On \(\alpha\)-convex functions of order \(\beta\) with \(m\)-fold symmetry (Q923750)

Let \(J_ m(\alpha,\beta)\) denote a class of functions \(f(z)=z+\sum^{\infty}_{n=1}a_{mn+1}z^{nm+1}\) which are regular in \(D=\{z:| z| <1\}\) and \[ \Re\{(1-\alpha)\frac{zf'(z)}{f(z)}+ \alpha(1+\frac{zf''(z)}{f'(z)})\}> \beta,\;z\in D, \] \(\alpha\geq 0\), \(0\leq \beta <1\), \(m=1,2,..\). The author proves among others: I. If \(g\in J_ 1(1-2\beta,\beta)\) and \(0\leq \lambda \leq 1\), then \[ (g'(z))^{\lambda}(g(z)/z)^{1-2\lambda}\prec \frac{1}{1-z} \] where \(\prec\) denotes subordination. In particular \[ g'(z)\prec \frac{1}{(1- z)^ 2}\frac{g(z)}{z}\prec \frac{1}{1-z}. \] II. If \(f\in J_ m(\alpha,\beta)\), \(| z| =r<1\), then \[ \frac{r^{1- \alpha}}{(1+r^ m)^{2(1-\beta)/m}}\leq | f'(z)|^{\alpha}| f(z)|^{1-\alpha}\leq \frac{r^{1-\alpha}}{(1-r^ m)^{2(1- \beta)/m}}, \] \(\Re\{(f'(z))^{\alpha} (f(z)/z)^{1-\alpha}\}> 2^{- 2(1-\beta /m}\), \[ | \arg \{f'(z)^{\alpha}(f(z)/z)^{1- \alpha}\}| \leq \frac{2(1-\beta)}{m}\arcsin r^ m. \] This results are a generalization of some results due to \textit{S. S. Miller}, \textit{P. T. Mocanu} and \textit{M. O. Reade} [Proc. Am. Math. Soc. 37, 553--554 (1973; Zbl 0258.30012)].