Radius Problems for Ratios of Janowski Starlike Functions with Their Derivatives

Let $$-1 \le B<A \le 1$$ -1≤B<A≤1 and $$F= f/f'$$ F=f/f′ for normalized locally univalent functions f. For analytic function pon the open unit disc $$\mathbb {D}$$ D with $$p(0)=1$$ p(0)=1 and satisfying the subordination $$p(z) \prec (1+A z)/(1+B z)$$ p(z)≺(1+Az)/(1+Bz) , we determine bounds for $$|p(z)- z p'(z)-1|$$ |p(z)-zp′(z)-1| and $$|z p'(z)|/{{\mathrm{Re\,}}}p(z)$$ |zp′(z)|/Rep(z) . As an application, we investigate the radius problem for Fto satisfy $$|F'(z)(z/F(z))^2-1| <1$$ |F′(z)(z/F(z))2-1|<1 , where fis a Janowski starlike function and the radius of univalence of Fwhen fis a starlike function of order $$\alpha $$ α . Also, we have discussed the radius of starlikeness of Fwhen fis a Janowski starlike function with fixed second coefficient. Apart from the radius problems, we give the sharp coefficient bounds of Fwhen fis a Janowski starlike function and a sufficient condition for starlikeness of fwhen Fis a Janowski starlike function. Our results generalize some of the earlier known results.