Radius of starlikeness of convex combinations of univalent starlike functions

Abstract

The radius of starlikeness of the convex combination \[ t f ( z ) + ( 1 − t ) g ( z ) , 0 > t > 1 , tf(z) + (1 - t)g(z),\quad 0 > t > 1, \] where f ( z ) f(z) and g ( z ) g(z) are normalized univalent starlike functions, is r u = 0.4035 … {r_u} = 0.4035 \ldots , the positive root of the equation r 6 + 5 r 4 + 79 r 2 − 13 = 0 {r^6} + 5{r^4} + 79{r^2} - 13 = 0 .