The first symmetrization transformation was introduced by Jacob Steiner in 1838 in his attempt to find a geometric proof of the classical Isoperimetric Problem.  It is amazing that for almost two centuries after its creation the method of symmetrization remains an incredibly powerful tool in many areas of mathematics and mathematical physics.  Several ramifications and generalizations of this method were suggested by George Pólya, Gabor Szegö, Walter Hayman, Igor P. Mityuk, Al Baernstein II, Vladimir Dubinin, Friedemann Brock and some other outstanding mathematicians.

I became acquanted with symmetrization in the late 70's, when my thesis adviser, Prof. Igor P. Mityuk, suggested several research problems in Geometric Function Theory.  One of those was the Pólya-Szegö problem on continuous Steiner symmetrization, which construction was suggested in one of my papers.  I also used some other symmetrization methods and in this talk, will discuss symmetrization-type transformations which I used in my work on problems suggested by Prof. Mityuk and other prominent mathematicians.

At the end of my talk I present some open questions on transformations of symmetrization type.