154 research outputs found
Some Characterizations on the Normalized Lommel, Struve and Bessel Functions of the First Kind
In this paper, we introduce new technique for determining some necessary and
sufficient conditions of the normalized Bessel functions , normalized
Struve functions and normalized Lommel functions of the
first kind, to be in the subclasses of starlike and convex functions of order
and type .Comment: arXiv admin note: text overlap with arXiv:1610.03233 by other author
Certain Geometric Properties of Some Bessel Functions
In this paper, we determine necessary and sufficient conditions for the
generalized Bessel function to be in certain subclasses of starlike and convex
functions. Also, we obtain several corollaries as special cases of the main
results, these corollaries gives the corresponding results of the familiar,
modified and spherical Bessel functions.Comment: The second author Changed his family name from Alaa H. Hassan to Alaa
H. El-Qadee
Order of Starlikeness and Convexity of certain integral transforms using duality techniques
For , and , the class
satisfies the condition \begin{align*}
{\rm Re\,} \left( e^{i\phi}\left((1-\alpha+2\gamma)f/z+(\alpha-2\gamma)f'+
\gamma zf''-\beta\right)\frac{}{}\right)>0, \quad \phi\in {\mathbb{R}},{\,}z\in
{\mathbb{D}}; \end{align*} is taken into consideration. The Pascu class of
-convex functions of order , having analytic
characterization \begin{align*} {\rm Re\,}\frac{\xi
z(zf'(z))'+(1-\xi)zf'(z)}{\xi zf'(z)+(1-\xi)f(z)}>\sigma,\quad 0\leq \sigma<
1,\quad z\in {\mathbb{D}}, \end{align*} unifies starlike and convex functions
class of order .The admissible and sufficient conditions on
are investigated so that the integral transforms \begin{align*}
V_{\lambda}(f)(z)= \int_0^1 \lambda(t) \frac{f(tz)}{t} dt, \end{align*} maps
the function from into
. Further several interesting applications, for specific
choice of are discussed which are related to the classical
integral transform.Comment: generalization of the previous articl
Applications of differential subordination for functions with fixed second coefficient to geometric function theory
The theory of first-order differential subordination developed by Miller and
Mocanu was recently extended to functions with fixed initial coefficient by R.
M. Ali, S. Nagpal and V. Ravichandran [Second-order differential subordination
for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci.
Soc. (2) 34 (2011), 611--629] and applied to obtain several generalization of
classical results in geometric function theory. In this paper, further
applications of this subordination theory is given. In particular, several
sufficient conditions related to starlikeness, convexity, close-to-convexity of
normalized analytic functions are derived. Connections with previously known
results are pointed out
Radii of starlikeness and convexity of some -Bessel functions
Geometric properties of the Jackson and Hahn-Exton -Bessel functions are
studied. For each of them, three different normalizations are applied in such a
way that the resulting functions are analytic in the unit disk of the complex
plane. For each of the six functions we determine the radii of starlikeness and
convexity precisely by using their Hadamard factorization. These are
-generalizations of some known results for Bessel functions of the first
kind. The characterization of entire functions from the Laguerre-P\'olya class
via hyperbolic polynomials play an important role in this paper. Moreover, the
interlacing property of the zeros of Jackson and Hahn-Exton -Bessel
functions and their derivatives is also useful in the proof of the main
results. We also deduce a sufficient and necessary condition for the
close-to-convexity of a normalized Jackson -Bessel function and its
derivatives. Some open problems are proposed at the end of the paper.Comment: 15 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1406.373
Convexity of the Generalized Integral Transform and Duality Techniques
Let be the class of normalized
analytic functions defined in the domain satisfying \begin{align*}
{\rm Re\,}
e^{i\phi}\left(\dfrac{}{}(1\!-\!\alpha\!+\!2\gamma)\!\left({f}/{z}\right)^\delta
+\left(\alpha\!-\!3\gamma+\gamma\left[\dfrac{}{}\left(1-{1}/{\delta}\right)\left({zf'}/{f}\right)+
{1}/{\delta}\left(1+{zf''}/{f'}\right)\right]\right)\right.\\
\left.\dfrac{}{}\left({f}/{z}\right)^\delta
\!\left({zf'}/{f}\right)-\beta\right)>0, \end{align*} with the conditions
, and .
