In this paper, we give sufficient conditions for the parameters of the normalized form of the generalized Bessel functions to be convex and starlike in the open unit disk. As an application of our main results, we solve a recent open problem concerning a subordination property of Bessel functions with different parameters. Moreover, we present a new inequality for the Euler gamma function, which we apply in order to have tight bounds for the generalized and normalized Bessel function of the first kind.

Some new inequalities for quotients of modified Bessel functions of the first and second kinds are deduced. Moreover, some developments on bounds for modified Bessel functions of the first and second kinds, higher-order monotonicity properties of these functions and applications to a special function that arises in finite elasticity, are summarized. The key tool in our proofs is a frequently used criterion for the monotonicity of the quotient of two Maclaurin series.

The Lambert $W$ function, giving the solutions of a simple transcendental equation, has become a famous function and arises in many applications in combinatorics, physics, or population dyamics just to mention a few. In this paper we construct and study in great detail a generalization of the Lambert $W$ which involves some special polynomials and even combinatorial aspects.

In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon, the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind.

Let and denote the modified Bessel functions of the first and second kinds of order In this note we prove that the monotonicity of <!-- MATH $u\mapsto I_{\nu}(u)K_{\nu}(u)$ --> on <!-- MATH $(0,\infty)$ --> for all <!-- MATH $\nu\geq -1/2$ --> is an almost immediate consequence of the corresponding Turán type inequalities for the modified Bessel functions of the first and second kinds of order Moreover, we show that the function <!-- MATH $u\mapsto I_{\nu}(u)K_{\nu}(u)$ --> is strictly completely monotonic on <!-- MATH $(0,\infty)$ --> for all <!-- MATH $\nu\in[-1/2,1/2].$ --> At the end of this note, a conjecture is stated.