Differential inequalities and Bessel functions

Let S(α,β,λ) denote the class of analytic functions f defined on the unit disk D with the normalization f(0)=f′(0)−1=0, z/f(z)≠0 in D and satisfy the condition |f′(z)(zf(z))2−βz3(zf(z))‴−(α+β)z2(zf(z))″−1|≤λ for all z∈D and for some real constants α>−1 and β such that α+β>−1. We find conditions on constants α>−1 and β such that functions in S(α,β,λ) are univalent in D. As a consequence of our investigation, we present univalence and starlikeness criteria. As applications, we present conditions such that z/up,b,c is in S(α,β,λ), where up,b,c denotes the suitably normalized form of the generalized Bessel functions of the first kind.