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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.
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Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz emph{et al.} [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. {bf 38} (2015), no.~3, 1255--1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results.
We will provide sufficient conditions for the shifted hypergeometric function $z_2F_1(a,b;c;z)$ to be a member of a specific subclass of starlike functions in terms of the complex parameters $a,b$ and $c.$ For example, we study starlikeness of order $alpha,$ $lambda$-spirallikeness of order $alpha$ and strong starlikeness of order $alpha.$ In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.
For $ngeq 4$ (even), the function $varphi_{nmathcal{L}}(z)=1+nz/(n+1)+z^n/(n+1)$ maps the unit disk $mathbb{D}$ onto a domain bounded by an epicycloid with $n-1$ cusps. In this paper, the class $mathcal{S}^*_{nmathcal{L}} = mathcal{S}^*(varphi_{nmathcal{L}})$ is studied and various inclusion relations are established with other subclasses of starlike functions. The bounds on initial coefficients is also computed. Various radii problems are also solved for the class $mathcal{S}^*_{nmathcal{L}}$.
In this paper, sums represented in (3) are studied. The expressions are derived in terms of Bessel functions of the first and second kinds and their integrals. Further, we point out the integrals can be written as a Meijer G function.
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.
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