mathbb{N}},$, such that $$Re\frac{f(z)}{z}>{\alpha},\;z{\in}{\mathbb{D}}:=\{z{\in}{\mathbb{C}}:{\mid}z{\mid}<1\}$$ for ${\alpha}=0$ and ${\alpha}=1/2$, respectively, where $$H_{3,1}(f):=\left|{\array{{\alpha}_1&{\alpha}_2&{\alpha}_3\\{\alpha}_2&{\alpha}_3&{\alpha}_4\\{\alpha}_3&{\alpha}_4&{\alpha}_5}}\right|$$ is the third Hankel determinant.
The sharp bound of the third Hankel determinant for some classes of analytic functions
Bogumi la Kowalczyk,Adam Lecko,Millenia Lecko,심영재 대한수학회 2018 대한수학회보 Vol.55 No.6
In the present paper, we have proved the sharp inequality $|H_{3,1}(f)|$ $\le 4$ and $|H_{3,1}(f)|\le 1$ for analytic functions $f$ with $a_n:=f^{(n)}(0)/n!,\ n\in\mathbb{N},$ such that $$\mathrm{Re}\, \frac{f(z)}{z}> \alpha,\quad z\in\mathbb{D}:=\{z \in\mathbb{C} : |z|<1\}$$ for $\alpha=0$ and $\alpha=1/2,$ respectively, where \begin{equation*} H_{3,1}(f):= \begin{vmatrix} a_1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 \end{vmatrix} \end{equation*} is the third Hankel determinant.
DIFFERENTIAL SUBORDINATIONS FOR FRACTIONAL-LINEAR TRANSFORMATIONS
KIM, Yong Chan,Lecko, Adam,Choi, Jae Ho,Saigo, Megumi 경북대학교 위상수학 기하학연구센터 1999 硏究論文集 Vol.7 No.-
For A and B such that -1≤B<A let h(A, B; z)=(1 + Az)/(1 + Bz), z ∈ u={z: |z|<1}. In this paper, we investigate some properties of differential subordinations of the forms ◁원문참고▷ and ◁원문참고▷ where γ≥0.
The Bounds of Some Determinants for Starlike Functions of Order Alpha
Cho, N. E.,Kowalczyk, B.,Kwon, O. S.,Lecko, A.,Sim, Y. J. Springer-Verlag 2018 BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SO Vol.41 No.1
<P>In the present paper, sharp estimates of some determinants over the class S*(alpha), alpha is an element of [0, 1), of analytic functions f such that Re(zf'(z)/f (z)) > alpha, z is an element of D := {z is an element of C : vertical bar z vertical bar < 1}, are computed.</P>