Dynamics of certain class of critically bounded entire transcendental functions

Let E denote the class of all transcendental entire functions for z∈C and an⩾0 for all n⩾0 such that f(x)>0 for x<0 and the set of all (finite) singular values of f forms a bounded subset of R. For each f∈E, one parameter family is considered. In this paper, we mainly study the dynamics of functions in the one parameter family S. If f(0)≠0, we show that there exists a positive real number λ∗ (depending on f) such that the bifurcation and the chaotic burst occur in the dynamics of functions in the one parameter family S at the parameter value λ=λ∗. If f(0)=0, it is proved that the Julia set of fλ is equal to the complement of the basin of attraction of the super attracting fixed point 0 for all λ>0. It is also shown that the Fatou set F(fλ) of fλ is connected whenever it is an attracting basin and the immediate basin contains all the finite singular values of fλ. Finally, a number of interesting examples of entire transcendental functions from the class E are discussed.