On convex functions of order \(\alpha\) and type \(\beta\) (Q1106971)

The particular cases of the classes K and \(S^*\) of convex and starlike functions in the unit disk \({\mathbb{D}}\) are considered. Namely, for \(0\leq \alpha <1\), \(0<\beta \leq 1\), the class \(K(\alpha,\beta)\) of functions \(f(z)=z+\sum^{\infty}_{2}a_ nz^ n,\) \((z\in {\mathbb{D}})\) satisfying \[ | zf''(z)/f'(z)| <| (2\beta -1)f''(z)/f'(z)+2\beta (1- \alpha)|,\quad (z\in {\mathbb{D}}) \] as well as the class \(S^*(\alpha,\beta)\) of functions \(g(z)=zf'(z)\), \(f\in K(\alpha,\beta).\) The sharp results in terms of subordination and Hadamard convolution are obtained. Examples: 1. \(p\in K=K(0,1)\), \(f\in K(\alpha,\beta)\) then \(f*p\in K(\alpha,\beta)\). 2. \(K(\alpha,\beta)\subset S^*(\alpha,\beta)\). 3. \(f\in K(\alpha,\beta)\) then \[ f'(z)\prec \begin{cases} (1+(1-2\beta)z)^{2\beta (1-\alpha)/(1-2\beta)},\quad &\beta \neq 1/2, \\ \exp\{(1-\alpha)z\},\quad &\beta =1/2.\end{cases} \]