Moreover, for , , the class
be the subclass of normalized analytic functions
such that \begin{align*} {\rm
Re}{\,}\left(1/\delta\left(1+zf''/f'\right)+(1-1/\delta)\left({zf'}/{f}\right)\right)>\zeta,\quad
|z|<1. \end{align*} In the present work, the sufficient conditions on
are investigated, so that the generalized integral transform
\begin{align*} V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t)
\left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta},\quad |z|<1, \end{align*}
carries the functions from into
. Several interesting applications are provided for
special choices of .Comment: Convexity results of the generalized integral operator, 25 page
Inclusion of generalized Bessel functions in the Janowski class
Sufficient conditions on , , , and are determined that will
ensure the generalized Bessel functions satisfies the
subordination . In particular this gives
conditions for , to be
close-to-convex. Also, conditions for which to be Janowski
convex, and to be Janowski starlike in the unit disk
are obtained.Comment: 13 page
Radius of fully starlikeness and fully convexity of harmonic linear differential operator
Let be a normalized harmonic mapping in the unit disk
\ID. In this paper, we obtain the sharp radius of univalence, fully
starlikeness and fully convexity of the harmonic linear differential operators
and
when
the coefficients of and satisfy harmonic Bieberbach coefficients
conjecture conditions. Similar problems are also solved when the coefficients
of and satisfy the corresponding necessary conditions of the harmonic
convex function . All results are sharp. Some of the results
are motivated by the work of Kalaj et al. \cite{Kalaj2014} (Complex Var.
Elliptic Equ. 59(4) (2014), 539--552).Comment: 14 pages; To appear in the Bulletin of the Korean Mathematical
Societ
Products of Bessel and modified Bessel functions
The reality of the zeros of the product and cross-product of Bessel and
modified Bessel functions of the first kind is studied. As a consequence the
reality of the zeros of two hypergeometric polynomials is obtained together
with the number of the Fourier critical points of the normalized forms of the
product and cross-product of Bessel functions. Moreover, the interlacing
properties of the real zeros of these products of Bessel functions and their
derivatives are also obtained. As an application some geometric properties of
the normalized forms of the cross-product and product of Bessel and modified
Bessel functions of the first kind are studied. For the cross-product and the
product three different kind of normalization are investigated and for each of
the six functions the radii of starlikeness and convexity are precisely
determined by using their Hadamard factorization. For these radii of
starlikeness and convexity tight lower and upper bounds are given via
Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given
for the parameters such that the six normalized functions are starlike and
convex in the open unit disk. The properties and the characterization of real
entire functions from the Laguerre-P\'{o}lya class via hyperbolic polynomials
play an important role in this paper. Some open problems are also stated, which
may be of interest for further research.Comment: 37 pages, 1 figur
Fully Starlike and Convex Harmonic Mappings of order \alpha
The hereditary property of convexity and starlikeness for conformal mappings
does not generalize to univalent harmonic mappings. This failure leads us to
the notion of fully starlike and convex mappings of order \alpha, (0\leq
\alpha<1). A bound for the radius of fully starlikeness and fully convexity of
order \alpha is determined for certain families of univalent harmonic mappings.
Convexity is not preserved under the convolution of univalent harmonic convex
mappings, unlike in the analytic case. Given two univalent harmonic convex
mappings f and g, the problem of finding the radius r_{0} such that f*g is a
univalent harmonic convex mapping in |z|<r_{0}, is being considered.Comment: 2 figure
- …