We find the radius of starlikeness order $\alpha$, $0\leq \alpha<1$, normalized analytic functions $f$ on unit disk satisfying either $Re(f(z)/g(z))>0$ or $\left|(f(z)/g(z))-1\right|<1$ for some close-to-star function $g$ with $Re(g(z)/(z+z^2/2))>0$ as well class $Re(f(z)/(z+z^2/2))>0$. Several other radii such univalence and parabolic are shown to be same app...

Let A consists of analytic functions f :

The asymptotic behaviour, with respect to the large order, of radii starlikeness two types normalised Bessel functions is considered. We derive complete expansions for and provide recurrence relations coefficients these expansions. proofs rely on notion Rayleigh sums inversion. techniques employed in paper could be useful treat similar problems where inversion involved.

This paper aims to provide sufficient conditions for starlikeness and convexity of Hadamard product (convolution) certain multivalent analytic functions with positive real parts. Moreover, the a integral operator other convolution results are also considered.