In the present article, we define and investigate a new subfamily of holomorphic functions connected with the cosine hyperbolic function with bounded turning. Further some interesting results like sharp coefficients bounds, sharp Fekete-Szegö estimate, sharp $2^{nd}$ Hankel determinant and non-sharp $3^{rd}$ order Hankel determinant. Moreover, the same estimates have been investigated for 2-fold, 3-fold symmetric functions, the first four initial sharp bounds of logarithmic coefficient and sharp second Hankel determinant of logarithmic coefficients fort his defined function family.

The study of expansions of certain mock theta functions in special functions theory has a long and quite significant history. Motivated by recent correlations between $q$-series and mock theta functions, we establish a new $q$-series transformation formula and derive the double-sum expansions for mock theta functions. As an application, we state new double-sum representations for certain mock theta functions.