Geometric Properties of function $az^{2}J_{ u }^{prime prime }(z)+bzJ_{ u }^{prime}(z)+cJ_{ u }(z)$

In this paper our aim is to find the radii of starlikeness and convexity for three different kind of normalization of the $N_ u(z)=az^{2}J_{ u }^{prime prime }(z)+bzJ_{ u }^{prime}(z)+cJ_{ u }(z)$ function, where $J_ u(z)$ is called the Bessel function of the first kind of order $ u.$ The key tools in the proof of our main results are the Mittag-Leffler expansion for $N_ u(z)$ function and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized $N_ u(z)$ function. Finally, we evaluate certain multiple sums of the zeros for $N_ u(z)$ function.

Geometric Properties of Bessel function derivatives

In this paper our aim is to find the radii of starlikeness and convexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for nth derivative of Bessel function and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized nth derivative of Bessel function. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind.

Radii of the $beta -$uniformly convex of order $alpha$ of Lommel and Struve functions

In this paper, we determine the radii of $beta -$uniformly convex of order $alpha $ for three kinds of normalized Lommel and Struve functions of the first kind. In the cases considered the normalized Lommel and Struve functions are $beta -$uniformly convex functions of order $alpha $ on the determined disks. The basic tool of this study is Lommel and Struve functions in series.

Radii of starlikeness and convexity of regular Coulomb wave functions

In this paper our aim is to determine the radii of univalence, starlikeness and convexity of the normalized regular Coulomb wave functions for two different kinds of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for regular Coulomb wave functions, and properties of zeros of the regular Coulomb wave functions and their derivatives. Moreover, by using the technique of differential subordinations we present some conditions on the parameters of the regular Coulomb wave function in order to have a starlike normalized form. In addition, by using the Euler-Rayleigh inequalities we obtain some tight bounds for the radii of starlikeness of the normalized regular Coulomb wave functions. Some open problems for the zeros of the regular Coulomb wave functions are also stated which may be of interest for further research.

Second Hankel determinant for bi-starlike and bi-convex functions of order b{eta}

In the present investigation the authors obtain upper bounds for the second Hankel determinant of the classes bi-starlike and bi-convex functions of order beta.

Starlikeness of Bessel functions and their derivatives

In this paper necessary and sufficient conditions are deduced for the starlikeness of Bessel functions of the first kind and their derivatives of the second and third order by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some Mittag-Leffler expansions for the derivatives of Bessel functions of the first kind, as well as some results on the zeros of these functions.

Close-to-convexity of normalized Dini functions

In this paper necessary and sufficient conditions are deduced for the close-to-convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag-Leffler expansions for Bessel functions of the first kind.

Certain subclasses of multivalent functions defined by new multiplier transformations

In the present paper the new multiplier transformations $mathrm{{mathcal{J}% }}_{p}^{delta }(lambda ,mu ,l)$ $(delta ,lgeq 0,;lambda geq mu geq 0;;pin mathrm{% }%mathbb{N} )}$ of multivalent functions is defined. Making use of the operator $mathrm{% {mathcal{J}}}_{p}^{delta }(lambda ,mu ,l),$ two new subclasses $mathcal{% P}_{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$ and $widetilde{mathcal{P}}% _{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$textbf{ }of multivalent analytic functions are introduced and investigated in the open unit disk. Some interesting relations and characteristics such as inclusion relationships, neighborhoods, partial sums, some applications of fractional calculus and quasi-convolution properties of functions belonging to each of these subclasses $mathcal{P}_{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$ and $widetilde{mathcal{P}}_{lambda ,mu ,l}^{delta }(A,B;sigma ,p)$ are investigated. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.

Subclasses of meromorphically multivalent functions defined by a differential operator

In this paper we introduce and study two new subclasses Sigma_{lambdamu mp}(alpha,beta)$ and $Sigma^{+}_{lambdamu mp}(alpha,beta)$ of meromorphically multivalent functions which are defined by means of a new differential operator. Some results connected to subordination properties, coefficient estimates, convolution properties, integral representation, distortion theorems are obtained. We also extend the familiar concept of $% (n,delta)-$neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions.