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Third Hankel determinants for starlike and convex functions of order alpha
Halit Orhan,Paweł Zaprawa 대한수학회 2018 대한수학회보 Vol.55 No.1
In this paper we obtain the bounds of the third Hankel determinants for the classes $\mathcal{S}^\ast(\alpha)$ of starlike functions of order $\alpha$ and $\mathcal{K}(\alpha)$ of convex functions of order $\alpha$. Moreover,we derive the sharp bounds for functions in these classes which are additionally 2-fold or 3-fold symmetric.
THIRD HANKEL DETERMINANTS FOR STARLIKE AND CONVEX FUNCTIONS OF ORDER ALPHA
Orhan, Halit,Zaprawa, Pawel Korean Mathematical Society 2018 대한수학회보 Vol.55 No.1
In this paper we obtain the bounds of the third Hankel determinants for the classes $\mathcal{S}^*({\alpha})$ of starlike functions of order ${\alpha}$ and $\mathcal{K}({\alpha}$) of convex functions of order ${\alpha}$. Moreover,we derive the sharp bounds for functions in these classes which are additionally 2-fold or 3-fold symmetric.
Fekete-Szegö Problem for a Generalized Subclass of Analytic Functions
Orhan, Halit,Yagmur, Nihat,Caglar, Murat Department of Mathematics 2013 Kyungpook mathematical journal Vol.53 No.1
In this present work, the authors obtain Fekete-Szeg$\ddot{o}$ inequality for certain normalized analytic function $f(z)$ defined on the open unit disk for which $$\frac{{\lambda}{\beta}z^3(L(a,c)f(z))^{{\prime}{\prime}{\prime}}+(2{\lambda}{\beta}+{\lambda}-{\beta})z^2(L(a,c)f(z))^{{\prime}{\prime}}+z(L(a,c)f(z))^{{\prime}}}{{\lambda}{\beta}z^2(L(a,c)f(z))^{{\prime}{\prime}}+({\lambda}-{\beta})z(L(a,c)f(z))^{\prime}+(1-{\lambda}+{\beta})(L(a,c)f(z))}\;(0{\leq}{\beta}{\leq}{\lambda}{\leq}1)$$ lies in a region starlike with respect to 1 and is symmetric with respect to the real axis. Also certain applications of the main result for a class of functions defined by Hadamard product (or convolution) are given. As a special case of this result, Fekete-Szeg$\ddot{o}$ inequality for a class of functions defined through fractional derivatives are obtained.
The Fekete-Szegö Problem for a Generalized Subclass of Analytic Functions
Deniz, Erhan,Orhan, Halit Department of Mathematics 2010 Kyungpook mathematical journal Vol.50 No.1
In this present work, the authors obtain Fekete-Szeg$\ddot{o}$ inequality for certain normalized analytic function f(z) defined on the open unit disk for which $\frac{(1-{\alpha})z(D^m_{{\lambda},{\mu}}f(z))'+{\alpha}z(D^{m+1}_{{\lambda},{\mu}}f(z))'}{(1-{\alpha})D^m_{{\lambda},{\mu}}f(z)+{\alpha}D^{m+1}_{{\lambda},{\mu}}f(z)}$ ${\alpha}{\geq}0$) lies in a region starlike with respect to 1 and is symmetric with respect to the real axis. Also certain applications of the main result for a class of functions defined by Hadamard product (or convolution) are given. As a special case of this result, Fekete-Szeg$\ddot{o}$ inequality for a class of functions defined through fractional derivatives is obtained. The motivation of this paper is to generalize the Fekete-Szeg$\ddot{o}$ inequalities obtained by Srivastava et al., Orhan et al. and Shanmugam et al., by making use of the generalized differential operator $D^m_{{\lambda},{\mu}}$.
SOME NOTES ON EXTENSIONS OF BASIC UNIVALENCE CRITERIA
Deniz, Erhan,Orhan, Halit Korean Mathematical Society 2011 대한수학회지 Vol.48 No.1
The object of the present paper is to obtain a more general condition for univalence of analytic functions in the open unit disk U. The significant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.
SOME NOTES ON EXTENSIONS OF BASIC UNIVALENCE CRITERIA
Erhan Deniz,Halit Orhan 대한수학회 2011 대한수학회지 Vol.48 No.1
The object of the present paper is to obtain a more general condition for univalence of analytic functions in the open unit disk U: The signicant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results.
On a subclass of certain starlike functions with negative coefficients
Muhammet Kamali,Halit Orhan 대한수학회 2004 대한수학회보 Vol.41 No.1
A certain subclass T_{Omega }(n,p,lambda ,alpha )of starlikefunctions in the unit disk is introduced. The object of thepresent paper is to derive several interesting properties offunctions belonging to the class T_{Omega }(n,p,lambda ,alpha). Coefficient inequalities, distortion theorems and closuretheorems of functions belonging to the class T_{Omega}(n,p,lambda ,alpha ) are determined. Also we obtain radii ofconvexity for the class T_{Omega }(n,p,lambda ,alpha ).Furthermore, integral operators and modified Hadamard products ofseveral functions belonging to the class T_{Omega }(n,p,lambda,alpha ) are studied here.
BOUNDS FOR RADII OF CONVEXITY OF SOME q-BESSEL FUNCTIONS
Aktas, Ibrahim,Orhan, Halit Korean Mathematical Society 2020 대한수학회보 Vol.57 No.2
In the present investigation, by applying two different normalizations of the Jackson's second and third q-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Also, the Laguerre-Pólya class of real entire functions plays an important role in this work.
Bounds for radii of convexity of some q-Bessel functions
Ibrahim Aktas,Halit Orhan 대한수학회 2020 대한수학회보 Vol.57 No.2
In the present investigation, by applying two different normalizations of the Jackson's second and third $q$-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Also, the Laguerre-P\'olya class of real entire functions plays an important role in this work.
ON PARTIAL SUMS OF NORMALIZED q-BESSEL FUNCTIONS
Aktas, Ibrahim,Orhan, Halit Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.2
In the present investigation our main aim is to give lower bounds for the ratio of some normalized q-Bessel functions and their sequences of partial sums. Especially, we consider Jackson's second and third q-Bessel functions and we apply one normalization for each of them.