On a subclass of uniformly starlike functions
Let \(A\) be the class of functions \(f\), analytic in the unit disc \(U=\{z:| z|<1\}\) normalized by the conditions \(f(0)=0=f'(0)-1\) and \(S\) be the class of functions \(f\in A\) which are univalent in \(U\). \(S^*(A,B, \beta)\) is defined as the class of functions \(f\in A\) satisfying the condition \[ {zf'(z)\over f(z)} \prec\left({1+Az \over 1+Bz} \right)^\beta,\;-1\leq B<A\leq 1,\;0<\beta\leq 1,\;z\in U. \] In addition \(K(A,B,\beta)\) is defined as class of functions \(f\in A\) satisfying \[ 1+{zf''(z)\over f'(z)} \prec\left({1+Az \over 1+Bz} \right)^\beta,\;-1\leq B<A\leq 1,\;0<\beta\leq 1,\;z\in U. \] In this paper the authors are studying certain properties of the classes \(S^*(A,B,\beta)\) and \(K(A,B\beta)\). Some analogous results has been obtained in 1986 by D. Blezu over so called \(n\)-close-to-convex functions of order \(\gamma\) with respect to a sector. See the papers of the reviewer in Math., Rev. Anal. Numér. Theor. Approximation, Math. 28(51), 9--19 (1986; Zbl 0613.30010) and ibid. 31(54), No. 1, 15--24 (1989; Zbl 0707.30011).
scientific article; zbMATH DE number 1267313