This paper aims to study differential subordination and superordination preserving properties for certain analytic univalent functions with in the open unit disk. In the present investigation, we obtain some subordination and superordination results involving Pascal distribution series for certain normalized analytic functions in the open unit disk. Also we estimate the sandwich results for the same class.

One of the statistical tools used in geometric function theory is the generalized distribution which has recently gained popularity due to its use in solving practical issues. In this work, we obtained a new subclass of holomorphic functions, which defined by the convolution of generalized distribution and incomplete beta function associated with subordination in terms of the bell number. Further, we estimate the coefficient inequality and upper bound for a subclass of holomorphic functions. Our findings show a clear relationship between statistical theory and geometric function theory.