On sharp radius estimates for <i>S*</i>(<i>β</i>) and a product function

Abstract In the present investigation, we determine various radius constants for the class S * ( β ) of starlike functions of order β : We define S λ,β to be the class of normalised analytic functions f satisfying Re(e i λ (1 − z ) 1+ β f ( z )/ z ) &gt; 0 and introduce a product function G ( z ) := (1 −z ) 1+ β g 1 ( z ) g 2 ( z ) =z with g 1 , g 2 ∈ S λ,β , to find radius constants for G ( z ) to be in certain desired classes. Notably, earlier known results are identified herein as special cases of our findings and all the results obtained are sharp.