On \(\alpha\)-quasi convex functions
Let \(\alpha\geq 0\). The class \(Q(\alpha)\) of all functions f(z), holomorphic in the unit disk U, normalized by \(f(0)=f'(0)-1=0\) is introduced as follows: Let \(M_{\alpha}\) denote the class of \(\alpha\)- convex functions introduced by \textit{P. T. Mocanu} [Mathematica, Cluj 11(34), 127-133 (1969; Zbl 0195.364)]. Then \(f\in Q(\alpha)\) if and only if for some \(g\in M_{\alpha}\), \[ | (1-\alpha)\arg (zf'(z))/(g(z))+\alpha \arg ([zf'(z)]')/(g'(z))|< \pi/2\quad (z\in U). \] If \(\alpha\) decreases from 1 to 0, \(Q(\alpha)\) is to vary between Q(1) the class of quasi-convex functions introduced by \textit{K. I. Noor} and \textit{D. K. Thomas} [Int. J. Math. Math. Sci. 3, 255-266 (1980; Zbl 0498.30017)], and Q(0) the class of close-to-convex functions. Results: (i) \(Q(\alpha)\leq Q(\beta)\), if \(\alpha\geq 1\), \(0\leq \beta \leq \alpha,\) (ii) \(\cap_{\alpha \geq 1}Q(\alpha)\) the class of convex mappings, (iii) \(\alpha\geq 1\), \(f\in Q(\alpha)\Rightarrow\) \(F(z)=\{(2/\alpha)/z^{2/\alpha}\int^{z}_{0}t^{(2-\alpha)/\alpha} f^{'1/\alpha}(t)dt \}^{\alpha}\in Q(\alpha)\).
scientific article; zbMATH DE number 4114949