Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian

It was established that if the left-hand side of the Gakhov equation is bounded by the constant 2, then this equation has exactly one root in the unit disk, where the constant is sharp and the root is not necessarily zero. We revealed two aspects arising with regard to this connection. The first aspect concerns the pre-Schwarzian immersion of the Gakhov class into the space of bounded holomorphic functions. It was shown that the width of this immersion is equal to 2; the full description was done for the intersection of the boundary of the immersion with the ball of the radius 2 centered at the origin. The second aspect is connected with maintenance of the uniqueness of the root when the linear or fractional linear actions on the pre-Schwarzian with multiplying by the unit disk variable are bounded. Some uniqueness conditions were constructed in the form of S.N. Kudryashov's univalence criteria.