The main objective of the present article is to define the class of bounded turning functions associated with modified sigmoid function. Also we investigate and determine sharp results for the estimates of four initial coefficients, Fekete-Szegö functional, the second-order Hankel determinant, Zalcman conjucture and Krushkal inequality. Furthermore, we evaluate bounds of the third and fourth-order Hankel determinants for the class and for the 2-fold and 3-fold symmetric functions.

The purpose of this present paper is to investigate some mapping properties of functions which map the unit disc onto a overlapped leaf-like curve, having real part greater than zero. Also we define a class of $\lambda$-pseudo starlike functions related to a leaf-like curve. Integral representation, inequalities for the initial Taylor-Maclaurin coefficients and Fekete-Szegö problem for subclasses of analytic functions related to various conic regions are obtained as our main results.

In this article, we introduce a new class of analytic functions in the open unit disc that are closely related to functions that are starlike with respect to a boundary point. For this new class of functions, we obtain representation theorem, interesting coefficient estimates and also certain differential subordination implications involving this new class.

In the present paper, we investigated some conditions to be in the class of Carathéodory functions by using the concept of the first-order differential subordinations. Moreover, various interesting special cases were considered in the geometric function theory as applications of main results presented here.