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 ) ( z f ( z ) ) 2 − β z 3 ( z f ( z ) ) ‴ − ( α + β ) z 2 ( z f ( 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 / u p , b , c is in S ( α , β , λ ) , where u p , b , c denotes the suitably normalized form of the generalized Bessel functions of the first kind.