On alpha-quasi-convex functions defined by convolution with incomplete beta functions (Q2708940)

In this paper it is defined a family \(Q_\alpha(a,c)\), \(\alpha\geq 0\) of functions \(f:f(z)= z+\sum^\infty_{n=2} a_nz^n\), analytic in the unit disc \(E\) by using a well-known convolution operator \(L(a,c)f= \varphi(a,c)*f\) where \(\varphi(a,c)\) is a incomplete beta function. The author investigates \(Q_\alpha (a,c)\) and gives some of its properties including integral representation, coefficient result, a covering theorem, and several inclusion results. These interesting results can be eventual extended by using the Sălğean operator \(D^nf\) and the ``differential subordination method'' introduced by \textit{P. T. Mocanu} and \textit{S. S. Miller}. [See \textit{D. Blezu} Math., No. 1, Rev. Anal. Numér. Théor. Approximation, 28(51), 9-19 (1986; Zbl 0613.30010) respectively ibid. 31(54), 15-24 (1989; Zbl 0707.30011)].NEWLINENEWLINEFor the entire collection see [Zbl 0947.00027].