Approximation by the Complex form of a Link Operator Between the Phillips and the Szasz-Mirakjan Operators

The link operator P-alpha(rho)(f, x) = Sigma(infinity)(k=1) s(alpha,k) (x) integral(infinity)(0) theta (alpha,k) (rho) (t) f(t) dt + e(-alpha x) f(0), alpha, rho > 0, x is an element of[0, +infinity), s(alpha,k) (x) = e(-alpha x) (alpha x)(k/)k !, theta(rho)(alpha,k)(t) = alpha rho/Gamma(k rho) e(-alpha rho t)(alpha rho t)(k rho-1), between the Phillips operator (obtained for rho=1), and the Szasz-Mirakjan operator (obtained for rho -> +infinity ), was introduced by Paltanea in (Carpathian J Math 24:378-385, 2008), for which he proved uniform convergence to f (as alpha -> +infinity ) in any compact subinterval [0,b] subset of [0, +infinity). In this paper, for entire functions f of some exponential growth in , quantitative estimates in approximation and in Voronovskaya-type result in any closed disk (D-r) over bar subset of C are obtained for P-alpha(rho)(f,z).