Angular estimations of certain analytic functions (Q1390185)

Let \({\mathcal A}\) denote the set of all normalized analytic functions in the open unit disc of the complex plane. Let \(f*g\) denote the Hadamard convolution of \(f\) and \(g\) in \({\mathcal A}\) and \(\prec\) denote the subordination symbol. Define for any real number \(a\), \(k_a(z)= z/(1- z)^a\). In this paper, the authors obtain the existence of \(\eta\) with \(0<\eta\leq 1\) such that \(f,g\in{\mathcal A}\), \(a\geq 1\), \(0\leq\alpha< 1\) and \[ \Biggl|\arg \Biggl[\pm\Biggl(z {(k_{a+ 1}* f)'(z)\over(k_{a+ 1}*g)(z)}- \alpha\Biggr)\Biggr]\Biggr|< {\pi\delta\over 2}\quad (0<\delta\leq 1) \] and \[ z{(k_{a+ 1}*g)'(z)\over (k_{a+ 1}*g)(z)}\prec {1+ Az\over 1+ Bz}\quad (-1\leq B<A\leq 1) \] imply \[ \Biggl|\arg\Biggl[\pm \Biggl(z {(k_a* f)'(z)\over (k_a* g)(z)}- \alpha\Biggr)\Biggr]\Biggr|< {\pi\eta\over 2}. \] Similar results are obtained involving \({(k_{a+ 1}* f)(z)\over z}\).