There constructed a new method of solving extremal problems on the class of univalent functions based on the parametric Loewner representation and optimization principles. This allowed us to solve some known and new problems. The Jakubowski conjecture was proved stating that the Pick functions are extremal in the problem of estimating even coefficients in the class of univalent functions $f$, $|f(z)|<M$ for sufficiently large $M$. We proved the conjecture that the maximum of the product $|a_2a_n|$ in the class of univalent functions close to identity is<br> attained for the Pick functions. We solved the Hengartner–Shober–Goodman–Saff–Brannan problem on the radius of a disk such that every function convex in a direction maps this disk onto a domain convex in another given direction.
Main publications:
Michalska M., Prokhorov D. V., Szynal J. The composition of hyperbolic triangle mappings // Complex Variable. 2000, 43, 179–186.
Jakubwski Z., Prokhorov D. V., Szynal J. Proof of a coefficient product conjecture for bounded univalent functions // Complex Variable, 2000, 43, 241–258.
