Some results for the family of holomorphic functions associated with the Babalola operator and combination binomial series

In this paper, we define a new class ${\mathcal{R}}_{t,\delta ,\upsilon }^{m,n,\sigma }\left(\mathcal{A},\mathcal{B}\right)$ of holomorphic functions in the open unit disk defined connected with the combination binomial series and Babalola operator using the differential subordination with Janowski-type functions. Using the well-known Carathéodory's inequality for function with real positive parts and the Keogh-Merkes and Ma-Minda's in equalities, we determined the upper bound for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Then, we found an upper bound for the Fekete-Szegö functional for the functions in this family. Further, a similar result for the first two coefficients and for the Fekete-Szegő inequality have been done the function ${\mathcal{G}}^{-1}$ when $\mathcal{G}\in {\mathcal{R}}_{t,\delta ,\upsilon }^{m,n,\sigma }\left(\mathcal{A},\mathcal{B}\right)$. Next, for the functions of these newly defined family we determine coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity. The novelty of the results is that we were able to investigate basic properties of these new classes of functions using simple methods and these classes are connected with the new convolution operator and the Janowski functions.