Some subclasses of close-to-convex functions

For α ∈ [0,1] and β ∈ (-π/2,π/2) we introduce the classes ${\displaystyle C_{\beta }(\alpha )}$ defined as follows: a function f regular in U = {z: |z| < 1} of the form ${\displaystyle f\left(z\right)=z+{\sum }_{n=1}^{\infty }{a}_n{z}^n}$, z ∈ U, belongs to the class ${\displaystyle C_{\beta }(\alpha )}$ if ${\displaystyle Re\{e^{i\beta }(1-\alpha ²z²)f\prime \left(z\right)\}<0}$ for z ∈ U. Estimates of the coefficients, distortion theorems and other properties of functions in ${\displaystyle C_{\beta }(\alpha )}$ are examined.