If a function \(f(z)=z+...,\) analytic in the unit disk \(| z| <1,\) satisfies \(\text{Re}\{zf'(z)/[f(z)-f(-z)]\}>0,\) then we say the function \(f(z)\) belongs to the class \(S_ s\). The author discusses the class \(S_ s\) and some subclasses of it and gets a product theorem. Many given results, including the Pólya-Schoenberg conjecture, can be deduced from the results obtained in the paper.