About the formula 6 and Cases Analysis in the paper

[hpuxing]

Thanks for your great work! I'm a little confused, how to get the upper and lower bounds of formula 6 in the paper? and are the two values i+(1-β)/2 and i+(β-1)/2 reversed in 3.3.4 Cases Analysis. Case1: β > 1. and Case1: β < 1. in the paper? there exists an overlap when the start i+(1-β)/2 is less than the end i+(β-1)/2, and exists an interval when the start i+(1-β)/2 is greater than the end i+(β-1)/2. Is it right?