Baernstein’s star-function, maximum modulus points and a problem of Erdős

The paper is devoted to the development of Baernstein's method of \(T^{*}\)-function. We consider the relationship between the number of separated maximum modulus points of a meromorphic function and the \(T^{*}\)-function. The results of Bergweiler, Bock, Edrei, Goldberg, Heins, Ostrovskii, Petrenko, Wiman are generalized. We also give examples showing that the obtained estimates are sharp.